1. Find lateral areas and surface areas of pyramids.
Find lateral areas and surface areas of cones.
Splash Screen

2. Mathematical Practices
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.
CCSS

3. regular pyramid
slant height
right cone
oblique cone
Vocabulary

4. Concept

5. Concept

6. Find the lateral area of the square pyramid.
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.
Example 1

7. Find the lateral area of the square pyramid.
A. 54 in2
B. 64 in2
C. 108 in2
D. 132 in2
Example 1

8. Concept

9. Find the surface area of the square pyramid to the nearest tenth.
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 = a = 6, b = 4, and c = ℓ
ℓ = Simplify.
Example 2

10. Step 2 Find the perimeter and area of the base.
Surface Area of a Square Pyramid
Step 2 Find the perimeter and area of the base.
P = 4 ● 8 or 32 m A = 82 or 64 m2
Step 3 Find the surface area of the pyramid.
S = Pℓ + B Surface area of a regular pyramid __ 1 2 __ 1 2
= (32) P = 32, ℓ = , B = 64
≈ Use a calculator.
Answer: The surface area of the pyramid is about square meters.
Example 2

11. Find the surface area of the square pyramid to the nearest tenth.
A. 96 in2
B in2
C in2
D. 156 in2
Example 2

12. Step 1 Find the perimeter of the base. P = 6 ● 10.4 or 62.4 cm
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
Example 3

13. Step 2 Find the length of the apothem and the area of the base.
Surface Area of a Regular Pyramid
Step 2 Find the length of the apothem and the area of the base.
A central angle of the hexagon is or 60°, so the angle formed in the triangle is 30°.
______
360° 6
Example 3

14. tan 30° = Write a trigonometric ratio to find the apothem a.
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 = Pa Area of a regular polygon
≈ (62.4)(9.0) Replace P with 62.4 and a width 9.0.
≈ Multiply.
So, the area of the base is approximately cm2.
Example 3

15. Step 3 Find the surface area of the pyramid.
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) P = 62.4, ℓ = 15, and B ≈ 280.8
≈ Simplify.
Answer: The surface area of the pyramid is about cm2.
Example 3

16. Find the surface area of the regular pyramid
Find the surface area of the regular pyramid. Round to the nearest tenth.
A. 198 in2
B in2
C in2
D in2
Example 3

17. Concept

18. Concept

19. 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 then the radius Use the altitude and the radius to find the slant height with the Pythagorean Theorem.
Example 4

20. Step 1 Find the slant height ℓ.
Lateral Area of a Cone
Step 1 Find the slant height ℓ.
ℓ2 = Pythagorean Theorem
ℓ2 ≈ Simplify.
ℓ ≈ 8.1 Take the square root of each side.
Step 2 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.
Example 4

21. 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 in.2
B in.2
C in.2
D in.2
Example 4

22. Find the surface area of the cone. Round to the nearest tenth.
Surface Area of a Cone
Find the surface area of the cone. Round to the nearest tenth.
Estimate: S ≈ 3 ● 1.5 ● ● 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 square centimeters. This is close to the estimate, so the answer is reasonable.
Example 5

23. Find the surface area of the cone. Round to the nearest tenth.
A cm2
B cm2
C cm2
D cm2
Example 5

24. Concept

25. Homework
Page 859 (7, 8, 11, 14-17)
Concept