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10-5

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Surface Area of

Pyramids and Cones

10-5

Warm Up

Lesson Presentation

Lesson Quiz

Holt Geometry

Warm Up

Find the missing side length of each right triangle with legs a and b and hypotenuse c.

1.a = 7, b = 24

2.c = 15, a = 9

3.b = 40, c = 41

4.a = 5, b = 5

5.a = 4, c = 8

c = 25

b = 12

a = 9

Objectives

Learn and apply the formula for the surface area of a pyramid.

Learn and apply the formula for the surface area of a cone.

Vocabulary

vertex of a pyramid

regular pyramid

slant height of a regular pyramid

altitude of a pyramid

vertex of a cone

axis of a cone

right cone

oblique cone

slant height of a right cone

altitude of a cone

The vertex of a pyramidis the point opposite the base of the pyramid. The base of a regular pyramidis a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramidis the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramidis the perpendicular segment from the vertex to the plane of the base.

The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid.

Example 1A: Finding Lateral Area and Surface Area of Pyramids

Find the lateral area and surface area of a regular square pyramid with base edge length 14 cm and slant height 25 cm. Round to the nearest tenth, if necessary.

Lateral area of a regular pyramid

P = 4(14) = 56 cm

Surface area of a regular pyramid

B = 142 = 196 cm2

The apothem is , so the base area is

Example 1B: Finding Lateral Area and Surface Area of Pyramids

Find the lateral area and surface area of the regular pyramid.

Step 1 Find the base perimeter and apothem.

The base perimeter is 6(10) = 60 in.

Example 1B Continued

Find the lateral area and surface area of the regular pyramid.

Step 2 Find the lateral area.

Lateral area of a regular pyramid

Substitute 60 for P and 16 for ℓ.

Example 1B Continued

Find the lateral area and surface area of the regular pyramid.

Step 3 Find the surface area.

Surface area of a regular pyramid

Substitute for B.

The apothem is so the base area is

Check It Out! Example 1

Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft.

Step 1 Find the base perimeter and apothem. The base perimeter is 3(6) = 18 ft.

Check It Out! Example 1 Continued

Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft.

Step 2 Find the lateral area.

Lateral area of a regular pyramid

Substitute 18 for P and 10 for ℓ.

Substitute for B.

Check It Out! Example 1 Continued

Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft.

Step 3 Find the surface area.

Surface area of a regular pyramid

The vertex of a coneis the point opposite the base. The axis of a coneis the segment with endpoints at the vertex and the center of the base. The axis of a right coneis perpendicular to the base. The axis of an oblique coneis not perpendicular to the base.

The slant height of a right coneis the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a coneis a perpendicular

segment from the vertex of the cone to the plane of the base.

Example 2A: Finding Lateral Area and Surface Area of Right Cones

Find the lateral area and surface area of a right cone with radius 9 cm and slant height 5 cm.

L = rℓ

Lateral area of a cone

= (9)(5) = 45 cm2

Substitute 9 for r and 5 for ℓ.

S = rℓ + r2

Surface area of a cone

= 45 + (9)2 = 126 cm2

Substitute 5 for ℓ and 9 for r.

Example 2B: Finding Lateral Area and Surface Area of Right Cones

Find the lateral area and surface area of the cone.

Use the Pythagorean Theorem to find ℓ.

L = rℓ

Lateral area of a right cone

= (8)(17)

= 136 in2

Substitute 8 for r and 17 for ℓ.

S = rℓ+ r2

Surface area of a cone

= 136 + (8)2

= 200 in2

Substitute 8 for r and 17 for ℓ.

ℓ

Check It Out! Example 2

Find the lateral area and surface area of the right cone.

Use the Pythagorean Theorem to find ℓ.

L = rℓ

Lateral area of a right cone

= (8)(10)

= 80 cm2

Substitute 8 for r and 10 for ℓ.

S = rℓ + r2

Surface area of a cone

= 80 + (8)2

= 144 cm2

Substitute 8 for r and 10 for ℓ.

Example 3: Exploring Effects of Changing Dimensions

The base edge length and slant height of the regular hexagonal pyramid are both divided by 5. Describe the effect on the surface area.

3 in.

12

12

2 in.

S = Pℓ + B

S = Pℓ + B

Example 3 Continued

base edge length and slant height divided by 5:

original dimensions:

3 in.

2 in.

Notice that . If the base edge length and slant height are divided by 5, the surface area is divided by 52, or 25.

Example 3 Continued

base edge length and slant height divided by 5:

original dimensions:

Check It Out! Example 3

The base edge length and slant height of the regular square pyramid are both multiplied by . Describe the effect on the surface area.

8 ft

12

12

10 ft

S = Pℓ + B

S = Pℓ + B

By multiplying the dimensions by two-thirds, the surface area was multiplied by .

Check It Out! Example 3 Continued

8 ft

original dimensions:

multiplied by two-thirds:

= 585 cm2

= 260 cm2

Example 4: Finding Surface Area of Composite Three-Dimensional Figures

Find the surface area of the composite figure.

Left-hand cone:

The lateral area of the cone is

L = rl = (6)(12) = 72 in2.

Right-hand cone:

Using the Pythagorean Theorem, l = 10 in.

The lateral area of the cone is

L = rl = (6)(10) = 60 in2.

Example 4 Continued

Find the surface area of the composite figure.

Composite figure:

S = (left cone lateral area) + (right cone lateral area)

= 60 in2 + 72 in2 = 132 in2

Check It Out! Example 4

Find the surface area of the composite figure.

Surface Area of Cube without the top side:

S = 4wh + B

S = 4(2)(2) + (2)(2) = 20 yd2

Check It Out! Example 4 Continued

Surface Area of Pyramid without base:

Surface Area of Composite:

Surface of Composite = SA of Cube + SA of Pyramid

The area of the pattern is the lateral area of the cone. The area of the pattern is also of the area of the large circle, so

Example 5: Manufacturing Application

If the pattern shown is used to make a paper cup, what is the diameter of the cup?

The radius of the large circle used to create the pattern is the slant height of the cone.

Example 5 Continued

If the pattern shown is used to make a paper cup, what is the diameter of the cup?

Substitute 4 for ℓ, the slant height of the cone and the radius of the large circle.

r = 2 in.

Solve for r.

The diameter of the cone is 2(2) = 4 in.

The area of the pattern is the lateral area of the

cone. The area of the pattern is also of the area of

the large circle, so

Check It Out! Example 5

What if…? If the radius of the large circle were 12 in., what would be the radius of the cone?

The radius of the large circle used to create the pattern is the slant height of the cone.

Check It Out! Example 5 Continued

What if…? If the radius of the large circle were 12 in., what would be the radius of the cone?

Substitute 12 for ℓ, the slant height of the cone and the radius of the large circle.

r = 9 in.

Solve for r.

The radius of the cone is 9 in.

Lesson Quiz: Part I

Find the lateral area and surface area of each figure. Round to the nearest tenth, if necessary.

1. a regular square pyramid with base edge length 9 ft and slant height 12 ft

2. a regular triangular pyramid with base edge length 12 cm and slant height 10 cm

L = 216 ft2; S = 297 ft2

L = 180 cm2; S ≈ 242.4 cm2

The surface area is multiplied by .

Lesson Quiz: Part II

4. A right cone has radius 3 and slant height 5. The radius and slant height are both multiplied by . Describe the effect on the surface area.

5. Find the surface area of the composite figure. Give your answer in terms of .

S = 24ft2