S U R FA C E A R E A S O F P Y R A M I D S A N D C O N E S

S E C T I O N 1 2 - 3

E S S E N T I A L Q U E S T I O N S

• How do you find lateral areas and surface areas of pyramids?

• How do you find lateral areas and surface areas of cones?

V O C A B U L A R Y

1. Regular Pyramid:

2. Slant Height:

3. Right Cone:

4. Oblique Cone:

V O C A B U L A R Y

1. Regular Pyramid:

2. Slant Height:

3. Right Cone:

4. Oblique Cone:

Has a base that is a regular polygon, and the altitude has an endpoint at the center of the base

V O C A B U L A R Y

1. Regular Pyramid:

2. Slant Height:

3. Right Cone:

4. Oblique Cone:

Has a base that is a regular polygon, and the altitude has an endpoint at the center of the base

The height of the lateral face

V O C A B U L A R Y

1. Regular Pyramid:

2. Slant Height:

3. Right Cone:

4. Oblique Cone:

Has a base that is a regular polygon, and the altitude has an endpoint at the center of the base

The height of the lateral face

A cone whose axis is also the altitude (perpendicular to the base)

V O C A B U L A R Y

1. Regular Pyramid:

2. Slant Height:

3. Right Cone:

4. Oblique Cone:

Has a base that is a regular polygon, and the altitude has an endpoint at the center of the base

The height of the lateral face

A cone whose axis is also the altitude (perpendicular to the base)

A cone whose axis is not an altitude

PA R T S O F A P Y R A M I D

PA R T S O F A P Y R A M I D

Vertex

PA R T S O F A P Y R A M I D

Vertex

PA R T S O F A P Y R A M I D

Vertex

Altitude

PA R T S O F A P Y R A M I D

Vertex

Altitude

PA R T S O F A P Y R A M I D

Vertex

Altitude

Lateral Face

PA R T S O F A P Y R A M I D

Vertex

Altitude

Lateral Face

PA R T S O F A P Y R A M I D

Vertex

Altitude

Lateral Face

Lateral Edge

PA R T S O F A P Y R A M I D

Vertex

Altitude

Lateral Face

Lateral Edge

PA R T S O F A P Y R A M I D

Vertex

Altitude

Lateral Face

Lateral Edge

Slant Height

PA R T S O F A P Y R A M I D

Vertex

Altitude

Lateral Face

Lateral Edge

Slant Height

PA R T S O F A P Y R A M I D

Vertex

Altitude

Lateral Face

Lateral Edge

Slant Height

Base

PA R T S O F A P Y R A M I D

Vertex

Altitude

Lateral Face

Lateral Edge

Slant Height

Base

PA R T S O F A P Y R A M I D

Vertex

Altitude

Lateral Face

Lateral Edge

Slant Height

Base

Base Edge

PA R T S O F A P Y R A M I D

Vertex

Altitude

Lateral Face

Lateral Edge

Slant Height

Base

Base Edge

F O R M U L A S F O R R E G U L A R P Y R A M I D S

F O R M U L A S F O R R E G U L A R P Y R A M I D S

Lateral Area of a Regular Pyramid

F O R M U L A S F O R R E G U L A R P Y R A M I D S

Lateral Area of a Regular Pyramid

L = 12Pl

F O R M U L A S F O R R E G U L A R P Y R A M I D S

Lateral Area of a Regular Pyramid

L = 12Pl

Surface Area of a Regular Pyramid

F O R M U L A S F O R R E G U L A R P Y R A M I D S

Lateral Area of a Regular Pyramid

L = 12Pl

Surface Area of a Regular Pyramid

SA = 12Pl +B

E X A M P L E 1Find the lateral area and the surface area of the

square pyramid. The base has a side length of 2.5 cm and a slant height of 5 cm.

E X A M P L E 1Find the lateral area and the surface area of the

square pyramid. The base has a side length of 2.5 cm and a slant height of 5 cm.

L = 12Pl

E X A M P L E 1Find the lateral area and the surface area of the

square pyramid. The base has a side length of 2.5 cm and a slant height of 5 cm.

L = 12Pl

L = 12(2.5)(4)(5)

E X A M P L E 1Find the lateral area and the surface area of the

square pyramid. The base has a side length of 2.5 cm and a slant height of 5 cm.

L = 12Pl

L = 12(2.5)(4)(5)

L = 25 cm2

E X A M P L E 1Find the lateral area and the surface area of the

square pyramid. The base has a side length of 2.5 cm and a slant height of 5 cm.

E X A M P L E 1Find the lateral area and the surface area of the

square pyramid. The base has a side length of 2.5 cm and a slant height of 5 cm.

SA = 12Pl +B

E X A M P L E 1Find the lateral area and the surface area of the

square pyramid. The base has a side length of 2.5 cm and a slant height of 5 cm.

SA = 12Pl +B

SA = 12(2.5)(4)(5)+ 2.52

E X A M P L E 1Find the lateral area and the surface area of the

square pyramid. The base has a side length of 2.5 cm and a slant height of 5 cm.

SA = 12Pl +B

SA = 12(2.5)(4)(5)+ 2.52

SA = 31.25 cm2

E X A M P L E 2Find the surface area of the square pyramid to the nearest hundredth.

E X A M P L E 2Find the surface area of the square pyramid to the nearest hundredth.

a2 +b2 = c 2

E X A M P L E 2Find the surface area of the square pyramid to the nearest hundredth.

a2 +b2 = c 2

42 + 62 = l2

E X A M P L E 2Find the surface area of the square pyramid to the nearest hundredth.

a2 +b2 = c 2

42 + 62 = l2

16 + 36 = l2

E X A M P L E 2Find the surface area of the square pyramid to the nearest hundredth.

a2 +b2 = c 2

42 + 62 = l2

16 + 36 = l2

52 = l2

E X A M P L E 2Find the surface area of the square pyramid to the nearest hundredth.

a2 +b2 = c 2

42 + 62 = l2

16 + 36 = l2

52 = l2

52 = l2

E X A M P L E 2Find the surface area of the square pyramid to the nearest hundredth.

a2 +b2 = c 2

42 + 62 = l2

16 + 36 = l2

52 = l2

52 = l2

l = 52

E X A M P L E 2Find the surface area of the square pyramid to the nearest hundredth.

E X A M P L E 2Find the surface area of the square pyramid to the nearest hundredth.

SA = 12Pl +B

E X A M P L E 2Find the surface area of the square pyramid to the nearest hundredth.

SA = 12Pl +B

SA = 12(4)(8)( 52)+ 82

E X A M P L E 2Find the surface area of the square pyramid to the nearest hundredth.

SA = 12Pl +B

SA = 12(4)(8)( 52)+ 82

SA = 16 52 + 64

E X A M P L E 2Find the surface area of the square pyramid to the nearest hundredth.

SA = 12Pl +B

SA = 12(4)(8)( 52)+ 82

SA = 16 52 + 64

SA ≈ 179.38 m2

PA R T S O F A C O N E

PA R T S O F A C O N E

Vertex

PA R T S O F A C O N E

Vertex

PA R T S O F A C O N E

Vertex

Altitude

PA R T S O F A C O N E

Vertex

Altitude

PA R T S O F A C O N E

Vertex

Altitude

Slant Height

PA R T S O F A C O N E

Vertex

Altitude

Slant Height

PA R T S O F A C O N E

Vertex

Altitude

Slant Height

Base

PA R T S O F A C O N E

Vertex

Altitude

Slant Height

Base

PA R T S O F A C O N E

Vertex

Altitude

Slant Height

Base

Radius

PA R T S O F A C O N E

Vertex

Altitude

Slant Height

Base

Radius

F O R M U L A S F O R C O N E S

F O R M U L A S F O R C O N E S

Lateral Area of a Cone

F O R M U L A S F O R C O N E S

Lateral Area of a Cone

L = πrl

F O R M U L A S F O R C O N E S

Lateral Area of a Cone

L = πrl

Surface Area of a Cone

F O R M U L A S F O R C O N E S

Lateral Area of a Cone

L = πrl

Surface Area of a Cone

SA = πrl +πr 2

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrl

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrlr = 1

2d

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrlr = 1

2d

r = 12(2.5)

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrlr = 1

2d

r = 12(2.5)

r = 1.25 in.

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrlr = 1

2d

r = 12(2.5)

r = 1.25 in.

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrlr = 1

2d

r = 12(2.5)

r = 1.25 in.

a2 +b2 = c 2

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrlr = 1

2d

r = 12(2.5)

r = 1.25 in.

a2 +b2 = c 2

82 +1.252 = l2

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrlr = 1

2d

r = 12(2.5)

r = 1.25 in.

a2 +b2 = c 2

82 +1.252 = l2

64 +1.5625 = l2

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrlr = 1

2d

r = 12(2.5)

r = 1.25 in.

a2 +b2 = c 2

82 +1.252 = l2

64 +1.5625 = l2

65.5625 = l2

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrlr = 1

2d

r = 12(2.5)

r = 1.25 in.

a2 +b2 = c 2

82 +1.252 = l2

64 +1.5625 = l2

65.5625 = l2

l = 65.5625

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrl

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrl

r = 1.25 in.

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrl

r = 1.25 in. l = 65.5625

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrl

r = 1.25 in. l = 65.5625

L = π1.25 65.5625

E X A M P L E 3An ice cream cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the cone to the nearest hundredth.

L = πrl

r = 1.25 in. l = 65.5625

L = π1.25 65.5625

L ≈ 31.80 in2

E X A M P L E 4Find the surface area of the cone. Round to the nearest hundredth.

E X A M P L E 4Find the surface area of the cone. Round to the nearest hundredth.

SA = πrl +πr 2

E X A M P L E 4Find the surface area of the cone. Round to the nearest hundredth.

SA = πrl +πr 2

SA = π (1.4)(3.2)+π (1.4)2

E X A M P L E 4Find the surface area of the cone. Round to the nearest hundredth.

SA = πrl +πr 2

SA = π (1.4)(3.2)+π (1.4)2

SA = 4.48π +1.96π

E X A M P L E 4Find the surface area of the cone. Round to the nearest hundredth.

SA = πrl +πr 2

SA = π (1.4)(3.2)+π (1.4)2

SA = 4.48π +1.96π

SA = 6.44π

E X A M P L E 4Find the surface area of the cone. Round to the nearest hundredth.

SA = πrl +πr 2

SA = π (1.4)(3.2)+π (1.4)2

SA = 4.48π +1.96π

SA = 6.44π

SA ≈ 20.23 in2

S O L I D M O D E LL AT E R A L

A R E AS U R FA C E A R E A

P R I S M

C Y L I N D E R

P Y R A M I D

C O N E

L =Ph

L = 2πrh

L = 12Pl

L = πrl

SA =Ph+ 2B

SA = 2πrh+ 2πr 2

SA = 12Pl +B

SA = πrl +πr 2

– R O N A L D R E A G A N

“We can’t help everyone, but everyone can help someone.”

P R O B L E M S E T

p. 843 #1-19 odd, skip #3