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Geometry
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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Nine Congruent Rectangles
- Annie Fetter
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Geometry, difficulty level 2. Nine congruent rectangles form a larger rectangle whose area is 720 units^2. What's the perimeter of the large rectangle?
... more>>
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Off on a Tangent
- Annie Fetter
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Geometry, difficulty level 3. Figure out how to construct the required figure.
... more>>
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Olympic Softball
- Annie Fetter
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Geometry, difficulty level 2. Should the pitcher's rubber in Olympic softball be behind the line drawn from third base to first base, or in front of it?
... more>>
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Olympic Target Shooting
- Annie Fetter
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Geometry, difficulty level 2. Given that a shooter is equally likely to hit any point on a 10-inch target, what are the odds that his shot will be in the 6-inch circle?
... more>>
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One Last Bit of Fun
- Annie Fetter
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Geometry, difficulty level 3. In parallelogram ABCD, the bisector of angle ABC intersects AD at P. If PD=5, BP=6, and CP=6, what is the value of AB?
... more>>
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Overlapping Squares
- Annie Fetter
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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?
... more>>
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Painting a Carousel
- Annie Fetter & Car Talk
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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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Painting a Cube
- Annie Fetter
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Geometry, difficulty level 3. A large cube, made up of smaller cubes, has some of its faces painted. When it's taken apart, 24 of the small cubes have no paint on them. How big was the large cube, and how many of its faces were painted?
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A Parallelogram Dissection
- Annie Fetter and Everyday Math Grade 3
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Geometry, difficulty level 3. Cut the given parallelogram along the dotted lines and rearrange the
three resulting triangles to form what seems to be a square. What
needs to be true of the original figure in order for the resulting
figure to actually be a square?
... more>>
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Partitioning the Plane
- Annie Fetter
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Geometry, difficulty level 3. Using six lines, what's the maximum number of regions into which you
can divide a plane?
... more>>
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Part of a Rectangle
- Annie Fetter
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Geometry, difficulty level 3. Given the area of part of a rectangle, find the area of the whole thing.
... more>>
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Patterning a Shotgun
- Annie Fetter
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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?
... more>>
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Pentagon Puzzle
- Annie Fetter
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Geometry, difficulty level 1. Explore the interior angles of pentagons.
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Pentomino Cubes
- Annie Fetter
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Geometry, difficulty level 3. Given the 12 pentominoes, explain which ones can't be used as the basis for the net of a cube. Then take three of the ones that will work and explain where you can attach the sixth square to make the full net.
... more>>
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The Perimeter of an Octagon
- Annie Fetter, with thanks to the Texas Education Agency
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Geometry, difficulty level 3. What's wrong with this picture of a regular octagon? Can you fix
what's wrong and find the perimeter?
... more>>
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Picking an Area Formula
- Annie Fetter
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Geometry, difficulty level 2. Find the area of a polygon on a grid and figure out which of four
formulas, incorporating points on the interior and points on the
boundary, calculates its area correctly.
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Picking a Point
- Steve Risberg
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Geometry, difficulty level 2. A point is randomly selected from within the rectangle having vertices
at (0,0), (2,0), (2,3) and (0,3). What's the probability that the
x-coordinate of the point will be less than the y-coordinate?
... more>>
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Picking Painted Cubes
- Annie Fetter
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Geometry, difficulty level 2. A large cube, composed of smaller cubes, is painted on all sides. The edge length is five small cubes. It's then broken apart and the smaller cubes are put in a bag. What are the odds that a chosen cube is painted on three sides? two? one? none?
... more>>
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A Picturesque Pythgorean Proof
- Annie Fetter
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Geometry, difficulty level 3. Write a proof to go with this picture.
... more>>
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Pisa's Lean
- Annie Fetter
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Geometry, difficulty level 4. Explore the angle and distance at which the Leaning Tower of Pisa actually leans.
... more>>
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Plane and Simple?
- Annie Fetter
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Geometry, difficulty level 3. Explore splitting the plane into regions using lines.
... more>>
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Play Ball!
- Annie Fetter
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Geometry, difficulty level 2. Figure out how to scale down parts of a basketball court so that it's "woman sized," assuming that the average men's college player is 6'3" and the average woman is 5'6".
... more>>
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Playing with a Pentadecagon
- Annie Fetter
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Geometry, difficulty level 2. Extend the sides AB and ED of the regular pentadecagon ABCDEFGHIJKLMNO
until they intersect. What is the measure of the angle at this
intersection?
... more>>
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Playing with Blocks
- Annie Fetter
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Geometry, difficulty level 4. A large cube is made up of 13 double cubes like the smaller one shown, plus a single cube. What color is the single cube, and where does it have to be?
... more>>
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Playing with Matchsticks
- Annie Fetter
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Geometry, difficulty level 2. Thirteen matchsticks are arranged to make six equal regions. Take away one of the matchsticks and arrange the remaining twelve so that they still make six equal regions.
... more>>
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Playing with Paint
- Annie Fetter
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Geometry, difficulty level 2. Explain what three-dimensional solid would make a given "trail" if it were dipped in paint and rolled on the floor.
... more>>
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Point P Perambulates
- Annie Fetter
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Geometry, difficulty level 3. Find the length of the path of a vertex of an equilateral triangle as that triangle rotates around the inside of a square.
... more>>
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Points, Lines, and Planes
- Annie Fetter
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Geometry, difficulty level 3. Determine the number of lines defined by 5 or 6 points, and the number of planes defined by 4 or 5 points. Expand this for N points.
... more>>
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Pondering Pantographs
- Annie Fetter and Lillian Ray
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Geometry, difficulty level 3. Why does a pantograph work?
... more>>
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Precision Paper Folding
- Annie Fetter
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Geometry, difficulty level 3. After folding down two corners of a rectangular piece of paper, explain what the resulting angle measure will always be.
... more>>
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Proofs Without Words II
- Annie Fetter
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Geometry, difficulty level 2. Based on a simple example that illustrates the distributive property, explain what the second picture illustrates.
... more>>
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The Protean Perimeter
- Annie Fetter
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Geometry, difficulty level 3. Given isosceles triangle ABC, with AC=AB=5 inches and angle CAB greater than or equal to 60, what's the perimeter of the triangle if all the edgelengths must be integers?
... more>>
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Proving PD
- Annie Fetter
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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.
... more>>
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The Puzzling Parallelogram
- Annie Fetter
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Geometry, difficulty level 3. Draw a parallelogram ABCD with AB=10. Draw EF with E between A and B and F between C and D such that EF divides the area of ABCD in half. If EF=4, what is FD?
... more>>
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Pythagorean Circles
- Lillian Ray
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Geometry, difficulty level 3. Figure out the relationship between the area of three regions in this
construction
of "Pythagorean circles".
... more>>
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Quad Query
- Annie Fetter
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Geometry, difficulty level 2. Given quadrilateral ABCD, with E the midpoint of AB and F the midpoint of AD. The area of FAEC is 13. What's the area of ABCD?
... more>>
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A Quickie Triangle Puzzle
- Sasha Sokolsky
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Geometry, difficulty level 2. Find the length of the unknown side of this triangle using knowledge of special right triangles.
... more>>
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A Rectangle Dissection
- Annie Fetter
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Geometry, difficulty level 3. Cut the given rectangle into three pieces as indicated by the dashed
lines. Rearrange the pieces to form a square, and explain why the
resulting shape actually is a square.
... more>>
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Regional Ratios
- Annie Fetter
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Geometry, difficulty level 2. A regular hexagon and an equilateral triangle have the same perimeter. What's the ratio of their areas?
... more>>
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Replacing the Football Field
- Annie Fetter
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Geometry, difficulty level 2. Will a regulation-size NCAA soccer field fit inside an equal-quadrant 400-meter track?
... more>>
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Returning to Napoleon's River
- Annie Fetter
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Geometry, difficulty level 3. A soldier stands on the bank of a river and uses his cap to measure its width. How far off will he be if he drops his head even one degree?
... more>>
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The Return of the Parallelogram II
- Annie Fetter
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Geometry, difficulty level 3. Draw a parallelogram ABCD with AB=10. Draw EF with E between A and B and F between C and D such that EF divides the area of ABCD in half. If EF=4, what is the area of the parallelogram?
... more>>
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Reza's Circle
- Reza Kassai
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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.
... more>>
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The Rolling Triangle
- Annie Fetter
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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?
... more>>
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Rotating a Triangle
- Annie Fetter
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Geometry, difficulty level 2. Given a triangle on a coordinate grid, find the new coordinates of the vertices if you rotate the triangle 90 degrees about the origin.
... more>>
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Running a Mile on a Metric Track
- Annie Fetter
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Geometry, difficulty level 4. How far out from the rail on a 400 meter track would you have to run to cover 440 yards?
... more>>
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Save the Lake!
- Annie & Tom Fetter
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Geometry, difficulty level 2. How far does the water level drop if you take 900,000 gallons of water out of a lake with a surface area of 334 acres?
... more>>
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Scintillating Similarity
- Annie Fetter
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Geometry, difficulty level 2. Given a triangle with sides of 3, 4, and 6. What is the perimeter of the smallest triangle that is similar to the first one and has one side with length 12?
... more>>
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Seeing Stars
- Annie Fetter
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Geometry, difficulty level 2. Find the measure of the angle at the "star points" of a "regular star"
when the star has five sides. How about when the star has eight sides?
... more>>
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