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Circles

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

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 912
Access to these problems requires a Membership.

All Around the World
 Terry Trotter & Annie Fetter

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?
... more>>

Arc to Area
 Annie Fetter

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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As Far as the Eye Can See
 Annie Fetter

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?
... more>>

Broken Pottery
 Annie Fetter

Geometry, difficulty level 2. Explain how to find the center of this sherd.
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Building a Vaulted Ceiling
 Annie Fetter

Geometry, difficulty level 4. An eyebrow window is a 15inchdeep slice off an 8.5foot circle. How wide will the base of the window be?
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Circle Derby
 Terry Trotter

Geometry, difficulty level 2. Determine which dot has the small angle to go and which has the
shortest distance to go in this race around a circle.
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A Circle Inscribed in an Isosceles Triangle
 Ilmar Vitsut

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

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

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, crisscrossing 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

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

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

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

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

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

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

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

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

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

Geometry, difficulty level 2. Figure out what's wrong with this picture that includes a circle and a rhombus.
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An Inscribed Circle
 Annie Fetter

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

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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It's Probably This Chord
 Annie Fetter

Geometry, difficulty level 3. Find the probability that a chord of a circle will have particular
characteristics.
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Learning About GPS
 Annie Fetter

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

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

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

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

Geometry, difficulty level 3. How would you change the recipe for a 9inch pie crust to make an 11inch pie crust?
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Measuring Cylinders
 Annie Fetter

Geometry, difficulty level 2. Help Ian measure his stove pipe and his propane tank.
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Measuring the Circumference of the Earth
 Annie Fetter

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 Problem
 Annie Fetter

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

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 9inch pan?
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More Tangents!
 Annie Fetter

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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Off on a Tangent
 Annie Fetter

Geometry, difficulty level 3. Figure out how to construct the required figure.
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Overlapping Squares
 Annie Fetter

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

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

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

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

Geometry, difficulty level 2. Will a regulationsize NCAA soccer field fit inside an equalquadrant 400meter track?
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Reza's Circle
 Reza Kassai

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

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

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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The Subtending Chord
 Annie Fetter

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?
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Voulez Vous des Voussoirs?
 Annie Fetter

Geometry, difficulty level 2. Voussoirs are trapezoidal chunks of stone used to build stone arches.
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What Do We Know About Tangents?
 Annie Fetter

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

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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