Download 12-3_Surface_Areas_of_Pyramids_and_Cones

Five-Minute Check (over Lesson 12–2)
CCSS
Then/Now
New Vocabulary
Key Concept: Lateral Area of a Regular Pyramid
Example 1: Lateral Area of a Regular Pyramid
Key Concept: Surface Area of a Regular Pyramid
Example 2: Surface Area of a Square Pyramid
Example 3: Surface Area of a Regular Pyramid
Key Concept: Lateral and Surface Area of a Cone
Example 4: Real-World Example: Lateral Area of a Cone
Example 5: Surface Area of a Cone
Concept Summary: Lateral and Surface Areas of Solids
Over Lesson 12–2
Find the lateral area of the cylinder.
A. 65.3 ft2
B. 80.2 ft2
C. 122.5 ft2
D. 130.7 ft2
Over Lesson 12–2
Find the lateral area of the prism.
A. 150 in2
B. 126 in2
C. 108 in2
D. 96 in2
Over Lesson 12–2
Find the surface area of the prism.
A. 320 mm2
B. 400 mm2
C. 494.2 mm2
D. 502 mm2
Over Lesson 12–2
Find the surface area of the cylinder.
A. 962.5 cm2
B. 1005.3 cm2
C. 1243.6 cm2
D. 1407.4 cm2
Over Lesson 12–2
The lateral area of a prism is 476 square meters.
The area of one of its bases is 25 square meters.
What is the surface area of the prism?
A. 426 m2
B. 451 m2
C. 501 m2
D. 526 m2
Content Standards
G.MG.1 Use geometric shapes, their
measures, and their properties to describe
objects (e.g., modeling a tree trunk or a
human torso as a cylinder).
Mathematical Practices
1 Make sense of problems and persevere in
solving them.
6 Attend to precision.
You found areas of regular polygons.
• Find lateral areas and surface areas of
pyramids.
• Find lateral areas and surface areas of
cones.
• regular pyramid
• slant height
• right cone
• oblique cone
Lateral Area of a Regular Pyramid
Find the lateral area of
the square pyramid.
Lateral area of a regular
pyramid
P = 2.5 ● 4 or 10, ℓ = 5
Answer: The lateral area is 25 square centimeters.
Find the lateral area of the square pyramid.
A. 54 in2
B. 64 in2
C. 108 in2
D. 132 in2
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.
Surface Area of a Square Pyramid
Step 2
Find the perimeter and area of the base.
P = 4 ● 8 or 32 m
Step 3
Find the surface area of the pyramid.
__
S = 1 Pℓ + B
2
1 (32)
= __
2
≈ 179.4
A = 82 or 64 m2
Surface area of a
regular pyramid
+ 64
P = 32, ℓ =
,
B = 64
Use a calculator.
Answer: The surface area of the pyramid is about
179.4 square meters.
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
Surface Area of a Regular Pyramid
Find the surface area of the regular pyramid.
Round to the nearest tenth.
Step 1
Find the perimeter of the base.
P = 6 ● 10.4 or 62.4 cm
Surface Area of a Regular Pyramid
Step 2
Find the length of the apothem and the
area of the base.
360° or
A central angle of the hexagon is ______
6
60°, so the angle formed in the triangle is
30°.
Surface Area of a Regular Pyramid
tan 30° =
Write a trigonometric ratio to
find the apothem a.
a=
Solve for a.
a ≈ 9.0
Use a calculator.
A=
Area of a regular polygon
≈
Pa
(62.4)(9.0) Replace P with 62.4 and a
width 9.0.
≈ 280.8
Multiply.
So, the area of the base is approximately 280.8 cm2.
Surface Area of a Regular Pyramid
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.
Answer: The surface area of the pyramid is about
748.8 cm2.
Find the surface area of the regular pyramid. Round
to the nearest tenth.
A. 198 in2
B. 228.5 in2
C. 255.5 in2
D. 316.3 in2
Lateral Area of a Cone
ICE CREAM A sugar cone has an altitude of
8 inches and a diameter of
inches. Find the
lateral area of the sugar cone.
If the cone has a diameter of
radius
then the
Use the altitude and
the radius to find the slant height with the Pythagorean
Theorem.
Lateral Area of a Cone
Step 1
Find the slant height ℓ.
ℓ2 = 82 + 1.252
Pythagorean Theorem
ℓ2 ≈ 65.56
Simplify.
ℓ ≈ 8.1
Step 2
Take the square root of
each side.
Find the lateral area L.
L = rℓ
Lateral area of a cone
≈ (1.25)(8.1) r = 1.25 and ℓ ≈ 8.1
≈ 31.8
Answer: The lateral area of the sugar cone is about
31.8 square inches.
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.
A. 39.7 in.2
B. 43.6 in.2
C. 48.2 in.2
D. 53.1 in.2
Surface Area of a Cone
Find the surface area of
the cone. Round to the
nearest tenth.
Estimate: S ≈ 3 ● 1.5 ● 3 + 3 ● 2 or 19.5 cm2
S = rℓ + r2
Surface area of a cone
= (1.4)(3.2) + (1.4)2 r = 1.4 and ℓ = 3.2
≈ 20.2
Answer: The surface area of the cone is about
20.2 square centimeters. This is close to the
estimate, so the answer is reasonable.
Find the surface area of the cone. Round to the
nearest tenth.
A. 58.2 cm2
B. 61.3 cm2
C. 63.6 cm2
D. 70.7 cm2