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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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Seven Congruent Rectangles
- Annie Fetter
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Geometry, difficulty level 2. Seven congruent rectangles are arranged to form a larger rectangle. If the area of the large rectangle is 336 units^2, what's the perimeter of the large rectangle?
... more>>
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Ships Ahoy!
- Annie Fetter
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Geometry, difficulty level 2. Examine some methods Thales might have used to measure the distance to
ships at sea.
... more>>
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The Shortest Possible Side
- Annie Fetter
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Geometry, difficulty level 2. Find the shortest possible side length of a triangle with a given
perimeter.
... more>>
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Shotgun Targets
- Annie Fetter
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Geometry, difficulty level 2. Figure out how large to make the inner circle of this shotgun target.
... more>>
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Sighting in a Rifle
- Annie Fetter
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Geometry, difficulty level 3. Analyze the given target of four shots, and tell me how far to move the scope on my rifle so that the group of shots will be centered around the bull's-eye.
... more>>
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Similar Perimeter
- Annie Fetter
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Geometry, difficulty level 2. Find the coordinates of the vertices of a triangle similar to the
given triangle with the designated perimeter.
... more>>
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Simple Signals
- Lisa Lavelle
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Trig/Calc, difficulty level 1. Determine the distance between Erin and Joey as well as how far Joey has travelled in "Perpendicular City."
... more>>
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Slicing a Cube
- Annie Fetter
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Geometry, difficulty level 3. Given a cube with a surface area of 54 cm^2, find the perimeter of the square cross section and that of the largest rectangular cross section of that cube.
... more>>
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Snail's Trail
- Annie Fetter, Lillian Ray
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Geometry, difficulty level 2. What should be the dimensions of the smallest piece of my "Snail's
Trail" quilt?
... more>>
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Snow Removal
- Annie Fetter
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Geometry, difficulty level 4. How many times would the snow in the Math Forum parking lot fill my office? How many pounds does all that snow weigh?
... more>>
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So Many Similar Triangles!
- Annie Fetter
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Geometry, difficulty level 3. Explain how to form pairs of similar triangles given four rods of
specific lengths and any other two rods from the extra supply.
... more>>
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The Speed Skating Oval
- Annie Fetter
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Geometry, difficulty level 2. Find the radii of the turns of a 400 meter speed skating oval.
... more>>
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Splitting a Plane
- Annie Fetter
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Geometry, difficulty level 3. What's the greatest number of regions into which you can split a plane using six lines?
... more>>
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Splitting Space
- Annie Fetter
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Geometry, difficulty level 4. What's the greatest number of regions into which you can divide space using four planes?
... more>>
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Splitting up a Triangle
- Annie Fetter
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Geometry, difficulty level 4. Take any triangle ABC. Construct D and E as the midpoints of BC and
AB, respectively. Now construct DF and EF, where F is any point on AC.
How are the areas of the triangles related to the area of the
quadrilateral?
... more>>
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Spring Up, Fall Back
- Annie Fetter
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Geometry, difficulty level 2. Help build a new rail for these solar panels so that they can be set
at three different angles.
... more>>
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Square in a Right Triangle
- Annie Fetter
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Geometry, difficulty level 3. In the given picture, square DEFG is inscribed in a right triangle ABC (with right angle at B) with DG on AC, E on AB, and F on BC. If AD is 14 and GC is 29, what's the area of the square?
... more>>
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Square Segments
- Annie Fetter
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Geometry, difficulty level 2. A square has been split into four rectangles, one of whose areas is 24
cm^2. The area of the square is no more than 125 cm^2. What are all of
the possible combinations of lengths for the four segments if the
lengths of all of the segments are integers?
... more>>
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Squares on Squares
- Annie Fetter
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Geometry, difficulty level 3. Let P be the center of a 16" x 16" square as well as a vertex of
another 16" x 16" square that pivots freely around P. What is the
largest possible value for the area where these two squares overlap?
... more>>
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Steam Up!
- Annie Fetter
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Geometry, difficulty level 3. Calculating the grade of an incline and the length of a train.
... more>>
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St. George's Banner
- Annie Fetter
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Geometry, difficulty level 2. Find out how wide the cross on this flag needs to be in order for the red and white regions to have the same area. If the flag is x feet by y feet, how wide is the red stripe?
... more>>
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Studying Stars
- Annie Fetter
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Geometry, difficulty level 2. Given a number of points spaced around a circle, how many lines does it take to connect each point to all the other points except for the ones next to it?
... more>>
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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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The Sun vs. the Earth
- Annie Fetter
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Geometry, difficulty level 1. Would one million Earths really fit inside the sun?
... more>>
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Taking Pictures of the Earth
- Annie Fetter
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Geometry, difficulty level 4. If you wanted to go out into space and take a picture of the earth, how far from the earth would you have to go to make sure that you got 1/4 of the equator in your picture? How about 1/3?
... more>>
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A Tale of Two Transformations
- Annie Fetter
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Geometry, difficulty level 3. Compare the results of these two transformations on the coordinate plane.
... more>>
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Talladega Afternoons
- Annie Fetter
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Geometry, difficulty level 1. Compare the corner banking of a track to the steepness of the roof of
a house.
... more>>
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A Tangent Square and Circle
- Reza Kassai
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Geometry, difficulty level 3. The square and circle below are tangent at one point and meet at four
other points, as shown in the picture. If the side of the square is 8
units long, what's the radius of the circle?
... more>>
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Tangrams
- Abram Falk
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Geometry, difficulty level 3. Explain which triangles in a tangram are congruent to each other and what the area is of each piece.
... more>>
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Taxicab Distances
- Annie Fetter
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Geometry, difficulty level 1. Find a general formula for distance in a taxicab geometry universe.
... more>>
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The North Hunterdon Triangle
- Annie Fetter
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Geometry, difficulty level 2. Given points O:(0,0) and Q:(a,0), describe the possible locations of
point P such that triangle OPQ is first isosceles and then equilateral.
... more>>
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Tied in Knots
- Annie Fetter
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Geometry, difficulty level 2. How many triangles can be made from the knotted rope?
... more>>
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Transforming a Triangle
- Annie Fetter
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Geometry, difficulty level 1. Given two triangles on a coordinate plane, find a pair of transformations that will map the first triangle onto the second triangle.
... more>>
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Triangle in Motion?
- Annie Fetter
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Geometry, difficulty level 2. Given triangle ABC, the triangle is translated and then reflected, and
the locations of A' and A'' are given. Find the locations of B'' and C''.
... more>>
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Triangles and String
- Annie Fetter and Steve Risberg
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Geometry, difficulty level 2. Given a loop of string 12" long, find all possible triangles of integer side lengths that it can form, and explain whether they are scalene, isosceles, or equilateral, as well as acute, right, or obtuse.
... more>>
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Triangle Transformation
- Annie Fetter
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Geometry, difficulty level 3. Given triangles ABC and DEF on the coordinate plane and a list of transformations, explain how to map ABC onto DEF in only two steps.
... more>>
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Triple Tango
- Annie Fetter
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Geometry, difficulty level 1. Sort six triangles into three pairs so that the two triangles in each pair can be put together to form a larger triangle without overlapping.
... more>>
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Trisecting a Square
- Annie Fetter
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Geometry, difficulty level 2. Explain how to use the given method to trisect the area of a square.
... more>>
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TV Dimensions
- Annie Fetter
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Geometry, difficulty level 2. If the aspect ratio of a TV is 16:9 and the diagonal measures 50
inches, what are the dimensions of the screen?
... more>>
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Two Rectangles
- Annie Fetter
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Geometry, difficulty level 3. Given the area of one rectangle, find the area of the other rectangle
in the picture.
... more>>
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Two Times Two Angles
- Annie Fetter
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Geometry, difficulty level 3. The measures of two angles are in the ratio 5:3. The measure of the larger angle is 30 greater than half the difference of the angles. Find the measure of each angle.
... more>>
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Up Periscope!
- Annie Fetter
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Geometry, difficulty level 2. Given a diagram of a broken submarine periscope, explain which angles need to be equal in order to for the image to reach the eyepiece.
... more>>
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Visiting Friends
- Annie Fetter
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Geometry, difficulty level 2. Annie and Suzanne live near the same train station. If you are told
how long it takes to walk from each of their houses to the train
station, what can we say about how long it takes to walk at this rate
directly from Annie's to Suzanne's?
... more>>
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Voulez Vous des Voussoirs?
- Annie Fetter
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Geometry, difficulty level 2. Voussoirs are trapezoidal chunks of stone used to build stone arches.
... more>>
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Waiting for Hot Water
- Annie Fetter
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Geometry, difficulty level 2. If a water pipe is 1/2" in diameter, the faucet is about 35 feet from the hot water tank, and the faucet runs at 3 gallons per minute, how long will it take the hot water in the tank to reach the faucet?
... more>>
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Walking Fast
- Annie Fetter
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Geometry, difficulty level 2. A woman with a 28" inseam takes 5 steps for every 4 steps I take. I have a 34" inseam and cover 20 yards in 20 steps. Am I covering more ground because my legs are longer, or because my legs form a larger angle with each step?
... more>>
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The Water in My Pool
- Leigh Nataro
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Geometry, difficulty level 2. Based on one measurement taken from the edge of my circular pool,
what is the volume of my pool?
... more>>
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Well, Well, Well
- Annie Fetter
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Geometry, difficulty level 1. Figure out the diameter of my cousin Frank's new well.
... more>>
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