Ch 12.4

V = Bh = [(12)(5)]/2 × (4) = 120

Find the volume of each prism.

V = Bh = π(52) × (10) = 250π

V = Bh = π(62) × (9) = 324π

V = Bh = (7.3)(6.2) × (4.5) = 203.7

Ch 12.4Surface Areas of Pyramids & Cones

Standard 9.0Students compute the surface areas of

pyramids and cones and commit to memory the formulas for pyramids.

Learning Target:I will be able to solve problems involving the surface area of pyramids and cones.

Ch 10.5Ch 12.4

regular pyramidThe altitude is perpendicular to the base at its center AND the base is a regular polygon.The lateral faces form congruent isosceles triangles.

slant heightThe height of each lateral face represented by l.This is different than the altitude.

Ch 12.4

Ch 12.4

Theorem 12-7

Lateral Area of a Regular Pyramid

Find the lateral area of the square pyramid.

Lateral area of a regularpyramid

P = 2.5 × 4 , ℓ = 5

Answer: The lateral area is 25 cm2.

Ch 12.4

Multiply.= 25

= (10)(5)12

A. 54 in2

B. 64 in2

C. 108 in2

D. 132 in2

Find the lateral area of the square pyramid.

Ch 12.4

Lateral area of a pyramid

P = (4 × 4) , ℓ = 8

Multiply.

= (16)(8)

= 64

L = Pℓ1212

Ch 12.4

Theorem 12-8

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.

Ch 12.4

Surface Area of a Square Pyramid

Step 2 Find the perimeter and area of the base.

P= 4 × 8 = 32 m B = 82 = 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.

Ch 12.4

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

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.

Ch 12.4

Use a calculator.

= (12) (√55) + 36

≈ 138.5

P = (6 × 4) , ℓ = √(82 - 32) , B = s2 , s = 6

= (24) (√55) + (62)

S = Pℓ + B Surface area of a pyramid

__1212

Simplify.

Example 3

Surface Area of a Regular Pyramid

Step 1 Find the perimeter of the base.

P = 6 × 10.4 = 62.4 cm

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

Ch 12.4

9

Step 2 Find the area of the base.

B = Pa Area of a regular polygon

= (62.4)(9.0) P = 10.4 × 6, a = 9.

= 280.8 Multiply.

Surface Area of a Regular Pyramid

Ch 12.4

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

9

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.

Ch 12.4

= 255.5

P = (6 × 6) , ℓ = 9, B = aP , a = 2.6 2

= (36) (9) + (93.5)

S = Pℓ + B Surface area of a pyramid

__1212

Simplify.

2.6

Ch 12.4

Theorem 12-9 & 12-10

Lateral Area of a Cone

ICE CREAM A sugar cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the sugar cone.

If the cone has a diameter of 2.5 inches then the radius is 2.5 ÷ 2. Use the altitude and the radius to find the slant height with the Pythagorean Theorem.

Ch 12.4

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.

Lateral Area of a Cone

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

= 10.1π Multiply

Answer: The lateral area of the sugar cone is about 31.8 in2.

Ch 12.4

ICE CREAM A sugar cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the sugar cone.

Step 1 Find the slant height ℓ: ℓ ≈ 8.1

A. 12.6π in.2

B. 13.9π in.2

C. 15.3π in.2

D. 16.9π 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.

Ch 12.4

Lateral area of a pyramid

P = 2π r, r = 2, ℓ = √(62 + 22)

Multiply.

= (2π)(2) (√52)

= 2π√40

L = Pℓ1212

Use a calculator.= 12.6π

Surface Area of a Cone

Find the surface area of the cone in terms of π.

Estimate: S ≈ (3 × 1.5 × 3) + (3 × 2) = 19.5 cm2

P = 2π r , B = π r2

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

= 6.56π Simplify.

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

Ch 12.4

S = Pℓ + B Surface area of a cone__12

12

= (2π r) ℓ + π r2

A. 18.5π cm2

B. 19.5π cm2

C. 20.2π cm2

D. 22.5π cm2

Find the surface area of the cone in terms of π

Ch 12.4

S = Pℓ + B Surface area of a cone__12

P = 2π r , B = π r2

= (3)(4.5) + (3)2 r = 3 , ℓ = 4.5

= 22.5π Simplify

12

= (2π r) ℓ + π r2

Ch 12.4