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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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Geometric Algebra?
- Annie Fetter
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Geometry, difficulty level 2. Given a geometric picture that illustrates the fact that a(a+b)=a^2+ab, explain what the second picture illustrates.
... more>>
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Geometry X-Games
- Annie Fetter
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Geometry, difficulty level 2. Given triangle ABC, with D on AB and E on BC. AC is 39 units, AD is x units, DB is 9 units, DE is 13 units, and BE is x units. If triangles ABC and EBD are similar, what's x, and what's the length of CE?
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Going Global
- Annie Fetter & NCTM's WLME
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Geometry, difficulty level 2. Look at Shakespeare's Globe Theater from a mathematical standpoint - figure out the interior angle of an icosagon and calculate how many people could stand in the "field" in front of and around the stage.
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Hippocrates' Lunes
- Annie Fetter
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Geometry, difficulty level 3. ABC is half a square inscribed in a semicircle. Then a semicircle is constructed on AB. BD is then constructed, the perpendicular bisector of AC, and triangle ABD is shaded, as is the part of the outer semi-circle that's not part of the original semicircle that's not part of the original semicircle. What did Hippocrates prove about the two shaded regions?
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How Deep Is It?
- Annie Fetter
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Geometry, difficulty level 2. How deep will the water be when this container is resting its largest
face?
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How Long is the Base?
- Annie Fetter
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Geometry, difficulty level 3. Find the length of the base of a particular isosceles triangle.
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How Many Angles?
- Annie Fetter
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Geometry, difficulty level 2. Given a picture of two parallel lines and two transversals, how many
different angle measures are represented by the 14 labeled angles?
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In a Pickle
- Lillian Ray
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Geometry, difficulty level 2. Figure out whether or not there will be enough pickle juice to make
the tartar
sauce.
... more>>
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An Inscribed Circle
- Annie Fetter
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Geometry, difficulty level 2. A circle is inscribed in a triangle with sides of 6, 6, and 4. What's the radius of the circle?
... more>>
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An Inscribed Circle
- Annie Fetter
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Geometry, difficulty level 2. Find the radius of a circle inscribed in a right triangle with legs of
5 units and 12 units.
... more>>
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Integer Triangle Angles
- Annie Fetter
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Geometry, difficulty level 1. The degree measures of a triangle are x, y, and z, and x, y, and z are
all integers where x < y < z. What's the largest possible value of z?
What's the smallest?
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Intersecting Circles
- Annie Fetter
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Geometry, difficulty level 3. What is the maximum number of times six circles of the same size can intersect? In other words, draw six circles of the same size on a piece of paper - what is the maximum number of intersections your drawing can have?
... more>>
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Is the Fair Fair?
- Annie Fetter
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Geometry, difficulty level 3. Figure out the probability of winning at this popular midway game.
... more>>
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It's Probably This Chord
- Annie Fetter
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Geometry, difficulty level 3. Find the probability that a chord of a circle will have particular
characteristics.
... more>>
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Journey Through Outer Space
- Annie Fetter
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Geometry, difficulty level 2. How many Earths can fit in Uranus?
... more>>
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Kyle's Similar Triangles
- Kyle Binder and Annie Fetter
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Geometry, difficulty level 3. Given two similar triangles and the lengths of two corresponding
sides. How do the areas of the two triangles compare?
... more>>
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Leaning Ladder
- Annie Fetter
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Geometry, difficulty level 3. A ladder is leaning against a wall. Explore what happens to the top
of the ladder when the bottom is moved closer or further from the wall.
... more>>
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Learning About GPS
- Annie Fetter
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Geometry, difficulty level 4. What percentage of the surface of the earth can be seen by a satellite orbiting at an altitude of 20,200 km? Are the Global Positioning System satellites "geosynchronous"?
... more>>
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The Length of a Belt
- Annie Fetter
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Geometry, difficulty level 3. The diameters of two wheels are 18 cm and 12 cm and their centers are 30 cm apart. A belt goes around them and crosses 18 cm from the center of the big wheel. How long is the belt? How do you know you're right?
... more>>
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A Line of Reflection
- Annie Fetter
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Geometry, difficulty level 2. Find the equation of the line of reflection for triangle ABC and its
image, A'B'C'.
... more>>
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Look West!
- Annie Fetter
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Geometry, difficulty level 2. How far away is the horizon when you're at the top of the Gateway Arch?
... more>>
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Lost at Sea?
- Annie Fetter
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Geometry, difficulty level 2. Help Darius and Skip figure out whether they're in the United States
or Canada using the bearings Skip takes.
... more>>
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Lotsa Popcorn!
- Annie Fetter
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Geometry, difficulty level 1. Can the World's Largest Popcorn Ball really be 50,000 times bigger than a
normal one? Wow!
... more>>
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Mad About Medians?
- Annie Fetter
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Geometry, difficulty level 3. The medians drawn from the vertices of the acute angles of a right triangle are 5 and sq root of 40. What is the length of the hypotenuse?
... more>>
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Magic Coins
- Annie Fetter
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Geometry, difficulty level 3. How come you can fit a quarter through a hole the size of a dime? Find a pair of coins in another system of money that could be used as "magic coins" - the "biggest" and "smallest" coins for which this would work.
... more>>
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Make a Square!
- Annie Fetter
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Geometry, difficulty level 3. Make two cuts in the given figure and rearrange the pieces to make a square.
... more>>
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Making a Map
- Annie Fetter
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Geometry, difficulty level 3. Given a list of relationships between five different objects, explain which ones you could uniquely place on a map.
... more>>
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Making a Six Pointed Star
- Annie Fetter
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Geometry, difficulty level 3. Take a circle of paper. Explain how to fold it so that you can make just one cut and get a "perfect" six-pointed star.
... more>>
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Making Pie Crust
- Annie Fetter
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Geometry, difficulty level 3. How would you change the recipe for a 9-inch pie crust to make an 11-inch pie crust?
... more>>
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Making Scalene Triangles
- Annie Fetter
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Geometry, difficulty level 2. A scalene triangle has sides of 8, x, and 19. The sides are listed in order from smallest to largest. What are all of the possible integer values for x?
... more>>
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Making the Biggest Triangle
- Lillian Ray
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Geometry, difficulty level 2. What length should the third side be to make the triangle with the
maximum possible area?
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Making Triangles I
- Annie Fetter
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Geometry, difficulty level 2. Take a scalene triangle. The lengths of the sides, in size order, are 5, x, and 15. What are the possible integral values for x?
... more>>
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Making Triangles II
- Annie Fetter
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Geometry, difficulty level 2. Given a scalene triangle with sides, listed from shortest to longest, of 6, x, and 16, what are all the possible integer values for x?
... more>>
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Measuring Cylinders
- Annie Fetter
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Geometry, difficulty level 2. Help Ian measure his stove pipe and his propane tank.
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Measuring the Circumference of the Earth
- Annie Fetter
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Geometry, difficulty level 3. At our office on the autumnal equinox, the angle formed by the sun's rays and a stick is 40 degrees. If our office is 2749 miles from the equator, what's the circumference of the Earth?
... more>>
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Measuring the Matterhorn
- Annie Fetter
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Geometry, difficulty level 2. Use some right triangle trigonometry to calculate the height of the
Matterhorn at Disneyland.
... more>>
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The "Middle" Triangle
- Annie Fetter
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Geometry, difficulty level 2. In rectangle ABCD, E is the midpoint of BD, F is the midpoint of ED, AD is 10 cm, and AB is 20 cm. What's the area of AEF?
... more>>
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A Minor Problem
- Annie Fetter
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Geometry, difficulty level 1. A chord of a circle is one side of an equilateral triangle. The other
two sides of the triangle are radii of the circle. Find the length of
the minor arc subtended by the chord, if the radius of the circle is
given.
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Mirror, Mirror
- Annie Fetter
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Geometry, difficulty level 2. In the given image, the green polygon is a reflection of the purple
polygon. Find the equation of the line of reflection.
... more>>
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More Pie, Anyone?
- Annie Fetter
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Geometry, difficulty level 2. How big a pie pan would you need if you made twice as much dough as you would need for a 9-inch pan?
... more>>
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More Tangents!
- Annie Fetter
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Geometry, difficulty level 3. Given a square ABCD inscribed in a unit circle. AB is extended to P, where PC is tangent to the circle. Find the length of PD.
... more>>
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Mowing My Lawn
- Annie Fetter
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Geometry, difficulty level 3. My lawn is 75 feet by 42 feet, my lawn mower cuts 18 inches at a time. I am mowing in a circular pattern, and have finished part of the lawn. That part took me 8 minutes. How much longer will I have to mow to finish the lawn?
... more>>
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My Mom Rocks!
- Annie Fetter
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Geometry, difficulty level 3. Help my mom decide if she's buried the stones in her new garden deep enough.
... more>>
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Name This Quadrilateral!
- Annie Fetter
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Geometry, difficulty level 3. A quadrilateral is drawn on a piece of paper. It has one pair of opposite sides congruent, the other pair not congruent, and a pair of opposite angles that are supplementary. What is this figure, and how do you know?
... more>>
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Napoleon's River
- Annie Fetter
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Geometry, difficulty level 2. A soldier uses the visor of his cap to measure the distance across the river.
... more>>
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Nautical Navigation
- Annie Fetter
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Geometry, difficulty level 2. Help the sailors of the Kaleidoscope figure out what time
they'll pass the lighthouse and how far from the lighthouse they'll be
at that time.
... more>>
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Nets of a Cube
- Annie Fetter
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Geometry, difficulty level 2. When you fold up the given net of a cube, what shape is formed by the segments on the faces of the cube?
... more>>
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