Geometry

Geometry Building Blocks Geometry words... Coordinate geometry... Pairs of lines... Classifying angles... Angles and intersecting lines... Circles...

PolygonsPolygon basics... Triangles... Quadrilaterals... Area of polygons and circles...

Relations and SizesCongruent figures... Similar figures... Squares and square roots... The Pythagorean Theorem and right triangle facts...

Three-dimensional FiguresSpace figures... Prisms... Pyramids... Cylinders, cones, and spheres...

GEOMETRY

Geometry is about the shape and size of things. Like algebra, geometry has its own special vocabulary. These pictures show the four basic concepts on which the rest of geometry is built.The world is made of many things. Some can be touched, counted, or seen. There are other things that exist only in the imagination, but these imaginary things can be very powerful tools! You can see geometry everywhere around you, in manmade structures, in nature, in sports, in manufacturing, and in art. In geometry there are only four imaginary itemssimple ideasupon which everything else is built: point, line, plane, and space.

The most important thing to remember about a point is that it has absolutely no dimensionsno length, no width, no depth. It is simply a location. We use a dot, like point N shown here, to symbolize a point, but a dot is not a true point because any written or printed dot, no matter how small, has dimensions.

A line is a set of points that has only one dimension, length. Think of multitudes of points lying on a single path. Points on the same line are called "collinear points." Two points are needed to define a line. Notice how to write the symbol for this line. The arrowheads on line DE show us that the line extends endlessly in both directionsit has no endpoints. When two lines meet at a point, they are called "intersecting lines."

The set of points called a plane lies all on one surface. In algebra, you've already used the coordinate plane for plotting points, so you're familiar with at least one example of a plane. The surface of a table is another example of a planebut a plane actually extends endlessly. Also, remember that the surface of a plane has no thickness. At least three points not on the same line are needed to define a plane.

Finally, there is space, the set of all points. It's difficult to illustrate the idea of space, since it's everywhere! You can think of it as an open, empty room, or the inside of an empty box. Space has no boundaries and extends endlessly in all directions. Of course, these four terms are just the beginning. You'll encounter many more words and their meanings as you work through this geometry course. Before you move on, here are three more basic terms to become familiar with. You'll be seeing them often. Line segment A line segment is simply a part of a line that has a specific length and specific endpoints. Notice the symbol for a line segment. How is it different from the symbol for a line? Two line segments that have the same length are called congruent segments. This symbol is used to show congruence.RayA ray is a part of a line that has only one endpoint. It extends endlessly in one direction. Notice the symbol for a ray. How is it different from the symbol for a line and a line segment?

Angle Two rays joined with a common endpoint form an angle. The common endpoint is called a vertex.

In algebra, you were introduced to the coordinate system, plotting ordered pairs, and graphing lines. These tools are used in geometry as well. Algebra and geometry are used hand-in-hand to solve many real-world math problems. Points, lines, line segments, rays, and angles, as well as other geometric shapes, can be graphed on the coordinate plane.Click each button to see a graph.

You used the coordinate plane in algebra for graphing and solving equations. This powerful tool is also used in studying geometry. Although the idea of graphing on a coordinate plane dates all the way back to the 1600's, it didn't appear in our high school math textbooks until the early 1900's. Since that time it's become more and more important for solving different types of math problems. We can graph lines, line segments, rays, and anglesas well as other geometric figureson the coordinate plane and use the graphs to help us determine their relationship to each other as well as other features such as length. Here's one way geometry is used in the real world. A team of archaeologists is studying the ruins of Lignite, a small mining town from the 1800's. They plot points on a coordinate plane to show exactly where each artifact is found.

They are using this coordinate plane as a map of a section of the town. It shows the location of a medicine bottle, a doorknob, and a pottery jug. Notice that each unit on the grid is equal to 5 meters. How far apart are the doorknob and the medicine bottle? The doorknob is at point (-3, -1) and the medicine bottle is at point (2, -1).If we count the units between the two points, we find that they are 5 units apart. We could also subtract the x coordinates of the two endpoints of the line segment to find the distance between them: "two minus negative 3 is equal to five." Since each unit is equal to 5 meters, we multiply the 5 units times 5 meters. The two artifacts are 25 meters apart. This example shows just one way that geometry can help people to solve real world problems. It's also used in building, in art, in clothing design and manufacture, in landscaping, in exploring space, and in organizing closetsin fact, there are so many uses for geometry that we could never name them all!

Pair of Lines

Two lines can be related to each other in four different ways. Click on each diagram to learn more about pairs of lines.

PAIRS OF LINES

In geometry, lines often occur in pairs. We can define four different types of line pairs: lines that intersect, lines that are perpendicular, lines that are parallel, and lines that are skew. We can use this picture of a box to investigate each of these line pairs. Intersecting lines are lines that have one, and only one, point in common. Look at the line segments formed by the edges of the box and find two line segments that intersect. There are several pairs of intersecting line segmentsfor example, line segment CG and line segment HG share one point in common, point G, so they are intersecting line segments. Perpendicular lines intersect in a special way, forming right angles. You'll learn more about right angles in the next lesson. The x and y axes on the coordinate plane are a good example of perpendicular lines. This box has many perpendicular line segments. The intersecting line segments AE and EH are perpendicular, and so are the line segments CG and HG. Parallel lines never touch, and they lie in the same plane. Can you find a pair of parallel lines on the box? Line segments AE and CG are parallel, and there are also several other sets of parallel edges. Like parallel lines, skew lines never touch, but unlike parallel lines, skew lines are not in the same plane. Look at edge CG on the right front side of the box, going up and down, and edge AB at the back, going left to right. Do these lines ever touch each other? No, they don't. Are they in the same plane, meaning the same surface of the box? No, they are not. These segments are an example of skew lines.

CLASSIFYING ANGLESThe diagrams show the four main types of angles that we work with in geometry: straight angle, right angle, acute angle, and obtuse angle. We use a protractor to measure an angle in degrees.

If you look around you, you'll see angles everywhere. Angles are measured in degrees. A degree is a fraction of a circlethere are 360 degrees in a circle, represented like this: 360. You can think of a right angle as one-fourth of a circle, which is 360 divided by 4, or 90. The right angle is the most common anglethe edges of your notebook paper, stair steps, door facings, the edges of picture framesall of these form right angles. Let's look at some other types of angles. A straight angle represents half of a circle. 360 divided by 2 equals 180that is the measure of a straight angle. An acute angle is any angle that measures less than 90. An obtuse angle measures greater than 90 but less than 180. We can estimate angle measures by looking at them and comparing them mentally with angles we know, such as the right angle. But how do we measure angles exactly?

This useful tool is called a protractor, and any student of geometry should have one! It is used both to measure angles and to draw angles. To use a protractor, place the arrow on the vertex of the angle you want to measure, and read off the measurement on the scale, in degrees. Two angles that have the same measure are called congruent angles.

Here are two angles that both measure 30. We say that angle x is congruent to angle y.Now let's look at some angle pairs. Two angles are called supplementary if their measures add up to 180. These two 90 angles are supplementary because 90 + 90 = 180.

Supplementary angles do not have to touch, or be in the same plane, or even be in the same room! Their measurements only are the secret of their relationship!

A pair of angles is called complementary if their measurements add up to 90. These two angles are complementary, because 60 + 30 = 90.

ANGLES AND INTERSECTING LINES

Two parallel lines intersected by a transversal form corresponding pairs of angles that are congruent.

Adjacent angles share a common vertex and a common ray. Two intersecting lines form pairs of adjacent angles that are supplementary. Also, two intersecting lines form pairs of congruent angles, called vertical angles.

Adjacent means "next to." But we use this word in a very specific way when we refer to adjacent angles. Study these two figures. Only the pair on the right is considered to be adjacent, angles c and d. Adjacent angles must share a common side and a common vertex, and they must not overlap each other.

Vertical angles are pairs of angles formed by two intersecting lines. Vertical angles are not adjacent anglesthey are opposite each other. In this diagram, angles a and c are vertical angles, and angles b and d are vertical angles. Vertical angles are congruent.

These two lines are parallel, and are cut by a transversal, which is just a name given to a line that intersects two or more lines at different points. Eight angles appear, in four corresponding pairs that have the same measure, so therefore are congruent. These four corresponding pairs are: angles a and e angles c and g angles b and f angles d and h The angles that lie in the interior area, or the area between the two lines that are cut by the transversal, are called interior angles. Angles c, d, e and f are interior angles. Angles a, b, g, and h lie in the exterior area, and they are called "exterior angles."

We call angles on opposite sides of the transversal alternate angles. Angles c and f, and d and e, are alternate interior angles. Angles a and h, and b and g, are alternate exterior angles. Note that these alternate pairs are also congruent. When a transversal cuts two lines that are not parallel, as shown here, it still forms eight anglesfour corresponding pairs. However, the corresponding pairs are not congruent as occurs with parallel lines.

CIRCLESA circle is a closed curve in a plane. All of its points are an equal distance from its center. That distance is called the radius of the circle. A diameter is a line segment that has both of its endpoints on the circle and passes through the center. A chord is a line segment with both endpoints on the circle. A portion of a circle is called an arc. A semicircle is an arc that is one-half of a circle.

The distance around a circle is called its circumference.Click each term to learn more.

Look around you and you'll find points, lines, planes, and angles of all sizes. Do you see any circular shapes among them? Perhaps the face of a clock, a lamp shade, a door knob, the base of a vase, or even a coffee mug? The rectangle is perhaps the "most seen" geometric figure, but the circle is a close second. By definition, a circle is all points in the same plane that lie an equal distance from a given center point. We might think of a dinner plate as a circle, but actually, it's notit's a disk. The outer rim is a circle. A circle is just the points that lie on the curve itself. The points that lie inside the curve are called "interior points." A circle is named by its center point. This circle is circle "A." The distance from the center point to a point on the circle is called the radius of the circle, shown in the diagram as r. The radius is a line segment with one endpoint on the circle and the other at the center of the circle. The plural of radius is "radii." Two circles that have congruent radii, or radii that are the same size, are called congruent circles. A diameter has both endpoints on the circle and must also contain the center of the circle. A diameter is twice the length of the radius.Segments with both points on the circle are called chords. A diameter is a special type of chord, which passes through the center of the circle. As with any segments, two chords are congruent if their lengths are equal. As you learned in an earlier lesson, a circle contains 360. A portion of a circle is called an arc. Arcs are measured in degrees, like angles, and are classified in a similar way. There are minor arcs, major arcs, and semicircles. A minor arc measures between 0 and 180 degrees. A major arc is between 180 and 360 degrees. The semicircle is exactly 180. Arcs also have length. Imagine placing a string along a portion of a circle, then picking it up and stretching it out, and measuring it. That is arc length. The sum of all the lengths of the non-overlapping arcs of a circle is called the circumference of that circle. This is the distance around the circle. Two circles that have the same center are called concentric circles. The circles on a bull's-eye target are concentric. When diameters intersect at the center of a circle, they form central angles. When you slice a round pizza, you usually slice along a diameter. If you slice a pizza into 8 congruent pieces, each section includes a central angle with a measure of 45.The measure of a central angle equals the measure of the arc that it "intercepts". An inscribed angle is an angle with its vertex on the circle and its sides containing chords of the circle. Angle RST is an inscribed angle.

Chords that intersect at any interior point form "interior angles." Angles r, s, t, and u are interior angles.

This tool, called a compass, is used to construct circles. Obviously, this is different from a magnetic compass that is used to show direction. Like a protractor, a compass is a tool that every geometry student should have. It has many other uses in addition to circle construction, which you'll see in later lessons.

POLYGONS BASICSPolygons are many-sided figures, with sides that are line segments. Polygons are named according to the number of sides and angles they have. The most familiar polygons are the triangle, the rectangle, and the square. A regular polygon is one that has equal sides. Polygons also have diagonals, which are segments that join two vertices and are not sides.

The table lists all the polygons having up to 10 sides.

The word polygon is a combination of two Greek words: "poly" means many and "gon" means angle. Along with its angles, a polygon also has sides and vertices. "Tri" means "three," so the simplest polygon is called the triangle, because it has three angles. It also has three sides and three vertices. A triangle is always coplanar, which is not true of many of the other polygons. A regular polygon is a polygon with all angles and all sides congruent, or equal. Here are some regular polygons.

We can use a formula to find the sum of the interior angles of any polygon. In this formula, the letter n stands for the number of sides, or angles, that the polygon has. sum of angles = (n 2)180Let's use the formula to find the sum of the interior angles of a triangle. Substitute 3 for n. We find that the sum is 180 degrees. This is an important fact to remember. sum of angles = (n 2)180= (3 2)180 = (1)180 = 180 To find the sum of the interior angles of a quadrilateral, we can use the formula again. This time, substitute 4 for n. We find that the sum of the interior angles of a quadrilateral is 360 degrees. sum of angles = (n 2)180= (4 2)180 = (2)180 = 360Polygons can be separated into triangles by drawing all the diagonals that can be drawn from one single vertex. Let's try it with the quadrilateral shown here. From vertex A, we can draw only one diagonal, to vertex D. A quadrilateral can therefore be separated into two triangles.

If you look back at the formula, you'll see that n 2 gives the number of triangles in the polygon, and that number is multiplied by 180, the sum of the measures of all the interior angles in a triangle. Do you see where the "n 2" comes from? It gives us the number of triangles in the polygon. How many triangles do you think a 5-sided polygon will have?

Here's a pentagon, a 5-sided polygon. From vertex A we can draw two diagonals which separates the pentagon into three triangles. We multiply 3 times 180 degrees to find the sum of all the interior angles of a pentagon, which is 540 degrees. sum of angles = (n 2)180= (5 2)180 = (3)180 = 540

TRIANGLES

As you learned in the last lesson, a triangle is the simplest polygon, having three sides and three angles. The sum of the three angles of a triangle is equal to 180 degrees. Triangles are classified by sides and by angles. Move your cursor over the triangles to learn more.

Just as the rectangle and the circle are very popular in the real world, so is the triangle! You'll find triangles at work bracing a structure or bridge, racking billiard balls, or holding up a shelf. Triangles are classified in two general ways: by their sides and by their angles. First, we'll classify by sides:

A triangle with three sides of different lengths is called a scalene triangle. An isosceles triangle has just two equal sides, called legs. The third side is called the base. The angles that are opposite the equal sides are also equal. An equilateral triangle has three equal sides. In this type of triangle, the angles are also equal, so it can also be called an equiangular triangle. Each angle of an equilateral triangle must measure 60 degrees, since the sum of the interior angles of any triangle must equal 180 degrees.

Now let's classify by angles. An acute triangle has three acute angles, or three angles with a measure of less than 90 degrees. An obtuse triangle has one angle that is greater than 90 degrees. If one of the angles in a triangle is a right angle, then the triangle is called a right triangle. Notice we draw a square at vertex C to show a right angle. You can use two labels for a triangle. For example, triangle MNO is both an acute and an isosceles triangle. Triangle PQR is an obtuse, scalene triangle.

QUADRILATERALS

The five most common types are the parallelogram, the rectangle, the square, the trapezoid, and the rhombus.

. There are many different kinds of quadrilaterals, but all have several things in common: all of them have four sides, are coplanar, have two diagonals, and the sum of their four interior angles equals 360 degrees. This is how they are alike, but what makes them different?

We know many quadrilaterals by their special shapes and properties, like squares. Remember, if you see the word quadrilateral, it does not necessarily mean a figure with special properties like a square or rectangle! In word problems, be careful not to assume that a quadrilateral has parallel sides or equal sides unless that is stated.Special QuadrilateralsA parallelogram has two parallel pairs of opposite sides.

A rectangle has two pairs of opposite sides parallel, and four right angles. It is also a parallelogram, since it has two pairs of parallel sides.

A square has two pairs of parallel sides, four right angles, and all four sides are equal. It is also a rectangle and a parallelogram.

A rhombus is defined as a parallelogram with four equal sides. Is a rhombus always a rectangle? No, because a rhombus does not have to have 4 right angles.

Trapezoids only have one pair of parallel sides. It's a type of quadrilateral that is not a parallelogram. (British name: Trapezium)

Kites have two pairs of adjacent sides that are equal.

We can use a Venn diagram to help us group the types of quadrilaterals.

A Venn diagram uses overlapping circles to show relationships between groups of objects. All "quadrilaterals" can be separated into three sub-groups: general quadrilaterals, parallelograms and trapezoids.Is a rectangle always a rhombus? No, because all four sides of a rectangle don't have to be equal. However, the sets of rectangles and rhombuses do intersect, and their intersection is the set of squaresall squares are both a rectangle and a rhombus. We can put squares in the intersection of the two circles. From this diagram, you can see that a square is a quadrilateral, a parallelogram, a rectangle, and a rhombus! Is a trapezoid a parallelogram? No, because a trapezoid has only one pair of parallel sides. That is why we must show the set of trapezoids in a separate circle on the Venn diagram.What about kites? Kites are quadrilaterals that can be parallelograms. If their two pairs of sides are equal, it becomes a rhombus, and if their angles are equal, it becomes a square.

AREA OF POLYGONS AND CIRCLESThe area of a shape is a number that tells how many square units are needed to cover the shape. Area can be measured in different units, such as square feet, square meters, or square inches. You can find an area by drawing a shape on graph paper, and counting the squares inside the shape. But this is not very practical, so we use area formulas instead. Every polygon and circle has a formula for finding its area..

Area is always a positive number. It represents the number of square units needed to cover a shape, such as a polygon or a circle. We generally use formulas to calculate areas.Area of: rectangle | square | parallelogram | triangle | trapezoid | circleArea of a RectangleA rectangle is a good, simple shape to begin with. The area of a rectangle is equal to the product of the length of its base and the length of its height. The height is a segment that is perpendicular to the base. For a rectangle, the base and height are often called the "length" and the "width", and sometimes the height is referred to as the "altitude."

Let's find the area of this rectangle, with a base measuring 4 feet and a height measuring 6 feet. Using the formula, we multiply 4 feet times 6 feet, to get 24 square feet.

Area of a SquareA square is a special rectangle, and you can find its area using the rectangle formula. However, since the base and height are always the same number for a square, we usually call them "sides." The area of a square is equal to the length of one side squared.

If the length of one side of this square is 4 centimeters, what is the area? We substitute the value "4 cm" into the formula, and we find the area to be 16 square centimeters.

Area of a ParallelogramTo find the area of a parallelogram, we can use the same formula that we used for the area of a rectangle, multiplying the length of the base times the length of the height.

Let's find the area of a parallelogram that has a base of 23 cm and a height of 7 cm. If we substitute the values into the formula, we find that the parallelogram has an area of 161 square centimeters.

Area of a TriangleWhat about triangles? This sketch of a quadrilateral shows us that one diagonal separates the interior area into two equal parts.

The area of a triangle is therefore one-half the area of the quadrilateral, which is base length multiplied by the height. What is the area of a triangle with a base length of 23 feet and a height of 16 feet? Substitute the values into the formula, and we find the area to be 184 square feet.

Area of a Trapezoid To find the area of a trapezoid, we can draw a diagonal so the trapezoid is divided into two triangles.

You can see that the area of the trapezoid DEFG is equal to the sum of the area of the two triangles EFG and DEG.Area of triangle EFG = bh = (base1)hArea of triangle DEG = bh = (base2)hArea of the trapezoid DEFG = (base1)h + (base2)h. Since the height of triangle EFG and DEG are the same, we can write the formula for the area of a trapezoid (either way is correct):

If trapezoid DEFG has a height of 8 inches, base DG measures 12 inches and base EF measures 7 inches, what is the area of the trapezoid? If we substitute the values into the formula, we find the area to be 76 square inches.

Area of a CircleFinally, we'll look at how to find the area of a circle. To help us understand circle area better, first we'll draw a square, and inscribe a circle in it, which means to draw the circle inside the square so that the circle just touches each side of the square.

We can find the area of this square by first finding the area of the four smaller squareseach with sides equal to r, the radius of the circleand adding them together.

Notice that the sides of the square are twice as long as the radius of the circle. You could also find the area of the square by multiplying the side times the side, or 2r x 2r, which also equals 4r2.You can see that the area of the circle must be less than the area of four of the squares. But how much less? We could make an educated guess and say that the area of the circle might be a little bit larger than three of the smaller squares.The actual number we're looking for, which is between 3 and 4, is the special number called pi, represented by the Greek letter . Pi is approximately equal to 3.14.

The symbol you see here means "approximately equal to." Pi actually has an unending number of decimal points, but 3.14 is usually close enough for our calculation purposes. Pi is the ratio between the diameter and circumference of a circle.The final formula for the area of a circle is shown here.

Let's say your watch face has a diameter of 1 inch, so its radius is or 0.5. We can find its area like this:

CONGRUENT

Two polygons are congruent if they are the same size and shape - that is, if their corresponding angles and sides are equal.

SIMILAR FIGURES

Let's look at the corresponding parts of these two triangles, triangle MHS and triangle ONE. Are the corresponding sides equal? How about the corresponding angles?

These triangles are not congruent, because the corresponding sides are obviously not equal. But they are somehow alike, aren't they? They have the same shape. What about their angles? Are they equal? If we put triangle ONE on top of triangle MHS, we could compare each angle, and we would find that the corresponding angles are all congruent. This gives us the definition of similar triangles: if the corresponding angles of two triangles are congruent, then the triangles are similar. Not only are the corresponding angles the same size in similar polygons, but also the sides are proportional. We can use the ratios called proportions to help us find missing values.

Here are two quadrilaterals: RUSH and GOLD. Are they similar? If we measured the angles, we would find that the corresponding angles are congruent. Therefore, these quadrilaterals are similar. Since we know that they are similar, we also know that their sides are proportional. We can set up proportions between the sides, like this:

Let's use a proportion to find the length of segment OL, given the other lengths shown here.

We can set up the problem like this. We read this proportion as "OL is to US as GO is to RU."

If we substitute the values of the segment lengths, we get this proportion.

Now we can use cross products to solve for the length of OL. We multiply the extremes and the means together, and solve for n.

SQUARE AND SQUARE ROOTS

What is the area of the large square? Do you have to count all the little squares to find out? How can you find the area of the square if you know the length of one side?

You worked with squares and square roots in Pre-Algebra and Algebra, and they are important in Geometry as well. Squaring a number and finding the square root are inverse operations. We can use geometric models to learn more about these important concepts.

How can you find the length of a side of the square if you know its area?

Any number raised to the power of 2 can be modeled using a polygon--the square! That's why we call raising a number to the second power "squaring the number." The perfect squares are squares of whole numbers. Here are the first five perfect squares. We've shown a geometric model to verify each of these squares. The square root of a number n is a number that, when multiplied by itself, equals n. Here are the square roots of the perfect squares above.

This model shows the number 169 as a square. From the model, what is the square root of 169? We can count the number of units making up each side of the square. We find 13 units to a side, so 13 is the square root of 169.

SPACE FIGURES

Space figures are figures whose points do not all lie in the same plane. In this unit, we'll study the polyhedron, the cylinder, the cone, and the sphere. Polyhedrons are space figures with flat surfaces, called faces, which are made of polygons. Prisms and pyramids are examples of polyhedrons.Cylinders, cones, and spheres are not polyhedrons, because they have curved, not flat, surfaces. A cylinder has two parallel, congruent bases that are circles. A cone has one circular base and a vertex that is not on the base. A sphere is a space figure having all its points an equal distance from the center point.Cone Cylinder Prism

Sphere PyramidThe space that we live in has three dimensions: length, width, and height. Three-dimensional geometry, or space geometry, is used to describe the buildings we live and work in, the tools we work with, and the objects we create. First, we'll look at some types of polyhedrons. A polyhedron is a three-dimensional figure that has polygons as its faces. Its name comes from the Greek "poly" meaning "many," and "hedra," meaning "faces." The ancient Greeks in the 4th century B.C. were brilliant geometers. They made important discoveries and consequently they got to name the objects they discovered. That's why geometric figures usually have Greek names! We can relate some polyhedrons--and other space figures as well--to the two-dimensional figures that we're already familiar with. For example, if you move a vertical rectangle horizontally through space, you will create a rectangular or square prism.

If you move a vertical triangle horizontally, you generate a triangular prism. When made out of glass, this type of prism splits sunlight into the colors of the rainbow.

Now let's look at some space figures that are not polyhedrons, but that are also related to familiar two-dimensional figures. What can we make from a circle? If you move the center of a circle on a straight line perpendicular to the circle, you will generate a cylinder. You know this shape--cylinders are used as pipes, columns, cans, musical instruments, and in many other applications.

A cone can be generated by twirling a right triangle around one of its legs. This is another familiar space figure with many applications in the real world. If you like ice cream, you're no doubt familiar with at least one of them!

A sphere is created when you twirl a circle around one of its diameters. This is one of our most common and familiar shapes--in fact, the very planet we live on is an almost perfect sphere! All of the points of a sphere are at the same distance from its center.

There are many other space figures--an endless number, in fact. Some have names and some don't. Have you ever heard of a "rhombicosidodecahedron"? Some claim it's one of the most attractive of the 3-D figures, having equilateral triangles, squares, and regular pentagons for its surfaces. Geometry is a world unto itself, and we're just touching the surface of that world. In this unit, we'll stick with the most common space figures.

PRISMS

A prism is a polyhedron, with two parallel faces called bases. The other faces are always parallelograms. The prism is named by the shape of its base. Here are some types of prisms. Move your mouse over each one to learn more. rectangular prism triangular prism hexagonal prism

A prism is a polyhedron that has two congruent parallel polygons as its bases. The other faces of a prism are parallelograms, most often rectangles. A prism is named for the shape of its base. For instance, this prism is called a hexagonal prism, because its two bases are hexagons.

Visualizing three-dimensional figures is an important part of geometry, and it helps to build mental math muscle. Study this two-dimensional pattern. If we folded its sides together to form a space figure in three dimensions, what figure would we get? Can you visualize how the pieces would fit together?This figure is the familiar cube, which is a type of rectangular prism. The cube is also called a "regular hexahedron," because it has 6 congruent faces. "Hexa" means six in Greek--there's one of those Greek names again! All space figures have volume. You can find the volume of any prism using this formula: V = Bhwhere B equals the area of the base of the prism and h is its height. Imagine the base passing through the entire prism like an elevator going up. The space that the base passes through is the volume of the prism.The units of volume are called "cubic units." Cubic feet, cubic meters, cubic inches, cubic yards, cubic centimeters--these are all examples of units of volume. They can all be written as powers of 3, because they have three dimensions. Examples of units of volume: cubic feet (ft3) cubic inches (in3) cubic yards (yd3) cubic meters (m3) cubic centimeters (cm3) Given the dimensions shown in the figure, let's compute the volume of this prism.

First we have to find the area of the triangle that forms the base of this prism. The area of a triangle is one-half the base of the triangle times the height of the triangle. Don't get the height of the triangle and the height of the prism mixed up! The triangle has a base of 14 inches and a height of 12 inches. If we substitute these values into the formula, we get 84 square inches for the area of the triangle.

The height of the prism is 20 inches. If we substitute the values into the volume formula, we find that the volume of the prism is 1,680 cubic inches.

PYRAMIDS

. A polyhedron is a pyramid if it has 3 or more triangular faces sharing a common vertex. The base of a pyramid may be any polygon. If the base is a triangle too, then the pyramid has four faces. This is the simplest polyhedron, also called a tetrahedron, from the Greek word "tetra", meaning "four".

triangular pyramid square pyramid(tetrahedron)

The pyramid is a famous shape primarily because of the ancient Egyptian pyramids. There are many mysteries surrounding these giant constructions, but one thing is certain--geometry was used to build them! A pyramid is a polyhedron that has only one base. (The base is the "bottom" of the Egyptian pyramids.) The other faces are all congruent triangles, and they share a common vertex, which is the top point. The base can be any type of polygon. If the base is a triangle, then the pyramid has a total of four faces. The Egyptian pyramids have square bases and four triangles as faces.

The volume of a pyramid is a measure of how much it would take to fill the shape. For a pyramid, the formula is:

Where B is the area of the base figure, and h is the height from the base to the vertex. The volume is expressed in measurement units, cubed, like cubic inches. See if you can imagine little cubes filling up the interior space of the shape.This formula is true for pyramids of any shape base. As long as you can find the area of the base and you know the height, you can calculate the volume.What is the volume of a pyramid with a square base with sides of 5 cm, and a height of 3 cm?

V = (5cm x 5cm) x 3 cmVolume = 25 cm3

CYLINDERS, CONES AND SPHERES

In this unit we'll study three types of space figures that are not polyhedrons. These figures have curved surfaces, not flat faces. A cylinder is similar to a prism, but its two bases are circles, not polygons. Also, the sides of a cylinder are curved, not flat. A cone has one circular base and a vertex that is not on the base. The sphere is a space figure having all its points an equal distance from the center point.

in this lesson, we study some common space figures that are not polyhedra. These figures have some things in common with polyhedra, but they all have some curved surfaces, while the surfaces of a polyhedron are always flat.

First, the cylinder. The cylinder is somewhat like a prism. It has parallel congruent bases, but its bases are circles rather than polygons. You find the volume of a cylinder in the same way that you find the volume of a prism: it is the product of the base area times the height of the cylinder:

Since the base of a cylinder is always a circle, we can substitute the formula for the area of a circle into the formula for the volume, like this:

Let's find the volume of this can of potato chips.

We'll use 3.14 for pi. Then we perform the calculations like this:

That's a lot of potato chips!

A cone has a circular base and a vertex that is not on the base. Cones are similar in some ways to pyramids. They both have just one base and they converge to a point, the vertex.The formula for the volume of a cone is:

Since the base area is a circle, again we can substitute the area formula for a circle into the volume formula, in place of the base area. The final formula for the volume of a cone is:

Let's find the volume of this cone.

We can substitute the values into the volume formula. When we perform the calculations, we find that the volume is 150.72 cubic centimeters.

Finally, we'll examine the sphere, a space shape defined by all the points that are the same distance from the center point. Like a circle, a sphere has a radius and a diameter. The shape of the earth is like a large sphere -- it has radius of about 4000 miles. A tennis ball is a sphere with a radius of about 2.5 inches. Since a sphere is closely related to a circle, you won't be surprised to find that the number pi appears in the formula for its volume:

Let's find the volume of this large sphere, with a radius of 13 feet. Notice that the radius is the only dimension we need in order to calculate the volume of a sphere. If we substitute 13 feet for the radius, then we get 9,198.11 cubic feet.

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