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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?
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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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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.
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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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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 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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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?
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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?
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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.
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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 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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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?
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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?
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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.
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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'.
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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.
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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.
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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?
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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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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?
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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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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?
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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?
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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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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?
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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).
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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.
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A Construction Puzzle
- Annie Fetter
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Geometry, difficulty level 3. Given angle PQR and point Y in the interior of the angle, construct XZ so that X lies on QP, Z lies on QR, and Y is the midpoint of XZ.
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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?
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Coordinated Triangle?
- Annie Fetter
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Geometry, difficulty level 3. Triangle ABC has an area of 24 units^2. Point A is at (6,0); point B is at (10,0). Where can point C be? What if the triangle is
isosceles?
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The Corner Banking of the Daytona Speedway
- Annie Fetter
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Geometry, difficulty level 2. Compare the corner banking at the Daytona Speedway, which is 31 degrees, with the roof on the house and garage that I helped build.
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Counting Squares
- Annie Fetter
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Geometry, difficulty level 2. Count the total number of squares in the given diagram.
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Cross Sections of a Cube
- Annie Fetter
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Geometry, difficulty level 3. You have a cube whose surface area is 150 cm^2. What would be the perimeter of a square cross section of this cube? What is the perimeter of the largest rectangle you could get as a cross section? How could you get an equilateral triangle as a cross section?
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Descartes' Triangle?
- Annie Fetter
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Geometry, difficulty level 3. Two vertices of a triangle are located at (0,6) and (0,12). If the area of the triangle is 12 units^2, where can the third vertex be? What if the triangle is also isosceles?
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Diagonal Dilemma
- Karen Cohen
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Geometry, difficulty level 1. What is the area of the square ABCD that has a diagonal of length 12 cm?
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The Diagonal Triangle
- Annie Fetter
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Geometry, difficulty level 1. A triangle is formed by connecting opposite vertices of a figure formed by two squares. Find the area of the triangle.
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Different Views
- Annie Fetter
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Geometry, difficulty level 3. Given a front, top, and side view of an object, find the volume of that object.
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Dilating a Quadrilateral
- Annie Fetter
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Geometry, difficulty level 3. Given two similar quadrilaterals on a coordinate grid, determine the ratio and center of dilation.
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The Dimensions of a Hunting Platform
- Annie Fetter
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Geometry, difficulty level 2. I have a picture of a hunting platform, and I need help figuring out some of the dimensions.
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Dissecting Shapes
- Annie Fetter
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Geometry, difficulty level 3. 1) Take the irregular shape shown and, making just two cuts, turn it into a square. 2) Take a square and, using four cuts, make five smaller equal squares whose total area is t he same as the original.
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Divided Rectangle
- Annie Fetter
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Geometry, difficulty level 1. A rectangle is divided into four rectangles with areas 45, 25, 15, and x. Find x.
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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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