Surface Areas of Pyramids and Cones

Lateral Area of a Regular Pyramid

Find the lateral area of the square pyramid.

Lateral area of a regularpyramid

P = 2.5 ● 4 or 10, ℓ = 5

Answer: The lateral area is 25 square centimeters.

A. 54 in2

B. 64 in2

C. 108 in2

D. 132 in2

Find the lateral area of the square pyramid.

Surface Area of a Square Pyramid

Find the surface area of the square pyramid to the nearest tenth.

Step 1 Find the slant height

c2 = a2 + b2 Pythagorean Theorem

ℓ2 = 62 + 42 a = 6, b = 4, and c = ℓ

ℓ = Simplify.

Surface Area of a Square Pyramid

Answer: The surface area of the pyramid is about 179.4 square meters.

Step 2 Find the perimeter and area of the base.

P = 4 ● 8 or 32 m A = 82 or 64 m2

Step 3 Find the surface area of the pyramid.S = Pℓ + B Surface area of a

regular pyramid

__12

__12

= (32) + 64 P = 32, ℓ = , B = 64

≈ 179.4 Use a calculator.

A. 96 in2

B. 124.3 in2

C. 138.5 in2

D. 156 in2

Find the surface area of the square pyramid to the nearest tenth.

Surface Area of a Regular Pyramid

Step 1 Find the perimeter of the base.

P = 6 ● 10.4 or 62.4 cm

Find the surface area of the regular pyramid. Round to the nearest tenth.

Surface Area of a Regular Pyramid

Step 2 Find the length of the apothem and the area of the base.

A central angle of the hexagon is or

60°, so the angle formed in the triangle is

30°.

______360°6

Surface Area of a Regular Pyramid

tan 30° = Write a trigonometric ratio to find the apothem a.

a = Solve for a.

a ≈ 9.0 Use a calculator.

A = Pa Area of a regular polygon

≈ (62.4)(9.0)Replace P with 62.4 and a width 9.0.

≈ 280.8 Multiply.

So, the area of the base is approximately 280.8 cm2.

Surface Area of a Regular Pyramid

Answer: The surface area of the pyramid is about 748.8 cm2.

Step 3 Find the surface area of the pyramid.

S = Pℓ + B Surface area of regular pyramid

≈ (62.4)(15) + 280.8 P = 62.4, ℓ = 15, and B ≈ 280.8

≈ 748.8 Simplify.

A. 198 in2

B. 228.5 in2

C. 255.5 in2

D. 316.3 in2

Find the surface area of the regular pyramid. Round to the nearest tenth.

Lateral Area of a Cone

ICE CREAM A sugar cone has an altitude of

8 inches and a diameter of inches. Find the

lateral area of the sugar cone.

If the cone has a diameter of then the

radius Use the altitude and

the radius to find the slant height with the Pythagorean

Theorem.

Lateral Area of a Cone

Step 1 Find the slant height ℓ.

ℓ2 = 82 + 1.252 Pythagorean Theorem

ℓ2 ≈ 65.56 Simplify.

ℓ ≈ 8.1 Take the square root ofeach side.

Step 2 Find the lateral area L.

L = rℓ Lateral area of a cone

≈ (1.25)(8.1) r = 1.25 and ℓ ≈ 8.1

≈ 31.8

Answer: The lateral area of the sugar cone is about 31.8 square inches.

A. 39.7 in.2

B. 43.6 in.2

C. 48.2 in.2

D. 53.1 in.2

HATS A conical birthday hat has an altitude of 6 inches and a diameter of 4 inches. Find the lateral area of the birthday hat.

Surface Area of a Cone

Find the surface area of the cone. Round to the nearest tenth.

Estimate: S ≈ 3 ● 1.5 ● 3 + 3 ● 2 or 19.5 cm2

S = rℓ + r2 Surface area of a cone

= (1.4)(3.2) + (1.4)2 r = 1.4 and ℓ = 3.2

≈ 20.2Answer: The surface area of the cone is about 20.2 square centimeters. This is close to the estimate, so the answer is reasonable.

A. 58.2 cm2

B. 61.3 cm2

C. 63.6 cm2

D. 70.7 cm2

Find the surface area of the cone. Round to the nearest tenth.