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Dr. Math FAQ:
Browse High School Triangles and Other Polygons
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Area of an irregular shape.
Pythagorean theorem proofs.
- Calculating How Much a Tablecloth Will Overhang the Table [07/27/2006]
Will a tablecloth that is 54' x 126' hang to the floor when put on a
table that is 8' x 3'?
- Calculating Polygon Area [01/11/2004]
How can you determine the area of an unusually shaped polygon?
- Calculating Square Footage or Area [01/08/2007]
In my job I am often asked to figure out square footage. What are the
formulas to find the areas of rectangles, triangles, and odd shapes?
- Calculating the Area of a Hexagon [2/10/1996]
Find the area of a hexagon given only its width between two parallel sides.
- Calculus Cylinder-Cone Problem [07/13/1999]
I have to find the altitude of a triangle...
- Can a Circle be a Polygon? [5/22/1996]
Could a circle be considered a polygon with an infinite number of sides?
- Carpet and Room Areas [10/26/2001]
A man buys a roll of carpet 9 ft. wide by 12 ft. long to fit a 10ft. by
10 ft. room. When the roll of carpet is unrolled, a hole is discovered in
the middle of the carpet...
- Carpeting an Irregular Hexagon [07/31/2003]
My office is an irregular hexagon. Is there a way to determine the
area of the office, given only the lengths of the sides and the angles
of each of the six corners?
- Carpet Problem [10/15/2001]
You have to carpet a 9x12 room, but when you go the store they only have
a 10x10 carpet and a 1x8 piece of carpet...
- Centroid, Circumcenter, Incenter, Orthocenter: Etymologies [01/20/2002]
Why are the points centroid, circumcenter, orthocenter, and incenter
named as they are, and are there any other special points associated with
- The Centroid of a Triangle [02/25/1998]
WHY is the centroid of any triangle the center of its balance?
- Ceva's Theorem [03/04/1999]
Prove Ceva's Theorem using vector methods and use it to prove the
concurrency of the medians, altitudes, or interior angle bisectors of a
- Chanukah hexagons [12/12/1994]
I gave the students the Star of David for Chanukah. We tried to find all
the triangles, quadrilaterals, and hexagons in this star. We were stumped
with the number of hexagons. Can you help?
- Characteristics of an Orthocenter [11/12/1999]
What are some characteristics of the orthocenter of a triangle?
- Chords From Inscribed Polygons [07/11/2002]
An regular polygon is inscribed in a circle of known radius. Each
side of the octagon is a chord of the circle. What is the length of
- Circle and Polygons: Lines of Symmetry [04/14/1997]
How many lines of symmetry are there in a circle?
- Circle Inscribed in a Right Triangle [09/09/1997]
What is the diameter of the circle if the legs of the triangle are known
to be A and B?
- Circle Inscribed in Triangle [04/04/1997]
What is the radius of a circle inscribed in a 3-4-5 right triangle?
- Circles in a Square [09/15/2001]
A circle of radius 1 is inside a square whose side has length 2. Show
that the area of the largest circle that can be inscribed between the
circle and the square is (pi(17 - 12sqrt(2))).
- Circles Inscribed in Triangles [11/14/1996]
Given two triangles, prove that r1 + r2 + r3 = r.
- Circumference of a Square [05/13/2002]
A circle has a circumference C. Find, in terms of C, the perimeter
of a square having the same area as the circle.
- Classical Geometry [04/16/2002]
Let ABC be a triangle with sides a, b, c. Let h be the perpendicular
from A to a, and m the median from A to the midpoint of a. Construct
the triangle using only ruler and compass if you know A, h, m.
- Classification of Quadrilaterals [02/14/2008]
Is there a correct hierarchy to classifying quadrilaterals by shape?
I've seen several different ones that each seem correct.
- Classifying Triangles [06/04/2002]
Given the lengths of the sides of a triangle, determine whether the
triangle is acute, right, or obtuse.
- Cleaning the Ice [09/09/1997]
The hockey rink is a rectangle, 120 ft. by 60 ft. The scraper cleans a 4-
ft.-wide strip... on which trip will it have cleaned half the area of the
- Coin With 11 Sides and a Constant Diameter [10/20/2007]
I was told that one of the reasons why a Canadian Loonie coin is
11-sided is that it rolls better than a 10-sided coin. Why is that?
- Collinearity [04/20/1999]
What points in a triangle are known to be collinear with the incenter?
- Compass, Ruler, and Radius — of a Sphere [09/16/2016]
A recreational mathematician seeks help calculating the radius of a sphere using only
compass and ruler. After considering several approaches, Doctor Peterson cracks the
case by invoking some triangle geometry and the law of cosines.
- Completing the Square: a Diagram [03/06/2002]
Show x^2 + 3x using a diagram.
- Complex Ratio Problem [07/31/1997]
If you randomly throw 3 points on a plane, you get a triangle... What is
the probability that the triangle will become obtuse...?
- Congruency Theorems for Triangles [3/13/1995]
Two triangles, one which has two sides that are of equal length to the
second triangle, and both having an angle (not contained) equal, cannot
be proved congruent. It seems to me that they are congruent, though. Any
thoughts on this?
- Congruent and Similar Triangle Theorems [05/14/1998]
Is there an Angle-Angle-Angle (AAA) triangle congruency theorem?
- Congruent Parts Congruent Triangles Congruent (CPCTC) [11/28/2001]
When should we use CPCTC, and how does it prove anything?
- Congruent Triangles [6/26/1996]
If two triangles have the same area and the same perimeter, must they be
- Congruent Triangles in a Rectangle [11/11/1999]
Given rectangle BART with AB parallel to RT, AR perpendicular to AB, BT
perpendicular to RT, AB congruent to RT, and AR congruent to TB, how can
I prove that triangle ABR is congruent to triangle TRB?
- Congruent Triangles in Parallelograms: Proof [06/29/2003]
O is a point inside triangle PQR. The parallelograms QORX, ROPY, and
POQZ are drawn. Prove that triangle PQR is congruent to triangle XYZ.
- Congruent Triangles - SSS Test [11/16/1998]
How do you know if two triangles are congruent?
- Construct a Trapezoid [08/28/2001]
I tried drawing two lines that are parallel to each other for b and f,
and I drew c, but then d didn't fit. How do I construct this?
- Constructible Angles and Regular Polygons [04/17/1998]
What angles and regular polygons are constructible?
- Constructing a Line to Divide Area of a Triangle in Half [05/13/1998]
Cutting a triangle into two pieces of equal area by drawing a a line
parallel to one of the sides.