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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.
- Pi and Polygons [03/14/1999]
Derive a formula to find the angle of an n-sided polygon with x sides.
- Pick's and Euler's Theorems [05/06/1999]
What is Pick's theorem and how can it be linked with Euler's theorem?
- Pick's Formula [05/22/1997]
What is Pick's Formula?
- Pick's Theorem [9/4/1996]
Could you explain Pick's Theorem?
- Picture Frame, Triangle Measurements [5/20/1996]
My teacher gave us ten questions to answer and I could do all except two:
1) A framed rectangular picture is 35cm long and 25cm wide... 2) The base
of a triangle is 9cm more than the perpendicular height...
- Picture of Icosahedron [7/31/1996]
Do you have a picture of an icosahedron?
- Pipe Needed for Natural Gas Line [09/03/1997]
Two houses are 130 metres apart at distances of 55 and 105 metres...
- Pixels in a Triangle [08/02/1999]
How can I find the number of pixels inside a triangle?
- Placing Coins That Touch [8/7/1996]
How many 20-cent coins can you put around a 20-cent coin so that all of
- A Point Inside a Square [09/18/2003]
Point P lies inside square ABCD such that P's distance from A is 1,
P's distance from B is 4 and P's distance from C is 5. What is the
area of the square?
- Point Inside or Outside Triangle? [08/02/2001]
I have coordinates of three vertices of the triangle and coordinates of
- A Point in the Triangle [02/12/1999]
Finding the point P in a plane of triangle ABC, where PA + PB +PC is
- Point within a Triangle [05/29/2003]
I have the coordinates of the three corners of a equilateral triangle
ABC. How can I decide whether an arbitrary point (X,Y) lies in the
plane of the triangle?
- Pole in a Box [02/09/1999]
Can a pole 6.5m long fit into a truck with dimensions of 3m, 3.5m, and
- Polygon Algorithms [05/10/2001]
Given a polygon as a set of points (X, Y) and a database table with X and
Y columns, select all records/points from the table that are inside the
polygon or belong to its border.
- Polygon Diagrams [05/11/2003]
I would like to see pictures of polygons with 11, 12, 13, 14, and 15
- Polygon Names II [12/11/2003]
Why is the triangle named "triANGL"', unlike all the other polygons, which have names like "quadriLATERAL" or "pentaGON"?
- Polygons and Triangles [03/09/1999]
Prove that after splitting a regular n-polygon into n triangles, the
isosceles triangles have greater area than the scalene triangles.
- Polygons, Infinite Sides, and Circles [04/03/1997]
Can a regular polygon with an infinite number of sides be a circle?
- Polyominoes [09/08/1997]
I am using polyominos, but I do not know how to tell my dad what they
are. How can I tell him so he will know?
- A Practical Use for the Orthocenter [03/07/2001]
Does the orthocenter of a triangle have any practical uses?
- Probability in the Infinite Plane [03/29/2003]
Three randomly drawn lines intersect so as to form a triangle on an
infinite plane. What is the probability that a randomly selected point
will fall inside that triangle?
- Probability of Making Triangle with Three Stick Pieces [02/24/2007]
Take a stick of unit length and break it into two pieces, choosing the
break point at random. Now break the longer of the two pieces at a
random point. What is the probability that the three pieces can be
used to form a triangle?
- Proof by Contradiction [09/25/1997]
Prove that no isoceles right triangle exists which has all three sides
- Proof: Median of a Trapezoid Theorem [03/24/2001]
Prove that the median of a trapezoid is: 1) parallel to its bases; 2)
length = 1/2 the sum of the bases.
- Proof of Congruency [10/13/1996]
Line PR bisects angles QPS and QRS; prove that segments RQ and RS are
- Proof of Hero's formula [09/08/1997]
Could you tell me where to find a proof of Hero's formula or help on how
to derive it?
- Proof of Morley's Theorem [08/09/2000]
How can I prove Morley's theorem (if every angle in a triangle is
trisected, each pair of trisectors meets in a point, and all three points
form the vertices of an equilateral triangle)?
- Proof of Pappus' Theorem [04/02/2001]
How can I prove Pappus' theorem of colinearity with the help of Menelaus'
- Proof of Pythagorean Theorem: Pythagoras' Reasoning [02/23/1998]
I would like to know how Pythagoras reasoned his theorem.
- Proof of the Feuerbach Theorem [03/14/2000]
Please submit the proof of the Feuerbach theorem (the nine-point circle
is tangent to the incircle and the circumcircle of a triangle.)
- Proof of the Parallelogram Law [08/07/1999]
How do you prove the parallelogram law geometrically, without using
- Proofs with Isosceles Triangles [10/28/1998]
What are altitudes, angle bisectors, and medians? How do you prove that
in an isoseles triangle, the altitude is a median and an angle bisector?
- Proof Using Coordinate Geometry [06/08/2001]
In equilateral triangle ABC, a segment is drawn from point A to the side
BC at a point (D)... prove that angle BFC (or EFC) is a right angle.
- Properties of Equilateral Triangles [07/20/1998]
If ABC is equilateral and AD is one of its heights, what are the measures
of the angles? Is ADB equal to ADC? If AB = 2 find BD and AD.
- Prove Triangle of Sides with Length... [8/1/1996]
Let a, b and c be the lengths of the sides of a triangle. Prove that the
square root of (a+b-c) plus the square root of (b+c-a) plus the square
root of (c+a-b) is equal to or less than the sum of the square roots of
a, b and c. Determine when equality occurs.
- Prove Triangles are Similar [12/16/1995]
How do you prove two triangles are similar?
- Proving Lines Congruent [03/29/2002]
Prove line AL is congruent to line CM.
- Proving Quadrilateral is a Parallelogram [11/30/2001]
We are having a problem with the idea of a quadrilateral having one pair
of opposite sides congruent and one pair of opposite angles congruent.
- Proving the Diagonals of a Rectangle Congruent [12/6/1995]
How would you prove that the diagonals of a rectangle are congruent?