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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.
- Irregularly Inscribing a Circle ... of What Radius? [12/24/2011]
A student struggles to determine the radius of a circle given the sidelengths of an
inscribed irregular polygon. First, Doctor Greenie offers a numerical approach; then,
Doctor Floor follows up with the desired analytical method, applying Ptolemy's theorem
and the rational root theorem to a key insight: the radius of the circumcircle remains
invariant under a re-ordering of the polygon's sidelengths.
- Is a Circle a Polygon? [03/07/1999]
A circle is not a polygon.
- Is an Equilateral Triangle Also Isosceles? [02/19/2008]
Can equilateral triangles also be classified as isosceles? In other
words, does an isosceles triangle have exactly two equal sides or at
least two equal sides?
- Is it Possible to Prove that... [10/28/1997]
... if the hypotenuse of a right angle triangle is divisible by 4, the
legs are also divisible by 4?
- Is Kite the True Name? [03/29/2002]
Is kite the true math name for this shape, or is there another?
- Isosceles Trapezoid [6/23/1996]
Bases AB and CD of an isosceles trapezoid ABCD are 12 units apart...
- Isosceles Trapezoid Proof [01/18/2002]
Given: ABCD is an isosceles trapezoid with bases BC and AD. Prove: ABCD
is an isosceles trapzoid.
- Isosceles Triangle - Angles [11/06/1996]
Given that one base angle of an isosceles triangle is 39 degrees, find
the measure of the other two angles in the triangle.
- Isosceles Triangle Maximizes Area? [09/11/2003]
How can you show that among all triangles having a specified base
and a specified perimeter, the isosceles triangle on that base has
the largest area?
- Isosceles Triangle Proof [05/14/2006]
Given triangle ABC, with D on BC and AD bisecting angle A. The center
of the circle circumscribing ABC is the same point as the center of
the circle inscribed in ADC. Prove that ABC is a isosceles triangle.
- Isosceles Triangles [2/8/1996]
A student asks how to find angle B of a given isosceles triangle.
- Is This a Square? [01/30/2001]
Given four points on a graph, what can I do to verify this is a square?
- Kitchen Tabletop [11/21/2001]
I need to determine the correct pivot point...
- A Ladder Puzzle [10/20/2000]
A 10-meter ladder is leaning against a wall just touching the corner of a
3-meter cube placed flat against the wall. At what height does the end of
the ladder touch the wall?
- Largest Triangle in a Square [10/31/1998]
If the area of a square is 1, what is the largest area of a triangle
constructed inside the square? How would you prove it?
- Lattice Points in a Rectangle [06/04/1999]
How can I prove that in any rectangle centered at (0,0) with an area
greater than 4, you can find lattice points other than (0,0)?
- Lattice Points on Hypotenuse [10/01/2001]
What is the number of lattice points on the hypotenuse of a right
- The Law of Margins [9/2/1996]
How do I figure out the margins of my mat using the Golden Ratio?
- Learning to Read Proofs [10/31/2003]
How do you know what step comes next in a proof?
- Least Perimeter [10/13/2002]
Finding a formula for least perimeter of a square or rectangle.
- Left-Sided Rhombuses in a Larger Rhombus [05/22/2000]
How many left-sided, right-sided, and vertical rhombuses can be found in
a larger NxN rhombus?
- Leg of a Triangle [04/02/2002]
I need to know where the name "leg" of a triangle comes from, or what its
- Length of a Triangle's Sides [1/23/1995]
I have a triangle problem for you to solve: The lengths of the three
sides of a triangle could be...
- Length of the Diagonals of a Parallelogram [05/22/2000]
A parallelogram has a 70-degree angle and sides 6cm and 10cm long. How
long are its diagonals?
- Limited Area, Unlimited Perimeter [11/27/1997]
What is the figure?
- Limit of Area [03/01/1998]
Limit approached by area of a square when its sides are repeatedly
divided into three congruent parts and squares are constructed outwardly
on the middle parts.
- Linear Systems of Equations in Two Variables [06/18/1999]
How can I find the length of AE, EB and DC, given that parallelogram ABCD
has a perimeter of 50, trapezoid AECD has a perimeter of 39, and AE = EC?
- Line Dividing a Plane [05/11/2001]
Given a square (graphed on the Cartesian coordinate system) and a point
in the square, draw a line through the point that will divide the square
into two regions: one the smallest area possible, the other the largest
- Lines Intersecting within a Polygon [10/24/1996]
Given an n-sided regular polygon with all vertices connected to each
other by straight line segments, how do you determine the number of
intersection points within the polygon?
- Lines of Symmetry [01/29/2001]
How do you find the number of lines of symmetry there are in a polygon?
- Lines of Symmetry in Regular Polygons [03/13/2001]
Is there a formula for finding all the lines of reflectional symmetry in
- Locus [05/03/1999]
What is a locus?
- Logarithms and the Area of a Triangle [6/3/1996]
Is is true that if A is the area of a triangle, then....log(A) = ...?
- Longest Ladder [11/30/2001]
Two hallways, one 8 ft. wide and other 4 ft. wide, meet to form a right
angle. What is the longest ladder that can go around the corner where the
- Ludolph van Ceulen and Pi [11/02/1998]
How did Ludolph van Ceulen estimate pi by inscribing and circumscribing a
circle with squares?
- Magic Triangle Puzzle [07/26/2002]
Where did the white square come from?
- Making a Pool Tarp [9/27/1995]
I'm trying to construct a pool frame out of pvc that will be placed over
a pool. We already have the pool tarp, but we need to build something
that will shed water so it has to have a slope. PVC only comes in 90
degree and 45 degree angles and straight sections. How do I figure out
the maximum height the frame can stick straight up in the air to utilize
- Math and Sports [02/26/1997]
Can you give me information on how math relates to sports?
- Maximize Area of Trapezoid [07/28/2003]
Given an isosceles trapezoid with three of its sides of length 10 cm,
find the fourth side so that the area is maximized.
- Maximizing Irregular Polygon Area: Which Circle? [05/01/2011]
How do you determine the radius of the circle that maximizes the area of an irregular
n-gon circumscribed on it? With the Pari computer algebra system, Doctor Vogler
approaches the question using numerical techniques such as Newton's Method and a
binary search, which suggests that no closed-form expression exists.