1. Warm Up Section 4.10
1. Find the area of a circle with diameter 12 in.
Find the radius of a circle with circumference 46 ft.
Find the surface area of a sphere with diameter 10 m.
Find the volume of a sphere with radius 8 cm.
Find the area of a sector of a circle whose central angle
measures 80o and whose radius measures 5 m.

2. 1. diameter 12 in. 2. C = 2r A= (6)2 = 36 46 = 2r
Answers to Warm Up Section 4.10
1. diameter 12 in C = 2r
A= (6)2 = 36  = 2r
A  in ft = r
diameter 10 m
SA = 4r2
SA = 4(5)2
SA = 100
SA  m2

3. Application of Circles Section 4.10
Standard: MM2G3 a-d MM2G4ab
Essential Question: How do I use concepts of circles and spheres to solve problems?

4. the measure of one section = = 30o
In the clock face shown at the right, the positions of the
numbers determine congruent arcs along the circle.
What is the measure of the arc
between any two consecutive numbers?
Since there are 12 equal
sections on the clock,
the measure of one section =
= 30o
An arc is traced out by the end of the second hand as it
moves from the 12 to the 4. Is this a minor arc or a major
arc?
Minor arc (less than 180o)

5. Reasoning: 180o = 6(30o), so count six
In the clock face shown at the right, the positions of the
numbers determine congruent arcs along the circle.
Starting at 2, what number does
the end of the second hand reach as
it completes a semicircle?
Answer: 8
Reasoning: 180o = 6(30o), so count six
sections around the clock from 2.

6. Reasoning: Count from 8 to 3. There are 7 sections, so 7(30o) = 210o.
In the clock face shown at the right, the positions of the
numbers determine congruent arcs along the circle.
d. When the second hand moves from
the 8 to the 3, what is the measure
of the arc?
Answer: 210o
Reasoning: Count from 8 to 3. There are 7
sections, so 7(30o) = 210o.

7. So, the clock moved from 15o to 24o which is a distance of 9o.
In the clock face shown at the right, the positions of the
numbers determine congruent arcs along the circle.
e. When the second hand moves from
halfway between the 1 and the 2
to 4/5 of the way from the 1 to the 2,
what is the measure of the arc?
Answer: 9o
Each section is 30o.
So, the clock moved from 15o to 24o
which is a distance of 9o.

8. From 3 to 7 there are 4 sections. 4(30o) = 120o.
In the clock face shown at the right, the positions of the
numbers determine congruent arcs along the circle.
The second hand moves from the 3
to the 7. What is the measure of
the corresponding major arc?
Answer: 240o
From 3 to 7 there are 4 sections. 4(30o) = 120o.
So, the major arc measures 360o – 120o = 240o.

9. 2. A play is being presented on a circular stage
2. A play is being presented on a circular stage. The two main characters are positions A and B at the back of the stage. Use the diagram to answer the following questions.
What angle of view between the main characters does an
actor at position C at the center stage have?
b. What angle of view between the main characters does the orchestra conductor at point D have?
c. What angle of view between the main characters does an audience member at point E have? A B D E
80°
38° C
80o
½ (80)o = 40o
½ (80 – 38)o = 21o

10. 50 cm AC = AD = 80 cm 50(x) = (80)(80) 50x = 6400 EA = 128 cm x = 128
3. An arch over a doorway is an arc that is 160 centimeters wide and 50 centimeters high. You are curious about the side of the entire circle containing the arch. By drawing and labeling a circle passing though the arc as shown in the diagram, you can use the following steps to find the radius of the circle.
a. Find AB.
b. Find AC and AD.
c. Use AB, AC, and AD to find EA.
160 cm B
50 cm 50 80 A 80
50 cm C D O
AC = AD = 80 cm x E
50(x) = (80)(80)
50x = 6400
x = 128
EA = 128 cm

11. 3. An arch over a doorway is an arc that is 160 centimeters wide and 50 centimeters high. You are curious about the side of the entire circle containing the arch. By drawing and labeling a circle passing though the arc as shown in the diagram, you can use the following steps to find the radius of the circle.
d. Find EB
e. Find the length of the radius, EO.
160 cm B
128 cm + 50cm = 178 cm 50 80 A 80
50 cm C D O
128
Radius = ½ (178 cm) = 89 cm E

12. 4. You want to estimate the diameter of a circular water tank.
You stand at a location 10.5 feet from the edge of the
circular tank. From this position, your distance to a point of
tangency on the tank is 23 feet.
a. Draw a diagram of the situation. Label your position as C
and the radius of the tank as r. r
10.5 r C 23

13. 4. You want to estimate the diameter of a circular water tank.
You stand at a location 10.5 feet from the edge of the
circular tank. From this position, your distance to a point of
tangency on the tank is 23 feet.
  b. Find the length of the radius to the nearest tenth of a foot. 
r2 + (23)2 = (r )2
r = r2 + 21r r
10.5 r
529 = 21r C 23
= 21r
19.9 = r
The radius of the tank is 19.9 feet.

14. 5. A satellite is about 100 miles above Earth’s surface
5. A satellite is about 100 miles above Earth’s surface. The satellite is taking photographs of Earth. Earth’s diameter is about 8000 miles.
a. Draw a diagram of the situation, representing Earth as a circle.
b. In the diagram, draw a segment to show one of the farthest possible points on Earth that can be photographed from the satellite. What type of geometric figure is the segment?
   
100 x
4000
8000
tangent

15. c. Find the length of the segment drawn in part b.x
4000
100
x2 + (4,000)2 = (4,100)2
x2 + 16,000,000 = 16,810,000
4000
x2 = 810,000
x = 900
One of the farthest points on earth that can photographed
by the satellite is 900 miles from the satellite.

16. 6. The chain of a bicycle travels along the front and rear
sprockets, as shown. The circumference of each sprocket is
given.
a. About how long is the chain?
10 in
160°
185°
rear sprocket
C = 12 in
front sprocket
C = 20 in
35.6 inches long
Rear:
Front:
Total: 10
5.3
35.6

17. 6. The chain of a bicycle travels along the front and rear
sprockets, as shown. The circumference of each sprocket is
given.
On a chain, the teeth are spaced in ½ inch intervals. About
how many teeth are there on this chains.
10 in
160°
185°
rear sprocket
C = 12 in
front sprocket
C = 20 in
There are 2 teeth for every inch. So,
There are about 71 teeth on the chain.

18. You need about 56.5 feet of fencing.
7. You have planted a circular garden with its center at one of the corners of your garage, as shown. You want to fence in your garden. About how much fencing do you need?
12 ft
You need about 56.5 feet of fencing.

19. A = r2 A = (21)2 A = 441 A 1385.4 ft2 d = 2r 42 = 2r 21 = r
8. A circular water fountain has a diameter of 42 feet.
Find the area of the fountain.
A = r2
A = (21)2
A = 441
A ft2
42 ft
d = 2r
42 = 2r
21 = r
The area of the fountain is
ft2.

20. a. What is the area of the lawn that is covered by the sprinkler?
9. The diagram below shows the area of a lawn covered by a water sprinkler.
a. What is the area of the lawn that is covered by the sprinkler?
135°
16 ft
The sprinkler covers about ft2 of lawn.

21. b. The water pressure is lowered so that the radius is 10 feet.
9. The diagram below shows the area of a lawn covered by a water sprinkler.
b. The water pressure is lowered so that the radius is 10 feet.
What is the area of lawn that will be covered?
135°
16 ft
The sprinkler now covers about ft2 of lawn.

22. 10. The window shown is in the shape of a semicircle. Find
the area of the glass in the shaded region.
45°
3 m
Area shaded = area large sector – area small sector
Large: r = 6
Small: r = 3
The area of the glass in
the shaded region is
10.6 m2