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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.
- Determining If a Given Point Lies inside a Polygon [12/27/2005]
I have a finite number of points that constitute a polygon, and a
point p(x,y). I want to know if point p lies inside the polygon.
- Determining Triangle Similarity [05/26/1998]
Given two triangles, how can you determine if they are similar?
- Diagonals and Axes of Symmetry [03/31/1998]
Could you explain the concepts behind the diagonals and axes of symmetry
in a regular octagon?
- Diagonals and Tiles [11/17/2001]
Jay tiled a 15x21' rectangular ballroom with 1 ft. sq. tiles. Then he
drew diagonals connecting opposite corners of the room. How many tiles
did the diagonals pass through?
- Diagonals of Polygons [05/21/1997]
How many diagonals does a polygon with n sides have?
- Diameter of a Circle Circumscribed Around a Triangle [05/13/1998]
Applying the Pythagorean Theorem to find the diameter of the circle
circumscribed around a triangle with side lengths 25, 39, and 40.
- The Diameter of an Octagon [7/25/1996]
What is the diameter of an octagon with each side length equal to 2
- Diameter of Flying Saucer [5/27/1996]
We are constructing an oval racetrack in Atlanta...
- Diameter of the Base of a Cone [08/12/1998]
How do you find the formula to calculate the diameter of the base of a
cone of nine degrees at various lengths?
- Different Answers with Sine Rule and Cosine Rule [08/07/2002]
Triangle ABC has measurements AB=8.2cm, BC=9.4cm and AC=12.8cm and
angle A=47 degrees. Find angle B. Why does using the Sine Rule here
fail to give the right answer?
- Different Triangles With the Same Base and Height [04/16/2004]
Why do two differently shaped triangles with the same base and height
have the same area?
- Dimensions of Rectangle [7/10/1995]
What are the dimensions of a rectangle if the area is equal to the
- Dissecting a Square into Acute Triangles [11/09/1999]
Can you dissect a square into a finite number (fewer than 14) of acute
- Distance between Two Men [04/24/2003]
Two men, starting at the same point, walk in opposite directions for 4
meters, then left, and walk another 3 meters. What is the distance
- Distances from a Point inside an Equilateral Triangle [12/09/2001]
Prove that the sum of the distances from a point inside an equilateral
triangle, measured parallel to the sides, is equal to the length of the
side of the triangle.
- Distance to an Object [04/07/2000]
Is there an easy way to measure the distance from a baseline to an object
if one knows the measurement of the baseline and both angles leading
toward the object?
- Distance to the Corner of a Rectangle [05/01/2001]
How can I find the distance from a point P inside a rectangle to the
fourth corner if the distance from P to one corner is 3, from P to the
opposite corner is 5, and from P to a third corner is 4?
- Distance to the Horizon [09/18/1997]
A 6-foot man is standing on the beach at sea level looking straight out
to sea. How far can he see - i.e. what is the distance from the man to
- Distance to the Horizon [05/23/1999]
How far is the horizon?
- Dividing a Square Cake into Five Equal Pieces [07/28/2001]
How can you divide a square-topped cake that is a rectangular solid and
is frosted on all faces into five pieces so that everyone receives the
same amount of cake and icing?
- Dividing a Square in Thirds [03/27/2001]
How can I divide a square into three equal pieces using three lines
radiating from the center of the square?
- Dividing Regular Shapes [10/22/1996]
If every vertex of a triangle is joined by straight lines to 6 points on
the opposite side of the triangle, how many regions are formed? If every
vertex of a regular pentagon is connected to every other vertex, how many
triangles are formed?
- Do External Angles of Polygons Always Sum to 360 Degrees? [02/13/2007]
I know that the sum of all external angles of all polygons is supposed
to be 360 degrees, but can you explain how a polygon with 16 right
angles can have a total external angle of 360 degrees?
- Do Figures with Equal Sides Have the Same Area? [11/20/2008]
If a rectangle is tipped slightly so the corners are not square but
the side lengths don't change, does the area of the figure change or
stay the same?
- Drawing Triangles [06/18/1997]
Is it possible to draw a triangle with more than 180 degrees?
- Ellipse Bounding A Rectangle [7/15/1996]
How do I calculate the ellipse bounding any given rectangle?
- Ellipses: Pythagorean Relationship [2/12/1996]
In an ellipse with major axis of 2a, minor axis of 2b, and foci c (on the
major axis), the relationship c squared = a squared - b squared holds
true... how do the three numbers fit into a Pythagorean relationship?
- Enlarging the Penguin Pond [06/03/1999]
By how much should its length and width be increased to double the area
of the 12 meter x 8 meter penguin pond at the zoo?
- Equable Polygons [10/29/2001]
I need a formula for finding equable polygons. I know the formula 4/
tan(90-180/n), but how do you get to this point?
- Equable Shapes: Triangle [08/27/2001]
I have been asked to do a piece of coursework on equable shapes, but I am
stuck on the triangle.
- Equal Area and Perimeter: Rectangles [09/09/2001]
There are only two rectangles whose area is exactly the same as their
perimeter if the dimensions of each are whole numbers. What are the
- Equal Parallelians Point [12/09/2002]
Within triangle ABC, draw three segments parallel to the sides of the
triangle, each touching two sides. The three segments meet at one
point, and they are all the same length, x. Find the length of x given
the length of the sides of triangle ABC.
- Equation of a Rectangle? [11/11/2012]
A student wonders if there is an equation for a rectangle in the Cartesian plane,
comparable to the one for the equation of a circle. Doctor Peterson confirms the student's
suspicions about the difficulties involved.
- Equilateral, Isosceles, Scalene - Word Origins [12/09/2001]
I need to find out about the origins of the scalene, isoceles, and
equilateral triangles. How they were named?
- Equilateral Shapes Inscribed in a Circle [04/07/2003]
Is there a general formula for the length of a side of an equilateral
shape that is inscribed in a circle?
- Equilateral Triangle: Area Formula and Proof [06/16/1998]
Is there a formula to find the area of an equilateral triangle given a
point on its interior and the lengths of the segments from the point to
the three vertices?
- Equilateral Triangle - Centroid/Incenter [09/27/2001]
Prove that if a triangle is equilateral, its centroid coincides with its
incenter, and vice versa.
- Equilateral Triangle Proof [11/05/2001]
Let a, b, and c be the lengths of the sides of a triangle. Show that if
a*a + b*b + c*c = bc + ca + ab, the triangle is equilateral.
- The Erdos-Mordell Theorem [10/13/2000]
Let P be a point in a triangle, D be the sum of the distances from P to
the 3 vertices, and E be the sum of the distances from P to the edges.
How can I prove that D is greater than or equal to 2*E?
- Etymology of the Word Hypotenuse [01/26/2003]
Why is the hypotenuse called 'hypotenuse'?