Topic 13 Surface Area & VolumeStudy Guide & Notes

Surface Area of Cylinders

A cylinder is a three-dimensional figure with two parallel circular bases that are the same size.

The base of a cylinder is one of a pair of parallel circular faces that are the same size.

The height of a cylinder is the length of a perpendicular segment joins the planes of the bases.

A right cylinder’s height is a segment that joins the centers of the bases.

The lateral surface of a cylinder is the curved surface not included in the bases.

Height Bases lateral surface

A net is a two-dimensional pattern that you fold to form a three-dimensional figure. You can use nets to help determine the surface area of a three-dimensional figure.

If we “unroll” the 3 dimensional cylinder, we can find the area of each surface and add up the totals to get the surface area of a cylinder. In this case you need to know the formulas for the area of circles and for the area of rectangles!

Surface Area of Cylinder FormulaUsing nets isn’t always practical or the most efficient method for determining the surface area of a cylinder. We can use what we know about rectangles and circles to find a more efficient method.

The lateral surface of a cylinder is the area of its lateral surface – which is the rectangular portion of the “unrolled” net.

The height of the cylinder = the height of the rectangle. To find the area of a rectangle we multiply b(base) • h(height) or bh.The circumference of the base of the cylinder = the base of the rectangle. So 2πr = b.So the lateral area of a cylinder is = 2πrh

Now we need to find the area of the bases! The area of a circle = πr2. Remember there are two bases, so the total area of the bases = 2πr2.

So the formula for the surface area of a cylinder = 2πrh + 2πr2

Volume of Cylinders

The volume of a cylinder is the number of cubic units needed to fill a cylinder. The formula for the volume of cylinder = Bh.B = the area of the base = πr2

h = the height of the cylinderSo the volume of a cylinder V = Bh = πr2h

Surface Area of a Cone

A cone is a three-dimensional figure with one circular base and one vertex.The base of a cone is a circle with radius r.The vertex of a cone is the point farthest from a baseThe lateral surface of a cone is the curved surface not included in the base.The slant height of a cone is the length of the lateral surface from base to vertex.The height of the cone h is the length of a segment from the vertex perpendicular to the plane of the base.A right cone connects the height from the vertex to the center of the base.

base vertex lateral surface slant height

height of cone

The formula for the surface area of a cone = lateral area + area of the base = L.A + B = πrl + πr2

Volume of a ConeThe formula for the volume of a cone is the same as the formula for the volume of a pyramid. The only difference is that the area of the base is calculated differently! Volume of a cone = 1/3 area of the base * height of the cone = 1/3Bh = 1/3 πr2h

Surface Area of Spheres

A sphere is the set of all points in space that are the same distance from a center point.

The radius of a sphere (r) is a segment that has one endpoint at the center of the sphere and the other endpoint on the sphere.

The surface area of a sphere is equal to the lateral area of a cylinder that has the same radius and a height of 2r. So if the surface area of the square is equal to the Lateral Area of a cylinder we can develop the equation. LA = 2πrh and h = 2r so LA = 4πr2 = surface area of a sphere!

Volume of Spheres

The volume of a sphere is the number of cubic units needed to fill the sphere. If you place a sphere with a diameter that is equal to a cylinders diameter and height into the cylinder, there will still be some room left over in the cylinder. Now divide the cylinder into 3 equal pieces and take the sphere out of the cylinder. Fill 1/3 of the cylinder with water and put the sphere back into the water. Now no space is left! We can use this information to develop a formula for the volume of a sphere.

Remember that the volume of cylinder, V = = πr2h, and in this case h = 2r so V = 2πr3, BUT the sphere only fills 2/3 of the cylinder so the volume of a sphere is = 2/3 the volume of the cylinder or V = 2/3(2πr3) =4/3(2πr3).