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Problems
Geometry
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Browse all Geometry Problems of the Week
Participation in the Geometry Problems of the
Week allows teachers and students to address the
NCTM
Problem Solving Standard for Grades 9-12, enabling students to
build new mathematical knowledge through problem solving; solve problems
that arise in mathematics and in other contexts; apply and adapt a
variety of appropriate strategies to solve problems; and monitor and
reflect on the process of mathematical problem solving.
For background information elsewhere on our site, explore the
High School Geometry
area of the Ask Dr. Math archives. To find relevant
sites on the Web, browse and search
Geometry
in our Internet Mathematics Library.
Access to these problems requires a Membership.
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Added Areas
- Annie Fetter
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Geometry, difficulty level 1. A circle and an isosceles triangle are inscribed in a square. If the areas of the three figures added together is 28 square units, what's the edgelength of the square?
... more>>
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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?
... more>>
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Analyzing Angles
- Annie Fetter
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Geometry, difficulty level 2. Given some information about angles in a picture, figure out some of
the angle measures.
... more>>
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Another Ambiguous Angle?
- Annie Fetter
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Geometry, difficulty level 3. Given isosceles triangle ABC, with BE perpendicular to AD. If angle ACB is x, what is angle CBE?
... more>>
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Ants Marching
- Annie Fetter
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Geometry, difficulty level 2. Traveling along the edges of a cross section of the cube, find the
length of the shortest possible path from one corner of a cube to the
opposite corner, as well as the longest possible path between the same
corners.
... more>>
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The Apprehendable Angle
- Annie Fetter
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Geometry, difficulty level 2. Triangle ABC is isosceles with a vertex angle (at the top) of 50 degrees. If D is on AC, E is on BC, and AD = BE, what is the measure of angle ADE?
... more>>
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Approximating Pi
- Annie Fetter
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Geometry, difficulty level 3. Approximate the value of pi using polygons instead of a circle.
... more>>
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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.
... more>>
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Arda and Brooke's Catapult
- Annie Fetter
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Geometry, difficulty level 2. Does Arda's method for measuring the distance across the river really
work?
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Area of a Rectangle
- Annie Fetter
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Geometry, difficulty level 2. Given rectangle ABCD with point E on CD. AE is 3, BE is 4, and AE is perpendicular to BE. What's the area of ABCD?
... more>>
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Area of a Rhombus
- Annie Fetter
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Geometry, difficulty level 2. Find the area of a rhombus, given one angle measure and the length of
the longer diagonal.
... more>>
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Arranging Rectangles
- Annie Fetter
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Geometry, difficulty level 3. Help Kasim decide if his arrangement of rectangles could ever form a square.
... more>>
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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?
... more>>
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Assessing Seismic Risk
- Craig Foster and Annie Fetter
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Geometry, difficulty level 3. Figure out how likely it would be for an earthquake to occur within
certain distances of a building.
... more>>
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Beach Boys
- Steve Risberg
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Geometry, difficulty level 3. Help Skip and Gil find the height of the cruise ship.
... more>>
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Beyond Exterior Angles
- Annie Fetter
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Geometry, difficulty level 2. Extend two sides of the given polygon and determine the angle at which
the new lines meet.
... more>>
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Bouncing Billiards
- Annie Fetter
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Geometry, difficulty level 1. Help Maria decide the angle that will sink her cue shot after four
bounces.
... more>>
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Boxing Up Harry's Broom
- Annie Fetter
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Geometry, difficulty level 1. Help Harry find the right size box for his broom.
... more>>
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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.
... more>>
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Building a Rain Gauge
- Annie Fetter
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Geometry, difficulty level 3. Given the depth of the water in a bottle, figure out how much rain really fell, considering that the opening of the bottle is much smaller than the main part.
... more>>
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Building a Regular Hexagon
- Annie Fetter and Steve Weimar
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Geometry, difficulty level 3. How do you form a regular hexagon by attaching rectangles to the sides
of an equilateral triangle?
... more>>
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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?
... more>>
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Building Bookshelves
- Steve Weimar
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Geometry, difficulty level 2. Merriem and Emmanuel's new bookshelf hits the ceiling when they try to
put it in place. Explain why, and help them figure out how much to
cut off the bottom so that it will fit.
... more>>
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Build Your Own Hexagon
- Annie Fetter
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Geometry, difficulty level 2. Put a square on each edge of an equilateral triangle (edgelength one unit) and connect the outside vertices of adjacent squares to form a hexagon. Is this hexagon equilateral? equiangular? What is its area?
... more>>
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Buying Paint and Shingles
- Annie Fetter
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Geometry, difficulty level 2. Figure out how many gallons of paint and how many bundles of shingles I need to buy to finish my new shed.
... more>>
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Calculating String Length
- Annie Fetter
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Geometry, difficulty level 2. Given the archery targets of two competitors, figure out which one has the shortest 'string length'.
... more>>
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A Carpenter's Trisection
- Annie Fetter
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Geometry, difficulty level 3. Explain whether or not the given method of trisecting an angle, using only a carpenter's square, really works.
... more>>
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Circle Derby
- Terry Trotter
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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.
... more>>
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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.
... more>>
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Circle Intersections
- Annie Fetter
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Geometry, difficulty level 2. What is the maximum number of times that six circles of the same size could intersect?
... more>>
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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.
... more>>
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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?
... more>>
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Clever Cuts
- Annie Fetter
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Geometry, difficulty level 2. Find three different ways to cut a hexagonal cake into six congruent
pieces.
... more>>
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Comparing the Areas of Two Squares
- Annie Fetter
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Geometry, difficulty level 3. Given two squares, one with the diameter of a circle as an edge, the
other inscribed in the circle. How do the areas of the squares
compare?
... more>>
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Congruence and Area
- Annie Fetter
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Geometry, difficulty level 3. "If two triangles have the same area, then they are congruent." Is this a true statement?
... more>>
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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.
... more>>
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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?
... more>>
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Congruent Rectangles
- Annie Fetter
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Geometry, difficulty level 1. Seven congruent rectangles are arranged to form a larger rectangle. If the area of the large rectangle is 756 units^2, what's the perimeter of the large rectangle?
... more>>
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Congruent Rectangles Help
- Annie Fetter
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Geometry, difficulty level 2. Find some possible areas for the large rectangle shown in the
given picture.
... more>>
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Constructing a Tangent
- Annie Fetter
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Geometry, difficulty level 3. Explain why this construction of a tangent works.
... more>>
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Constructing a Window
- Annie Fetter
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Geometry, difficulty level 4. Explain the shapes of the other four sides of a regular hexagonal window that's built between two wall studs 22.5" apart.
... more>>
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Constructing the Great Pyramid
- Annie Fetter & Suzanne Alejandre
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Geometry, difficulty level 2. Figure out if the instructions provided will result in a scale
model of the Great Pyramid of Giza.
... more>>
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The Coordinated Quadrilateral
- Annie Fetter
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Geometry, difficulty level 3. A quadrilateral has vertices at (-3,6), (-1,-2), (7,-4), and (5,4). What kind of quadrilateral is it?
... more>>
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