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Polygons
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The problems in this category require knowledge of the properties
of regular and non-regular polygons. Measures of interior,
exterior, and central angles form the basis of many of the problems.
Many of the problems in this category will require advanced knowledge
and an integration of several geometric topics.
Problems about
triangles and
quadrilaterals
are listed in their own categories.
- Related Resources
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Interactive resources from our Math Tools project:
Geometry: Polygons
The closest match in our Ask Dr. Math archives:
High School: Triangles/Polygons
NCTM Standards:
Geometry Standard for Grades 9-12
Access to these problems requires a Membership.
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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?
... more>>
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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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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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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.
... more>>
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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.
... more>>
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Drawing Out a Decagon
- Annie Fetter
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Geometry, difficulty level 2. Extend sides AB and IH of the regular decagon ABCDEFGHIJ until they intersect. What is the measure of the acute angle at this intersection?
... more>>
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Examining an Octagon
- Annie Fetter
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Geometry, difficulty level 3. Describe the rectangles that you would need to put on the sides of a square in order to be able to end up with a regular octagon when you connect the outside vertices of the figure.
... more>>
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Extending the Enneagon
- Annie Fetter
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Geometry, difficulty level 3. Extend the sides AB and ED of the regular enneagon ABCDEFGHI until they intersect. What is the measure of the acute angle at this
intersection? Extend sides AB and EF. Now what's the measure of that acute angle at the intersection?
... more>>
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Fix This Picture
- Annie Fetter
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Geometry, difficulty level 2. Figure out what's wrong with this picture that includes a circle and a rhombus.
... more>>
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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.
... more>>
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The Hexagonal Cross Section of a Cube
- Annie Fetter
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Geometry, difficulty level 3. Explain how to get a hexagonal cross section of a cube. If the cube has edgelength one, what are the area and perimeter of the hexagon?
... more>>
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A Hexagon and a Triangle in a Circle
- Annie Fetter
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Geometry, difficulty level 2. A regular hexagon and an equilateral triangle are both inscribed in the same circle so that the hexagon and the triangle share three vertices. The radius of the circle is 10 units. What is the area of the region between the two polygons?
... 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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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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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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Splitting a Hexagon
- Annie Fetter
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Geometry, difficulty level 2. Split a regular hexagon into three identical parts. What shape is each part? Split a regular hexagon into six identical parts, at least two different ways. What shapes are you r pieces? Split a regular hexagon into six identical kites.
... 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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What's This Polygon?
- Annie Fetter
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Geometry, difficulty level 4. Triangle PQR is inscribed in a circle. If PR is a side of a 21-sided inscribed regular polygon, PQ is a side of a 28-sided incribed regular polygon, and QR is a side of an n-sided inscribed regular polyon, find a possible value for n.
... more>>
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