NAME _____________________________________________ DATE ____________________________ PERIOD _____________
Lesson 12.8 Study Guide
Surface Area of Prisms
Every prism has two parallel bases. Faces that are not bases are called lateral faces. The lateral area is the sum of
the areas of the lateral faces. The surface area is the total area of all the faces, or the sum of the lateral area plus the
area of the bases. To find the lateral area L of a prism with a height h and base with a perimeter P, use the formula
L = Ph. To find the surface area S of a prism with a lateral area L and a base area B, use the formula S = L + 2B.
This can also be written as S = Ph + 2B.
Example 1
Find the lateral and surface area of the rectangular prism.
Find the lateral area.
Find the surface area.
L = Ph
S = L + 2B
= (2ℓ + 2w)h
= L + 2ℓw
= (2 ⋅ 2.1 + 2 ⋅ 5.8)2.8
= 44.24 + 2 ⋅ 2.1 ⋅ 5.8
= 44.24 ft 2
= 68.6 ft 2
Example 2
Find the lateral and surface area of the triangular prism.
Find the lateral area.
Find the surface area.
L = Ph
= (5 + 5 + 6)7
S = L + 2B
= 112 ft
= 112 + 2 ⋅
2
1
2
⋅ 6⋅ 4
= 136 ft 2
Exercises
Find the surface area of each prism. Show your work.
1.
2.
3.
4. cube: side length 8.3 centimeters
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
Lesson 12.9 Study Guide
Surface Area of Cylinders
As with a prism, the surface area of a cylinder is the sum of the lateral area and the area of
the two bases. If you unroll a cylinder, its net is a rectangle (lateral area) and two circles
(bases).
The lateral area L of a cylinder with radius r and height h is the product of the circumference
of the base (2πr) and the height h. This can be expressed by the formula L = 2πrh.
The surface area S of a cylinder with a lateral area L and a base area B is the sum of the
lateral area and the area of the two bases. This can be expressed by the formula S = L + 2B
or S = 2πrh + 2π𝑟 2
Example
Find the lateral and surface area of the cylinder.
Find the lateral area.
Find the surface area.
L = 2πrh
S = L + 2πr2
= 35π + 2π(3.5)2
= 2⋅ π ⋅ 3.5 ⋅ 5
= 35π in2
≈ 110.0 in2
exact answer
= 59.5π in2
approximate answer
≈ 186.9 in2
Exercises
Find the surface area of each cylinder. Round to the nearest tenth. Show your work!
1.
2.
4. diameter of 20 yards and a height of 22 yards
3.
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
Lesson 12.10 Study Guide
Surface Area of Pyramids and Cones
Regular pyramids have bases which are a regular polygon and lateral faces which
are congruent isosceles triangles. The height of each lateral face is called the slant
height of the pyramid.
The lateral area L of a regular pyramid is half the perimeter P of the base times the
1
slant height ℓ or L = Pℓ. The total surface area S of a regular pyramid is the lateral
2
1
area L plus the area of the base B or S = L + B, or S = Pℓ + B.
2
Example
Find the lateral and total surface area of the square pyramid.
Find the lateral area.
1
L = Pℓ
Write the formula.
2
1
S=L+B
Write the formula.
= 27 + (3 ⋅ 3) Replace L with 27
= (3 ⋅ 4)4.5
2
= 27cm
Find the surface area.
2
Replace P with 3 ⋅ 4 and ℓ with 4.5.
and B with 3 · 3.
= 36 cm
Simplify.
2
Simplify.
The lateral surface area is 27 cm2, and the total surface area is 36 cm2 .
The lateral area L of a cone is the product of π, the radius r, and the slant height ℓ.
This can be represented by the formula L = πrℓ.
The surface area S of a cone is the lateral area L plus the area of the base or π𝑟 2 .
This can be represented by the formula S = L + π𝑟 2 , or S = πrℓ + π𝑟 2 .
Exercises
Find the surface area of each figure. Round to the nearest tenth.
1.
Hint: find the radius first!!
3.
2.