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Circles
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Problems in this section deal with the basic definitions of parts
of a circle. Algebraic problems are based on the theorems that
describe the relations among chords, radii, and tangents, or between
arcs and angles. Many of the problems in this category will require
advanced knowledge and an integration of several geometric topics.
- Related Resources
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Interactive resources from our Math Tools project:
Geometry: Circles
The closest match in our Ask Dr. Math archives:
High School: Conic Sections/Circles
NCTM Standards:
Geometry Standard for Grades 9-12
Access to these problems requires a Membership.
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Aim at This Target!
- Annie Fetter
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Geometry, difficulty level 4. Figure out the areas of different parts of this target, and the relation between the different areas. Can you explain this relation?
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All Around the World
- Terry Trotter & Annie Fetter
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Geometry, difficulty level 2. If a wire wrapped around the equator of the Earth is lengthened by 100 meters, how far above the surface of the Earth would the wire now be if lifted off the surface an equal distance all the way around?
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Approximating the Circumference and Area of a Circle
- Annie Fetter
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Geometry, difficulty level 2. Take a circle and circumscribe a square and inscribe a square. Construct a square between the other two squares, and compare the area and the perimeter of the middle square with the area and circumference of the circle.
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Arc to Area
- Annie Fetter
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Geometry, difficulty level 4. Given an arc with a measure of 40 degrees whose endpoints are at (1,5) and (5,3), find the area of the circle that contains the arc.
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Areas of Circles in a Target
- Annie Fetter
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Geometry, difficulty level 2. Given one circle inside another circle. If the outer circle is 36 inches in diameter, how big must the inner circle be so that the area of the inner circle equals the area of the outer circle NOT covered by the inner circle?
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As Far as the Eye Can See
- Annie Fetter
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Geometry, difficulty level 4. If I'm 5'10" tall, can I see the entire length of Lake Memphramagog, which is about 23 miles long?
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Broken Pottery
- Annie Fetter
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Geometry, difficulty level 2. Explain how to find the center of this sherd.
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Building a Vaulted Ceiling
- Annie Fetter
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Geometry, difficulty level 4. An eyebrow window is a 15-inch-deep slice off an 8.5-foot circle. How wide will the base of the window be?
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Chords and Arcs
- Annie Fetter
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Geometry, difficulty level 3. A chord of a circle is the hypotenuse of an isosceles right triangle whose legs are radii of the circle. The radius of the circle is 6 times the square root of 2. What is the length of the minor arc subtended by the chord?
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Circle and Rhombus
- Annie Fetter
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Geometry, difficulty level 3. In the picture, we have a circle and a rhombus. BC is 6, AE is 4, angle DAE is 45 degrees, and AD is a diameter of the circle. How far is it around the perimeter of the whole figure?
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A Circle Inscribed in an Isosceles Triangle
- Ilmar Vitsut
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Geometry, difficulty level 4. Find the radius of a circle inscribed in an isosceles triangle with sides 12, 12, and 8.
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Circles and Tangents
- Annie Fetter
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Geometry, difficulty level 3. AOD is a diameter of circle O. B is any point on the circle. At B, a tangent is drawn to the circle. From the center, O, a line is drawn parallel to AB, meeting the tangent at P. Prove that PD is tangent to
the circle.
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Circumnavigating Circles
- Annie Fetter
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Geometry, difficulty level 3. The radii of two wheels are 10 vershoks and 5 vershoks and their centers are 30 vershoks apart. A belt goes around both of the wheels, criss-crossing between the two centers to form two internal tangents. What's the length of the belt?
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Congruent Chords and Circles
- Annie Fetter
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Geometry, difficulty level 4. Prove that two congruent chords in a circle are equal distances from the center of the circle.
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Congruent Chords
- Annie Fetter
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Geometry, difficulty level 3. Two congruent circles are drawn, and four congruent chords are drawn, two in each circle, all perpendicular to the diameter through both circles. The distance between the two furthest chords is 20, and the distance between two chords of the same circle is 8. What's the area of one of the circles?
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Dendrochronology
- Annie Fetter
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Geometry, difficulty level 2. Figure out about how old a large yellow birch is, given its diameter.
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Drawing a Kite Plan
- Annie Fetter
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Geometry, difficulty level 3. Given a picture of a kite plan, figure out the radius of the circle that creates the curved section.
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Drawing Out a Decagon
- Annie Fetter
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Geometry, difficulty level 2. Extend sides AB and IH of the regular decagon ABCDEFGHIJ until they intersect. What is the measure of the acute angle at this intersection?
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Equilateral Triangles and Circles
- Annie Fetter
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Geometry, difficulty level 2. Two vertices of an equilateral triangle are on the diameter of a circle whose area is 49 pi. The other vertex is on the circumference of the circle. What's the largest possible area of the triangle?
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Extending the Enneagon
- Annie Fetter
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Geometry, difficulty level 3. Extend the sides AB and ED of the regular enneagon ABCDEFGHI until they intersect. What is the measure of the acute angle at this
intersection? Extend sides AB and EF. Now what's the measure of that acute angle at the intersection?
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Find the Largest Equilateral Triangle
- Annie Fetter
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Geometry, difficulty level 3. Two vertices of an equilateral triangle lie on a diameter of a circle whose area is 36pi cm^2, and the third vertex lies on the circle. What is the largest possible area of the triangle?
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Find the Perimeter
- Annie Fetter
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Geometry, difficulty level 3. In the given picture, DB and DC are both 20 inches. What's the perimeter of triangle FDG?
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Fix This Picture
- Annie Fetter
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Geometry, difficulty level 2. Figure out what's wrong with this picture that includes a circle and a rhombus.
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A Hexagon and a Triangle in a Circle
- Annie Fetter
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Geometry, difficulty level 2. A regular hexagon and an equilateral triangle are both inscribed in the same circle so that the hexagon and the triangle share three vertices. The radius of the circle is 10 units. What is the area of the region between the two polygons?
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An Inscribed Circle
- Annie Fetter
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Geometry, difficulty level 2. Find the radius of a circle inscribed in a right triangle with legs of
5 units and 12 units.
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Intersecting Circles
- Annie Fetter
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Geometry, difficulty level 3. What is the maximum number of times six circles of the same size can intersect? In other words, draw six circles of the same size on a piece of paper - what is the maximum number of intersections your drawing can have?
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Jill and Jimmy - Soap Bubbles
- Annie Fetter
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Geometry, difficulty level 3. You're given two spheres with centers 14 cm apart. If the two spheres intersect in a circle whose radius is 12 cm, and the radius of one sphere is 13 cm, what's the radius of the other sphere?
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Learning About GPS
- Annie Fetter
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Geometry, difficulty level 4. What percentage of the surface of the earth can be seen by a satellite orbiting at an altitude of 20,200 km? Are the Global Positioning System satellites "geosynchronous"?
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The Length of a Belt
- Annie Fetter
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Geometry, difficulty level 3. The diameters of two wheels are 18 cm and 12 cm and their centers are 30 cm apart. A belt goes around them and crosses 18 cm from the center of the big wheel. How long is the belt? How do you know you're right?
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Look West!
- Annie Fetter
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Geometry, difficulty level 2. How far away is the horizon when you're at the top of the Gateway Arch?
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Magic Coins
- Annie Fetter
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Geometry, difficulty level 3. How come you can fit a quarter through a hole the size of a dime? Find a pair of coins in another system of money that could be used as "magic coins" - the "biggest" and "smallest" coins for which this would work.
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Making Pie Crust
- Annie Fetter
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Geometry, difficulty level 3. How would you change the recipe for a 9-inch pie crust to make an 11-inch pie crust?
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Measuring the Circumference of the Earth
- Annie Fetter
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Geometry, difficulty level 3. At our office on the autumnal equinox, the angle formed by the sun's rays and a stick is 40 degrees. If our office is 2749 miles from the equator, what's the circumference of the Earth?
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A Minor Arc Subtended by a What??
- Annie Fetter
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Geometry, difficulty level 2. A chord of a circle is the hypotenuse of an isosceles right triangle whose legs are radii of the circle. The length of the chord is 6 times the square root of 2. What is the length of the minor arc subtended by the chord?
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A Minor Problem
- Annie Fetter
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Geometry, difficulty level 1. A chord of a circle is one side of an equilateral triangle. The other
two sides of the triangle are radii of the circle. Find the length of
the minor arc subtended by the chord, if the radius of the circle is
given.
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More Pie, Anyone?
- Annie Fetter
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Geometry, difficulty level 2. How big a pie pan would you need if you made twice as much dough as you would need for a 9-inch pan?
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More Tangents!
- Annie Fetter
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Geometry, difficulty level 3. Given a square ABCD inscribed in a unit circle. AB is extended to P, where PC is tangent to the circle. Find the length of PD.
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Overlapping Squares
- Annie Fetter
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Geometry, difficulty level 3. Two squares, one 8 cm on a side and the other 10 cm, overlap. A corner of the 10 cm square is anchored at the middle of the 8 cm square and can rotate around that point. What are the maximum and minimum areas they'll overlap?
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Painting a Carousel
- Annie Fetter & Car Talk
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Geometry, difficulty level 2. Help Liam and Thomas figure out the area of the deck of a carousel
using only one measurement.
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Patterning a Shotgun
- Annie Fetter
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Geometry, difficulty level 2. Construct a circular target with nine regions of equal area, with one a circle in the middle and spokes going out to the edge. If the whole circle is 6 feet in diameter, what will the diameter of the inner circle be?
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Proving PD
- Annie Fetter
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Geometry, difficulty level 3. AOD is a diameter of the circle with center O. B is any point on the
circle that isn't A or D. A tangent is drawn to the circle at point B.
A line is drawn through O parallel to AB, meeting the tangent at P.
Prove that PD is a tangent to the circle.
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Replacing the Football Field
- Annie Fetter
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Geometry, difficulty level 2. Will a regulation-size NCAA soccer field fit inside an equal-quadrant 400-meter track?
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Reza's Circle
- Reza Kassai
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Geometry, difficulty level 2. OC is the radius of a circle and has a length of sqrt2 units. DE is
the perpendicular bisector of OC. BA is tangent to the circle at A. DC
= CB. Find the length of AB.
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The Rolling Triangle
- Annie Fetter
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Geometry, difficulty level 3. Equilateral triangle ABC sits on top of unit square DEFG so that AC is
coincident with DE. The triangle is rotated around the outside of the
square until it returns to the top edge. How far did point A travel?
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Running a Mile on a Metric Track
- Annie Fetter
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Geometry, difficulty level 4. How far out from the rail on a 400 meter track would you have to run to cover 440 yards?
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So, What Do You Know About Tangents?
- Annie Fetter
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Geometry, difficulty level 3. AB and CB are tangents to a circle, with A and C on the circle and B their intersection. D is a point on the minor arc AC, and a tangent is drawn at D, which intersects the AB at E and CB at F. What is the perimeter of EFB?
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The Subtending Chord
- Annie Fetter
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Geometry, difficulty level 3. A chord of a circle is the hypotenuse of an isosceles right triangle whose legs are radii of the circle. The length of the chord is 8 times the square root of 2. What is the length of the minor arc subtended by the chord?
... more>>
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What Do We Know About Tangents?
- Annie Fetter
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Geometry, difficulty level 2. Given two externally tangent congruent circles, construct a segment from the center of one circle tangent to the other circle. If the length of this segment is 10, what's the radius of the circles?
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Which Pizza?
- Annie Fetter
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Geometry, difficulty level 2. Which is a better deal - a 16" diameter pizza for $11 or a 10" diameter pizza for $7?
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