modul-perimeter area volume

Perimeter, area, volume – JBH 2011

TOPIC 3 PERIMETER, AREA AND VOLUME

Introduction

When you think of the size of a line, you measure its length and perimeter. When you think of the size of a 2-dimensional surface, you measure its area. When you think of the size of a 3-dimensional space, you measure its volume. Apart from length, area and volume as well as perimeter are three other attributes closely related to 2-dimensional shapes and 3-dimensional space. Thus, this topic focuses on helping you to understand the basic concepts related to these three attributes.

Learning Outcomes

By the end of this topic, you will be able to:

1. understand the concept of perimeter, area and volume;2. understand the concept of measuring units for area and volume; 3. solve problems involving perimeter, area and volume.

3.1 Perimeter and Area

What is perimeter? What is area? Write down your ideas.

The perimeter of a region is the length of its boundary whreas its area refers to the amount of surface inside the given boundary. You have to take note that a region is a closed figure. In other words, you can only measure the perimeter and area of closed shapes. Figure 1(a), (b) and (c) show examples of closed shapes. Note that all these shapes have the same perimeter of 12 units but have different areas.

area = 8 square units area = 5 square units area = 9 square units

(a) (b) (c)

Figure 1. Closed shapes with the same perimeter but different areas.

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Figure 2(a), (b) and (c) show more examples of closed shapes. Note that these shapes have different perimeters, but the same area of 12 units.

perimeter = 14 units perimeter = 16 units perimeter = 20 units

(a) (b) (c)

Figure 2. Closed shapes with the same area but different perimeters.

After knowing the meaning of perimeter and area, you naturally will want to know how these two attributes are measured. You should realise that perimeter is a special kind of length. Thus the measuring units for perimeter are the same as length. However, you may need to spend a little bit more time to think about the measuring units for area.

Measuring Perimeter of Polygons

The perimeter of a polygon can be found easily if you know the lengths of all its sides. Table 1 shows the formulae to find the perimeter of some common polygons quickly.

Table 1Perimeter of Common Polygons

Polygon Perimeter

Equilateral triangle with sides of length s 3s

Square with sides of length s 4s

Rhombus with sides of length s 4s

Rectangle with sides of lengths a and b 2a + 2b

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SS

S

aa

b

b

s

s

s

s


Table 1 (Cont’d)Perimeter of Common Polygons

Polygon Perimeter

Parallelogram

with sides of lengths a and b2a + 2b

Kite with sides of lengths a and b 2a + 2b

Circle with radius r 2 r

You must take note that the perimeter of a circle is given a special name circumference. In every circle, the ratio of the circumference C to the diameter d,

namely , is a constant, called . So, C = d = 2 r.

Measuring Area of Polygons

To measure the area of a suface, you need a new type of unit, one that can be used to cover a surface. The area of a surface will be the number of units it takes to cover it. Theoretically, any shape can be used as a unit for measuring area. However, squares have been found to be the most convenient shapes for measuring area. If we use the square in Figure 3(a) as the unit square, then the area of the triangle in Figure 3(b) is ½ square unit. Similarly, the area of the shaded region in Figure 3(c) is 5 square units, because it can be covered by 4 squares and 2 half-squares.

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b

b

a a

r


(a) (b) (c)

Figure 3. A square as the measuring unit of area.

The metric units for measuring area will depend on the unit of length used for the unit square. So, 1 square metre (written as m2) is a square whose sides have lengths of 1 metre. Similarly, 1 square centimetre (1 cm2) is a square whose sides have lengths of 1 centimetre. Figure 4 illustrates the ideas of 1 m2 and 1 cm2.

Figure 4. One square metre and one square centimetre.

How many cm2 equal 1 m2 ?

How many mm2 equal 1 cm2 ?

How many m2 equal 1 km2?

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1 square metre (m2)

1 m

1 m

1 cm

1 cm

1 square centimetre

(cm2)


TIDBIT

Estimate the area of the cat in terms of square units.

The area of a polygon can be found by counting the number of square units required to cover it completely. In Figure 5, the number of square units required to cover the rectangle can be easily found by multiplying the lengths of its two sides, 4 x 3. Thus, the area A of of a rectangle with sides of lengths a and b is A = ab.

Area = 12 unit squares Area = ab

3 b

4 a

Figure 5. Area of a rectangle.

Once you understand how the area of a rectangle is calculated, then you will be able to use this knowledge to find the area of most other polygons easily.

Area of a square. Since a square is a special type or rectangle, so you can easily deduce that the area A of a square whose sides have length s is A = s2 as shown in Figure 6.

Figure 6. Area of a square.

Area of a triangle. The area of a triangle can be obtained from the formula to find the area of a rectangle. As illustrated in Figure 7, the area A of the triangle ABC is half the area of the rectangle ABCD. So, it is obvious that A = ½ bh.

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s

s

Area of square = s x s = s2


Area of ABC = ½ x Area of ABCD = ½ bh

Figure 7. Area of a triangle.

How would you show that the area of each of these triangles is ½ bh ?

Area of a parallelogram. Similar to the area of a triangle, the area of a parallelogram can also be obtained from the formula for finding the area of a rectangle. As illustrated in Figure 8, the area A of the parallelogram ABCD with base b and height h is equal to the area of the rectangle BCNM . So, it is obvious that the area A = bh.

Area of ABCD = Area of BCMN = bh

Figure 8. Area of a parallelogram.

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h

b

h

b

A

B C

DM

h

bM

B C

D N

h

bA

B C

D


Area of a trapezoid. Suppose that you have a trapezoid PQRS whose bases have lengths a and b and whose height is h, the distance between the parallel bases. You may play a simple trick on the trapezoid. Duplicate another trapezoid and turn it 180o

as shown in Figure 9 in order to form a parallelogram.

Area of PMNS = (a + b)h

Area of PQRS = ½ x Area of PMNS = ½ (a + b)h

Figure 9. Area of a trapezoid.

As you can see, the figure PMNS is a parallelogram with base (a + b) and height h. Since PQRS is ½ of PMNS, then the area of PQRS = ½ (a+b)h.

Area of a circle. Suppose you have a circle whose radius is r as shown in Figure 10. How do you find the area of the circle? As you have noticed, the area of each square is r 2. The area of the circle is less than 4 times r 2. A good estimation for the area of the circle is 3.1416 times r 2. The value 3.1416 is an estimation for the value of , a very special and famous constant in mathematics.

Area of circle < 4 x (r 2)Area of circle 3.1416 r 2 = r 2

Figure 10. Area of a circle.

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b

a

h

S

P Q

R

b

a

h

S

P Q

R b

a

b

a

h

S

P Q

R b

a M

N

r

r

A B

CD

E

F

G


Exercise 11. Explain why the perimeter of ABCD in the figure below cannot be

calculated.

2. Given that PQRS is a rectangle and MQ = QR = 9 cm. Calculate the perimeter and area of the shaded region MNRQ.

3. In the figure below, BFGC is a square with sides 10 cm long. ABCD is a parallelogram. Given that AB = 13 cm , calculate the perimeter and area for the whole figure.

4. In the figure below, ABCD is a rectangle and EFG is a semicircle with a radius of 7 cm. Given that AB = 20 cm and BC = 15 cm. Calculate the perimeter and area of the shaded region.

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6 cm

B

C D

A

8 cm10 cm

7 cm

P

Q R

S

M

N

12 cm

5 cm

F

A

B C

D

G


Exercise 1 (Cont’d)

5. In the figure below, ABEF is a parallelogram and BCDE is a trapezoid. Calculate the perimeter and area of

a) ABEFb) The whole figure.

Time to take your tidbit and relax!Come back again to learn more about volume.

TIDBIT

The Great Pyramid of Egypt was completed in 2580 B.C. and is one of the seven wonders of the ancient world. Its original height was 146.5 m and the base was 230 m square. It contains about 5.8 million tonnes of sandstones blocks. It is still standing today.

What is the volume of this pyramid?

(Duffy, Murty & Mottershead, 1989, p.147)

3.2 Volume

What is volume? Write down your ideas.

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A

F

B

C

D

E

15cm

12cm

6cm10cm


The volume of a three-dimensional figure is a measure of the amount of space that it occupies. To measure the amount of space, we need units that can be used to fill up a space. Cubes have been found to be the most convenient three-dimensional figures for measuring volume because they pack together without gaps or overlapping. If we use the cube in Figure 11(a) as the unit cube, as you can see, the box in Figure 11(b) can be filled with 12 cubes on its base and 12 more cubes can be placed above these to fill the box. Thus, the volume of the box is 24 cubic units.

Figure 11. A cube as the measuring unit for volume.

The metric units for measuring volume will depend on the unit of length used for the unit cube. So, 1 cubic metre (written as m3) is a cube whose sides have lengths of 1 metre. Similarly, 1 cubic centimetre (1 cm3) is a cube whose sides have lengths of 1 centimetre. Figure 12 illustrates the ideas of 1 m3 and 1 cm3.

1 cm3 1 m3

Figure 12. One cubic metre and one cubic centimetre.

How many cm3 equal 1 m3 ?

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TIDBIT

How many small cubes make up this block?

If the outer layer of cubes is removed, how many small cubes are left in the smaller block?

(Daly, et al., 1994, p. 384)

The volume of a 3-dimensional shape (or solid) can be found by counting the number of unit cubes required to fill its space compeletely. In Figure 13, the number of centimetre cubes required to fill the cuboid can be easily found by multiplying its length with its breadth and its height, 20 cm x 10 cm x 6 cm = 1200 cm3. Thus, the volume V of a cuboid with length l, breadth b, and height h is V = lbh.

Volume = 20 cm x 10 cm x 6 cm = 1 200 cm3 Volume = l x b x hFigure 13. Volume of a cuboid.

Volume of a prism. Cuboid is a rectangular prism. You should have notice that the number of unit cubes that cover the base of the prism is the same as the area of the base. Therefore, the volume of the prism can be calculated by multiplying the area of the base by the height of the prism. In general, the volume of any right prism can be computed by the formula Volume of prism = area of base x height. Figure 14 shows some examples of prisms.

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lb

h


Volume = [Area of base x height] = B x h

Figure 14. Area of prisms.

Volume of a pyramid. How do you find the the volume of a pyramid? The answer lies in knowing the relationship between the pyramid and the smallest prism containing it. Figure 15 shows the relationship of a square pyramid with the smallest prism (a cube) containing it. As you can see, the prism can be divided into three congruent square pyramids. Hence, the volume of a pyramid is one-third of the smallest prism containing it.

Volume of pyramid = x volume of the smallest prism containing it

= x area of base x height

Figure 15. Volume of a square pyramid.

Similarly, we will expect the area of a cone, which is a circular pyramid to equal to

of the volume of the smallest prism (a cylinder) containing it as shown in Figure 16.

Volume of cone = x volume of cylinder

Figure 16. The volume of a cone.

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B

h

B

h

B

h


Volume of a sphere. The volume of a sphere is two-third the volume of the smallest cylinder containing the sphere as shown in Figure 17.

Volume of cylinder = ( r 2 ) x h = ( r2 ) x 2r = 2r3

Volume of sphere = x volume of cylinder = x 2r 3 = r 3

Figure 17. Volume of a sphere.

Exercise 2

1. The figure on the right shows a solid object formed from a combination of a hemisphere and a cylinder. The radius of the hemisphere is 7 cm.

Using ,

Calculate the volume of the solid object.

2. The figure below shows a cuboid measures 8cm x 10cm x 10cm. A pyramid was taken out from the cuboid.

Calculate the volume of the remaining solid.

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10 cm

10 cm

9 cm

h

7 cm

7 cm


Time for dessert and relax!

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Exercise 3 (Cont’d)

3. The figure shows a cylinder with a radius of 7 cm. Two identical cones were taken out as shown in the figure below.

Taking and calculate the volume of the remaining solid.

4. The figure below shows a wooden craft formed from a combination of two pyramids with square bases and a cube with each side measuring 5 cm.

Calculate the volume of the solid.

12 cm

9 cm

5 cm

6 cm


DESSERT

The area of the inscribed square is what percent of the area of the circle?

(Bennett & Nelson, 2004, p.675)

Summary

1. The perimeter of a region is the length of its boundary.

2. Area is the amount of 2-dimensional space inside a given boundary. A unit square is a common unit of measure for area because it is the most convenient shape for covering a region without gaps or overlapping.

3. Volume is the amount of space occupied by a 3-dimensional shape (or solid). A unit cube is a common unit of measure for volume because it is the most convenient 3-dimensional shape to fill a space without gaps or overlapping.

References

Bennett, A. B. Jr. & Nelson, T. L. (2004). Mathematics for elementary teachers. A conceptual approach. 6th ed. New York, NY: McGraw Hill.

Daly, T; Ardley, J; Buruma, J; Cody, M; Tomlinson, P. (1994). Mathematiacs today Year 9. 2nd ed. Roseville, NSW Australia: McGraw Hill Book.

Musser, G. L.; Burger, W. F. & Peterson, B. E. (2001). Mathematics for elementary teachers. A contemporary approach. 5th ed. New York, NY: John Wiley & Sons.

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modul-perimeter area volume

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