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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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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?
... 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.
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Angle Trisection with a Carpenter's Square
- Annie Fetter
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Geometry, difficulty level 3. Prove how a carpenter's square can be used to trisect an angle.
... 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?
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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?
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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.
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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.
... 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.
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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.
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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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Arranging Six Squares
- Annie Fetter
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Geometry, difficulty level 3. How many ways can you arrange six squares in the plane so that they all share an edge with at least one other square? How many of these configurations could be folded up to form a cube?
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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.
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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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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.
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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 Bridge Across a Chasm
- Annie Fetter
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Geometry, difficulty level 2. So they're standing on the edge of this chasm. It isn't really that far across, but they need to find a log or tie together enough rope to reach a tree on the other side. So they mark a spot on the edge of the chasm directly across from the tree. Then they walk 20 paces along the chasm, and she stays there. He walks another 20 paces and marks the spot. Then he turns ninety degrees and walks away from the chasm until she's directly in the line of sight between him and the tree on the other side. Then he claims that the distance he just walked from the edge of the chasm is the same as the distance across the chasm.
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Building a Dog Enclosure
- Annie Fetter
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Geometry, difficulty level 3. Two people are arguing about whether three sections of fencing 6, 8, and 10 feet long will enclose more or less area than three sections 6, 8, and 12 feet long.
... 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.
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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?
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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.
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Building Polygons
- Annie Fetter
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Geometry, difficulty level 2. Put squares on the edges of an equilateral triangle and connect their outside corners to form a hexagon. Do the same thing with a regular hexagon. Are the resulting figures equilateral? Are they equiangular?
... 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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The Center of Gravity of a Quadrilateral
- Annie Fetter
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Geometry, difficulty level 3. Find the center of gravity of a quadrilateral.
... more>>
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Checking a Building Foundation for Square
- Annie Fetter
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Geometry, difficulty level 2. How can you determine whether a rectangle is square when all you have is string and a tape measure?
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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?
... more>>
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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?
... more>>
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Circle, Hexagon, & Triangle
- Annie Fetter
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Geometry, difficulty level 2. A regular hexagon and an equilateral triangle share three vertices and are inscribed in a circle with a radius of 8 units. What is the area of the region between the two polygons?
... 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?
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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?
... 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.
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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.
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Construct an Isosceles Triangle
- Annie Fetter
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Geometry, difficulty level 2. Give at least three different ways to construct an isosceles triangle (a construction can be repeated over and over and in this case will always yield an isosceles triangle).
... 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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