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Study Guide
Unit 3 – Circles and Volume
MCC9-12.G.C.1 Prove that all circles are similar.
1. Given a circle of a radius of 3 and another circle with a radius of 5, compare the ratios of the two radii, the two
diameters, and the two circumferences.
2. What is the ratio of circumference to diameter of any circle? ___________
MCC9-12.G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship
between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle
is perpendicular to the tangent where the radius intersects the circle.
3. In circle P,
𝑚∠𝐴𝑃𝐵 = 80°.
̂ = _____
m𝐴𝐵
̂ = _____
m𝐴𝐶𝐵
4. In the figure on the right,
𝑚∠𝐵𝑂𝐶 = 85°, 𝑚∠𝐴𝑂𝐷 =
̂ = 80°
120°, and 𝑚𝐸𝐷
Find:
̂ = _____
𝑚𝐵𝐶
̂ = _____
𝑚𝐴𝐷
𝑚∠𝐴𝐵𝐷 = _____
𝑚∠𝐸𝐵𝐷 = _____
̂ = 85°.
5. ∠𝑃𝑁𝑄 is inscribed in circle O and 𝑚𝑃𝑄
a. What is the measure of
∠𝑃𝑂𝑄?
6. In circle P, ̅̅̅̅
𝐴𝐵 is a diameter.
If 𝑚∠𝐴𝑃𝐶 = 94°, find the
following:
a. 𝑚∠𝐵𝑃𝐶
b. What is the measure of
∠𝑃𝑁𝑄 ?
̂
b. 𝑚𝐵𝐶
c. 𝑚∠𝐵𝐴𝐶
̂
d. 𝑚𝐴𝐶
7. In this circle, ̅̅̅̅
𝐴𝐵 is tangent to the
̅̅̅̅ is tangent to
circle at point B, 𝐴𝐶
the circle at point C, and point D
lies on the circle.
What is 𝑚∠𝐵𝐴𝐶?
8. In the circle shown, ̅̅̅̅
𝐵𝐶 is a
̂
diameter and 𝑚𝐴𝐵 = 120°.
What is ∠𝐴𝐵𝐶 ?
̅̅̅̅and 𝑍𝑌
̅̅̅̅ and chords 𝑊𝑋
̅̅̅̅̅
9. Circle P has tangents 𝑋𝑌
̅̅̅̅̅
and 𝑊𝑍, as shown in this figure.
The measure ∠𝑋𝑊𝑍 = 60° .
10. Solve for x.
What is ∠𝑋𝑌𝑍 ?
11. Find the m∠ABD, the inscribed
angle of ○C.
13. Find the m∠ABD, the
inscribed angle of ○C.
15. In circle P, ̅̅̅̅
𝐷𝐺 is a tangent.
12. Find the m∠ABD, the
inscribed angle of ○C, if
̂ = 240°
𝑚𝐵𝐸𝐷
14. A diameter of a circle is perpendicular to a chord
whose length is 12 inches. If the length of the
shorter segment of the diameter is 4 inches, what is
the length of the longer segment of the diameter?
16. Chords AB and CD intersect inside a circle at
point E.
AE= 2, CE =4 , and ED =3 .
Find EB.
AF = 8, EF = 6, BF = 4, and EG = 8.
a. Find CF
b. Find DG
17. Chords AB and CD intersect inside a circle at
point E.
AE= 5, CE =10, EB = x, and ED = x-4.
18. Two secant segments are drawn to a circle from a
point outside the circle. The external segment of
the first secant segment is 8 centimeters and its
internal segment is 6 centimeters.
If the entire length of the second secant segment is
28 centimeters, what is the length of its external
segment?
19. A tangent segment and a secant segment are drawn 20. The diameter of a circle is 19 inches. If the
to a circle from a point outside the circle.
diameter is extended 5 inches beyond the circle
The length of the tangent segment is 15 inches. The
to point C, how long is the tangent segment from
external segment of the secant segment measures 5
point C to the circle?
inches. What is the measure of the internal secant
segment?
Find EB and ED.
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21. A satellite orbits the earth so that it remains at the same point above the
Earth’s surface as the Earth turns.
a. If the satellite has a 50° view of the equator, what percent of the
equator can be seen from the satellite?
b. The average radius of the Earth is approximately 3959 miles. How
far above the Earth’s surface is the satellite?
c. What is the length of the longest line of sight from the satellite to
the Earth’s surface? Identify this line of sight using the diagram.
MCC9-12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
22. ABCD is an inscribed quadrilateral. If 𝑚∠𝐶𝐷𝐴 = 130° and 𝑚∠𝐷𝐴𝐵 = 75°, then…
Find: 𝑚∠𝐴𝐵𝐶 = ______ and 𝑚∠𝐵𝐶𝐷 = ______
MCC9-12.G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the
radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a
sector.
23. Complete the table and consider the ratio of arc length to radius for different radii.
Angle
Radius
45°
45°
45°
45°
2
4
9
13
Exact Arc Length
Exact
Arc Length / Radius
What is the radian measure of 45°?
What do you notice about the ratio of the arc length to radius?
̂ ? Leave your answer in
25. What is the length of 𝐶𝐷
terms of pi.
24. If x = 50°, what is the area of the shaded sector of
circle A? Leave your answer in terms of pi.
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26. What is area of the shaded part of the circle below?
Leave your answer in terms of pi.
27. The spokes of a bicycle wheel form 10
congruent central angles. The diameter of the
circle formed by the outer edge of the wheel is
18 inches.
What is the length, to the
nearest 0.1 inch, of the outer
edge of the wheel between two
consecutive spokes?
MCC9-12.G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
MCC9-12.G.GMD.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a
sphere and other solid figures.
MCC9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★
28. What is the volume of the cone shown below?
29. A cylinder has a radius of 10 cm and a height of 9
cm. A cone has a radius of 10 cm and a height of 9
cm. Show that the volume of the cylinder is three
times the volume of the cone.
30. A sphere has a radius of 3 feet. What is the volume
of the sphere?
31. What is the volume of a cylinder with a radius of 3
9
in. and a height of 2 in.?
32. Let’s investigate the relationship between a Cone
and its corresponding Cylinder with the same
height and radius. LABEL the height, h and
radius, r on each diagram below.
33. Let’s investigate the relationship between a
Pyramid and its corresponding Rectangular Prism
with the same height, length, and width. LABEL
the height, h, length, l, and width, w, on each
diagram below.
a. If the cylinder was full of water and you
poured it into the cone, how many times would
it fill up the cone completely?
b. If the cone was full of water, how much of the
cylinder would it fill up?
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c. If the rectangular prism was full of water and
you poured it into the pyramid, how many
times would it fill up the pyramid completely?
d. If the pyramid was full of water, how much of
the rectangular prism would it fill up?
34. Approximate the Volume of the Backpack that is
17 in x 12 in x 4 in.
35. Find the Volume of the Grain Silo shown below
that has a diameter of 20 ft and a height of 50 ft.
36. Cylinder A and cylinder B are shown below. What
is the volume of each cylinder?
37. The volume of a cylindrical watering can is 100
cm3. If the radius is doubled, then how much
water can the new can hold?
38. Jason constructed two cylinders using solid metal washers. The cylinders have the same height, but one of
the cylinders is slanted as shown.
Which statement is true about Jason’s cylinders?
A. The cylinders have different volumes because they have different radii.
B. The cylinders have different volumes because they have different surface areas.
C. The cylinders have the same volume because each of the washers has the same height.
D. The cylinders have the same volume because they have the same cross-sectional area at every plane
parallel to the bases.
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