geometry section 12-3
TRANSCRIPT
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:
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
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.
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.
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
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.
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