1. Five-Minute Check (over Lesson 11–5) Mathematical Practices Then/Now
New Vocabulary
Example 1: Identify Similar and Congruent Solids
Theorem 11.1 Similar solid
Example 2: Use Similar Solids to Solve Problems
Example 3: Find the Volume of Similar Solids by Using Scale Factor
Example 4: Real-World Example: Use Similar Solids to Find Unknown Values
Lesson Menu

2. Name a line not containing point P on the sphere.B. C. D.
5-Minute Check 1

3. Name a triangle in the sphere.
A. ΔVQS
B. ΔRTU
C. ΔPQR
D. ΔPXW
5-Minute Check 2

4. Name a segment containing point Q in the sphere.B. C. D. TU
5-Minute Check 3

5. A. Yes, through 2 points there is exactly one line.
Tell whether the following statement from Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain why. If B is between A and C, then AB + BC = AC.
A. Yes, through 2 points there is exactly one line.
B. Yes, the points on any great circle or arc of a great circle can be put into one to one correspondence with real numbers.
C. No, AC may not be the distance from A to C through B. It may be the distance the other direction around the sphere.
5-Minute Check 4

6. Which of the following is represented by a line in spherical geometry?
A. triangle
B. great circle
C. radius
D. diameter
5-Minute Check 5

7. Mathematical Practices
8 Look for and express regularity in repeated reasoning.
Content Standards
G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of cylinder, pyramid, and cone.
G.GMD.2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. MP

8. You compared surface areas and volumes of spheres.
Identify scale factor by using dilation.
Find surface areas and volumes of similar solids by using scale factors.
Then/Now

9. dilation
scale factor
Vocabulary

10. Identify Similar and Congruent Solids
Determine whether the pair of rectangular prisms is similar, congruent, or neither. If the prisms are similar, state the scale factor.
Substitution.
Simplify.
Example 1

11. Identify Similar and Congruent Solids
Substitution.
Simplify.
Compare the ratios between the corresponding parts of the rectangular prisms.
Since the ratios are not the same, the prisms are neither similar nor congruent.
Answer: Because the ratios of corresponding measures are not equal, the prisms are neither congruent nor similar.
Example 1

12. A. Determine whether the pair of solids is similar, congruent, or neither.
A. similar
B. congruent
C. neither
Example 1

13. B. Determine whether the pair of solids is similar, congruent, or neither.
A. similar
B. congruent
C. neither
Example 1

14. Theorem

15. Use Similar Solids to Solve Problems
The surface area of a small pyramid is 40 square centimeters. If the scale factor between the small pyramid and a larger pyramid is , what is the surface area of the larger pyramid?
Theorem Similar Solid
Theorem Similar Solid
Example 2

16. Use Similar Solids to Solve Problems
Set up a proportion of the scale factor and the area of the small pyramid to the large pyramid and solve for x.
x = 360
Cross multiply.
Answer: 360 cm2
Example 2

17. Two similar cones have radii of 5 inches and 15 inches
Two similar cones have radii of 5 inches and 15 inches. What is the ratio of the volume of the smaller cone to the volume of the larger cone?
A. 1:3
B. 1:9
C. 1:27
D. 1:81
Example 2

18. Find the Volume of Similar Solids by Using Scale Factor
Circular cone A and circular cone B are similar. The cones have radii of 10 millimeters and 15 millimeters, respectively. The volume of cone A is approximately cubic millimeters. Find the volume of cone B.
Theorem Similar Solid
Theorem Similar Solid
Example 3

19. Find the Volume of Similar Solids by Using Scale Factor
Set up a proportion of the scale factor and the area of the small pyramid to the large pyramid and solve for x.
8x = 28,274.4
Cross multiply.
x =
Divide both sides by 8.
Answer: ≈ mm3
Example 3

20. Understand You know the volume of the softballs.
Use Similar Solids to Find Unknown Values
SOFTBALLS The softballs shown are similar spheres. If the radius of the larger softball is 1.9 inches, find the radius of the smaller softball.
Understand You know the volume of the softballs.
Plan Use Theorem 12.1 to write a ratio comparing the volumes. Then find the scale factor and use it to find r.
Example 4

21. Write a ratio comparing volumes.
Use Similar Solids to Find Unknown Values
Solve
Write a ratio comparing volumes. =
Simplify. ≈
Write as
Example 4

22. Find the cross products.
Use Similar Solids to Find Unknown Values
Ratio of radii
Scale factor
Find the cross products.
r ≈ 1.45
Solve for r.
Answer: So, the radius of the smaller softball is about 1.45 inches.
Example 4

23. Use Similar Solids to Find Unknown Values
Check 
Example 4

24. CONTAINERS The containers below are similar cylinders
CONTAINERS The containers below are similar cylinders. Find the height h of the smaller container.
A. 2 in.
B. 3 in.
C. 4 in.
D. 5 in.
Example 4