CONFIDENTIAL 1

GeometryGeometry

Surface Area of Surface Area of Pyramids and Cones Pyramids and Cones

CONFIDENTIAL 2

Warm UpWarm Up

15 in.

10 in.

17 in.

10 cm

15 cm

8 cm

Find the surface area of each right prism or right cylinder. Round your answer to the nearest tenth.

2 cm

3 cm

1. 2. 3.

1) 490 in2 2) 62.83 cm2 3) 700 cm2

CONFIDENTIAL 3

Surface Area of Pyramids and Cones Surface Area of Pyramids and Cones

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

regular pyramid nonregular pyramid

Slant height

Vertices

Lateral faces

Altitude

Bases

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CONFIDENTIAL 4

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.

s

s s

sl

P = 4s

l

ssss

CONFIDENTIAL 5

Lateral and surface Area of a Lateral and surface Area of a regular pyramidregular pyramid

The lateral area of a regular pyramid with perimeter P and slant height l is L = 1/2Pl.

The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = ½ pl + B.

l

CONFIDENTIAL 6

Finding Lateral and surface Area of Pyramids

Find the lateral area and surface area of each pyramid.a.) a regular square pyramid with base edge length 5 in. and slant height 9in.

L = ½ Pl Lateral area of a regular pyramid = ½(20)(9) = 90 in P = 4(5) = 20 in.S = ½ Pl + B Surface area of a regular pyramid = 90 + 25 = 115 in B = 5 = 25 in

2

2 2 2

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CONFIDENTIAL 7

b.)Find the lateral area and surface area of each regular pyramid.

7 m

4 m

find the base perimeter and apothem.The base perimeter is 6(4) = 24 m.The apothem is 2 3 m, so the base area is ½ aP = ½ (2 3) (24) = 24 3 m 2

Step 1

Step 2 Find the lateral area.L = ½ Pl Lateral area of a regular pyramid = ½ (24)(7) = 84 m Substitute 24 for P and 7 for l.

Step 3 Find the surface area.S = 1/2 Pl + B Surface area of a regular pyramid = 84 + 24 3 = 125.6 cm Substitute 24 3 for B. 2

CONFIDENTIAL 8

Now you try!

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

1) lateral area = 90 ft2 surface area = 105 ft2

CONFIDENTIAL 9

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

Slant height

right cone oblique cone

Vertices

Lateral surfaces

axis axis

Base

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CONFIDENTIAL 10

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

CONFIDENTIAL 11

Lateral and Surface Area of a right cone

Lateral and Surface Area of a right cone

r

l

rl

The lateral area of a right cone with radius r and slant height l is L = rl.

The surface area of a right cone with lateral area L and base area B is S = L + B, or S = rl + r 2

CONFIDENTIAL 12

Finding Lateral Area and Surface Area of right cones

Find the lateral area and surface area of each cone. Give your answers in terms of .

A) A right cone with radius 2 m and slant height 3 m.

L = rl Lateral area of a cone = (2) (3) = 6 m2 Substitute 2 for r and 3 for l.S = rl + r2 Surface area of a cone = 6 + (2)2 = 10 m2 Substitute 2 for r and 3 for l.

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CONFIDENTIAL 13

B)

5 ft

l12 ft

Step 1 Use the Pythagorean Theorem to find l.

l = 5 + 12 = 13 ft2 2

Step 2 Find the lateral area and surface area.

L = rl Lateral area of a right cone = (5) (13) = 65 ft2 Substitute 5 for r and 13 for l.S = rl + r2 Surface area of a right cone = 65 + (5)2 = 90 ft2 Substitute 5 for r and 13 for l.

CONFIDENTIAL 14

Now you try!

16 ft

6 ft

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

2) lateral area = 251.43 ft2 surface area = 452.57 ft2

CONFIDENTIAL 15

Exploring Effects of Changing Dimensions

3 cm

5 cmThe radius and slant height of the right cone tripled. Describe the effect on the surface area.

Notice that 216 = 9(24). I f the radius and slant heightare tripled, the surface area is multiplied by 32, or 9.

S = rl + r2 = (9)(15) + (9)2 = 216 cm2

S = rl + r2 = (3)(5) + (3)2 = 24 cm2

Original dimensions: Radius and slant height tripled:

CONFIDENTIAL 16

Now you try!

3) The base edge length and slant height of the regular square pyramid are both multiplied by 2/3 . Describe the effect on the surface area.

12 ft

15 ft

3) The surface area is multiplied by 4/9.

CONFIDENTIAL 17

Finding Surface Area of Composite Three-Dimensional Figures

Finding Surface Area of Composite Three-Dimensional Figures

Find the surface area of the composite figure.

28 cm

90

cm

45

cm

The height of the cone is 90 - 45 = 45 cm.By the Pythagorean Theorem,l = 282 + 452 = 53 cm. the lateral area of thecoan isL = rl = (28)(53) = 1484 cm2.

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CONFIDENTIAL 18

28 cm

90

cm

45

cmThe lateral area of the cylinder is L = 2rh = 2(28)(45) =2520 cm2.

The base area is B = r2 = (28)2 = 784 cm2

S = (cone lateral area) + (cylinder lateral area) + (base area) = 2520 + 784 + 1484 = 4788 cm2

CONFIDENTIAL 19

Now you try!

4) Find the surface area of the composite figure.

2 yd

2 yd

2 yd4) 20.94 yd2

CONFIDENTIAL 20

Electronics Application

Tim is replacing the paper cone of an antique speaker. He measured the existing cone and created the pattern for the lateral

surface from a large circle. What is the diameter of the cone?

10 in.

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

The area of the pattern is the lateral area of the cone. The area of the pattern is also ¾ of the area of the large circle, so rl = ¾ r . r(10) = ¾ (10) = 7.5 in.The diameter of the cone is 2(7.5) = 15 in.

2

2Substitute 10 for l, the slant height of the cone and the radius of the large circle.Solve it

CONFIDENTIAL 21

Now you try!

5) If the radius of the large circle were 12 in., what would be the radius of the cone?

5) 9 in.

CONFIDENTIAL 22

Now some problems for you to practice !

CONFIDENTIAL 23

Assessment

1) Describe the endpoints of an axis of a cone.

1) The endpoints at the vertex and the center of the base of the cone joins to form the axis of the cone.

CONFIDENTIAL 24

2) Find the lateral area and surface area of each regular pyramid.

a. b.

8 cm

12 cm15 ft

16 ft

16 ft

2b) lateral area = 480 ft2 surface area = 1472 ft2

2a) lateral area = 288 cm2 surface area = 576 cm2

CONFIDENTIAL 25

3) Find the lateral area and surface of each right cone. Given your answer in terms of .

24 in.

25 in.

14 m.

22 m.

a. b.

3b) lateral area = 308∏ m2 surface area = 504∏ m2

3a) lateral area = 175∏ in2 surface area = 224∏ in2

CONFIDENTIAL 26

4) Describe the effect of each change on the surface area of the given figure.

a. The dimensions are cut in half.

b. The dimension are tripled.

9 cm.

15 cm.10 in.

6 in.

6 in.

4b) lateral area = 308∏ m2 surface area = 504∏ m2

4a) lateral area = 175∏ in2 surface area = 224∏ in2

CONFIDENTIAL 27

5) Find the surface area of each composite figure.

15 m.

12 m.

32 m.

26 m.15 ft.

8 ft.

18 ft.

a. b.

5b) 3498.95 m2 5a) 1184 ft2

CONFIDENTIAL 28

6) Anna is making a birthday hat from a pattern that is ¾ of a circle of colored paper. If Anna’s head is 7 inches in diameter, will the hat fit her? Explain.

6 in.

6 ) Anna’s head will fit in the hat as the diameter of the cone will be 9 inches.

CONFIDENTIAL 29

Let’s review

CONFIDENTIAL 30

Surface Area of Pyramids and Cones Surface Area of Pyramids and Cones

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

regular pyramid nonregular pyramid

Slant height

Vertices

Lateral faces

Altitude

Bases

Next page:

CONFIDENTIAL 31

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.

s

s s

sl

P = 4s

l

ssss

CONFIDENTIAL 32

Lateral and surface Area of a Lateral and surface Area of a regular pyramidregular pyramid

The lateral area of a regular pyramid with perimeter P and slant height l is L = 1/2Pl.

The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = ½ pl + B.

l

CONFIDENTIAL 33

Finding Lateral and surface Area of Pyramids

Find the lateral area and surface area of each pyramid.a.) a regular square pyramid with base edge length 5 in. and slant height 9in.

L = ½ Pl Lateral area of a regular pyramid = ½(20)(9) = 90 in P = 4(5) = 20 in.S = ½ Pl + B Surface area of a regular pyramid = 90 + 25 = 115 in B = 5 = 25 in

2

2 2 2

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CONFIDENTIAL 34

b.)Find the lateral area and surface area of each regular pyramid.

7 m

4 m

find the base perimeter and apothem.The base perimeter is 6(4) = 24 m.The apothem is 2 3 m, so the base area is ½ aP = ½ (2 3) (24) = 24 3 m 2

Step 1

Step 2 Find the lateral area.L = ½ Pl Lateral area of a regular pyramid = ½ (24)(7) = 84 m Substitute 24 for P and 7 for l.

Step 3 Find the surface area.S = 1/2 Pl + B Surface area of a regular pyramid = 84 + 24 3 = 125.6 cm Substitute 24 3 for B. 2

CONFIDENTIAL 35

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

Slant height

right cone oblique cone

Vertices

Lateral surfaces

axis axis

Base

Next page:

CONFIDENTIAL 36

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

CONFIDENTIAL 37

Lateral and Surface Area of a right cone

Lateral and Surface Area of a right cone

r

l

rl

The lateral area of a right cone with radius r and slant height l is L = rl.

The surface area of a right cone with lateral area L and base area B is S = L + B, or S = rl + r 2

CONFIDENTIAL 38

Finding Lateral Area and Surface Area of right cones

Find the lateral area and surface area of each cone. Give your answers in terms of .

A) A right cone with radius 2 m and slant height 3 m.

L = rl Lateral area of a cone = (2) (3) = 6 m2 Substitute 2 for r and 3 for l.S = rl + r2 Surface area of a cone = 6 + (2)2 = 10 m2 Substitute 2 for r and 3 for l.

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CONFIDENTIAL 39

B)

5 ft

l12 ft

Step 1 Use the Pythagorean Theorem to find l.

l = 5 + 12 = 13 ft2 2

Step 2 Find the lateral area and surface area.

L = rl Lateral area of a right cone = (5) (13) = 65 ft2 Substitute 5 for r and 13 for l.S = rl + r2 Surface area of a right cone = 65 + (5)2 = 90 ft2 Substitute 5 for r and 13 for l.

CONFIDENTIAL 40

Exploring Effects of Changing Dimensions

3 cm

5 cmThe radius and slant height of the right cone tripled. Describe the effect on the surface area.

Notice that 216 = 9(24). I f the radius and slant heightare tripled, the surface area is multiplied by 32, or 9.

S = rl + r2 = (9)(15) + (9)2 = 216 cm2

S = rl + r2 = (3)(5) + (3)2 = 24 cm2

Original dimensions: Radius and slant height tripled:

CONFIDENTIAL 41

Finding Surface Area of Composite Three-Dimensional Figures

Finding Surface Area of Composite Three-Dimensional Figures

Find the surface area of the composite figure.

The height of the cone is 90 - 45 = 45 cm.By the Pythagorean Theorem,l = 282 + 452 = 53 cm. the lateral area of thecoan isL = rl = (28)(53) = 1484 cm2.

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28 cm

90

cm

45

cm

CONFIDENTIAL 42

28 cm

90

cm

45

cmThe lateral area of the cylinder is L = 2rh = 2(28)(45) =2520 cm2.

The base area is B = r2 = (28)2 = 784 cm2

S = (cone lateral area) + (cylinder lateral area) + (base area) = 2520 + 784 + 1484 = 4788 cm2

CONFIDENTIAL 43

Electronics Application

Tim is replacing the paper cone of an antique speaker. He measured the existing cone and created the pattern for the lateral surface from a large circle. What is the diameter of the cone?

10 in.

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

The area of the pattern is the lateral area of the cone. The area of the pattern is also ¾ of the area of the large circle, so rl = ¾ r . r(10) = ¾ (10) = 7.5 in.The diameter of the cone is 2(7.5) = 15 in.

2

2Substitute 10 for l, the slant height of the cone and the radius of the large circle.Solve it

CONFIDENTIAL 44

You did a You did a greatgreat job job today!today!