224 Part 3: The Shape of the World
CHECK POINT
Find the surface area and volume of each cylinder. SA = Sr^2 + 2Srh and V = Sr^2 h- A cylinder 14 cm high with a radius of 5 cm.
- A cylinder 8 inches high with a diameter of 6 inches.
- A cylinder 2 m high with a circumference of 2S m.
- A cylinder 82 cm high with a diameter of 90 cm.
- A cylinder 20 inches high with a circumference of 20S inches.
Cones
If cylinders behave a lot like prisms, it’s not surprising that cones
have a lot in common with pyramids. Like cylinders, cones have a
circle as their base. Like pyramids, their lateral surface slopes to a
point. Trying to unroll a cone is a bit trickier than taking the label
off a can, as you did with the lateral area of a cylinder. If you try to
unroll the cone’s lateral area, you’ll find it’s part of a circle, but how
big a part depends on the size of the base circle.A cone is a solid with a
circular base and a lateral
surface that slopes to
a point.Surface Area
The surface area of a pyramid is equal to the area of the base plus
the combined area of all the surrounding triangles. That lateral
area, you learned, was equal to
1
2 v perimeter v slant height. For
a cone, the surface area is still the area of the base plus the lateral
area. The base is a circle, so the area of the base is Sr^2. If the slant
height is l, the lateral area is a fraction of the area of a circle with
area Sl^2. What fraction? The fraction you create when you put the
circumference of the base, 2Sr, over the circumference of a circle
with radius equal to the slant height.^2
2r 22
ll r
l lrl
. The
lateral area is Srl, so the surface area is SA = Sr^2 + Srl.
To find the surface area of a cone with a radius of 8 cm and a slant
height of 10 cm, find the area of the base. Sr^2 = S(8^2 ) = 64S, and
then calculate Srl = S(8)(10) = 80S. The surface area is 64S + 80S
= 144S square centimeters.