Section 12.2Surface Areas of Prisms and Cylinders

Lateral Areas and Surface Areas of Prisms In a solid figure, faces that are not bases are called lateral faces. Lateral faces intersect each other at the lateral edges, which are all parallel and congruent. The lateral faces intersect the base at the base edges. The altitude is a perpendicular segment that joins the planes of the bases. The height is the length of the altitude. Recall that a prism is a polyhedron with two parallel congruent bases.

The lateral area L of a prism is the sum of the areas of the lateral faces. The net below shows how to find the lateral area of a prism.

From this point on, you can assume that solids in the text are right solids. If a solid is oblique, it will be clearly stated. Example 1: Find the lateral area of the regular octagonal prism. Round your answers to the nearest hundredth.

The bases are regular octagons. So the perimeter of one base is 8(3) or 24 centimeters.

L = Ph Lateral area of a prism = (24)(9) P = 24, h = 9 = 216 Multiply

Answer: The lateral area is 216 square centimeters.

Example 2: Find the lateral area of the regular hexagonal prism. Round your answers to the nearest hundredth.

The bases are regular hexagon. So the perimeter of one base is 6(5) or 30 centimeters.

L = Ph Lateral area of a prism = (30)(12) P = 30, h = 12 = 360 Multiply

Answer: The lateral area is 360 square centimeters.

The surface area of a prism is the sum of the lateral area and the areas of the bases.

Example 3: Find the surface area of the rectangular prism.

Answer: The surface area is 360 square centimeters.

T = L + 2B Surface area of a prism = Ph + 2B L = Ph = (24)(12) + 2(36) Substitution = 360 Simplify

Example 4: Find the surface area of the regular hexagonal prism

Answer: The surface area is 360 + square centimeters.

Since the bases are regular hexagons, to find the

area of one base, use the formula: a =

and P = 30.

So, A =

T = L + 2B Surface area of a prism

= Ph + 2B L = Ph

= (30)(12) + Substitution

1.

2A aP 2.5 3

12.5 3 30

2

75 3

75 3

Example 5: Find the surface area of the triangular prism.

Answer: The surface area is 416 square units.

Since the bases are triangles, to find the area of one base,

use the formula: So, A = 48. To find the

perimeter of the base use the right triangle with a

base of 6 and height of 8 to find the hypotenuse

of 10. Therefore P = 32.

T = L + 2B Surface area of a prism

= Ph + 2B L = Ph

= (32)(10) + 2(48) Substitution

1.

2A bh

Lateral Areas and Surface Areas of Cylinders The axis of a cylinder is the segment with endpoints that are centers of the circular bases. If the axis is also an altitude, then the cylinder is a right cylinder. If the axis is not an altitude, then the cylinder is an oblique cylinder.

The lateral area of a right cylinder is the area of the curved surface. Like a right prism, the lateral area, L equals Ph. Since the base is a circle, the perimeter is the circumference of the circle C. So, the lateral area is Ch or 2πrh.

The surface area of a cylinder is the lateral area plus the areas of the bases.

Example 6: Find the lateral area and the surface area of the cylinder. Round to the nearest thousandth.

L = 2rh Lateral area of a cylinder

= 2(14)(18) Replace r with 14 and h with 18.

≈ 1583.363 Use a calculator.

S = 2rh + 2r2 Surface area of a cylinder

≈ 1583.363 + 2(14)2 Replace 2rh with 1583.363and r with 14.

≈ 2814.867 Use a calculator.

Answer: The lateral area is about 1583.363 square feet and the surface area is about 2814.867 square feet.

Example 7: A can of soup is covered with the label shown. What is the radius of the soup can?

L = 2rh Lateral area of a cylinder

125.6 = 2r(8) Replace L with 15.7 ● 8 and h with 8.

125.6 = 16r Simplify.

2.5 ≈ r Divide each side by 16.

Answer:The radius of the soup can is about 2.5 inches.

Example 4

Find Missing Dimensions