Ask Dr. Math

High School Archive

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.
    Classifying quadrilaterals.
    Heron's formula.
    Polygon diagonals.
    Pythagorean theorem proofs.
    Triangle congruence.

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Measuring by Shadows [05/22/2001]

How can I measure a tree using its shadow and mine?

Measuring the Height of a Building Using Shadows [05/24/2000]

What time of day is best to use a shadow to measure the height of a building by using triangles?

Median and Altitude Constructions [07/16/2003]

How to draw a median and an altitude from the three sides of a triangle.

Medians of a Triangle [02/16/1999]

Prove that the 3 medians of a triangle divide themselves up into a ratio of 1:2.

Medians of Triangles Proof [05/29/2000]

Prove that in any triangle, the sum of the medians is more than 3/4 of its perimeter, but less than the whole perimeter.

Menelaus's Theorem [01/25/1999]

A straight line intersects sides AB, BC and the extension of side AC of a triangle ABC at points D, E and F respectively. Prove that the midpoints of the line segments DC, AE and BF lies on a straight line.

Mid-segment Theorem [02/02/1999]

Can you help me prove the Mid-Segment Theorem?

Minimal Distances to a Point in a Triangle [01/05/2001]

How can I prove that the smallest value of PA + PB + PC occurs when angle APB = angle BPC = angle CPA = 120 degrees, for a triangle ABC and a point P?

Minimum Angle Proof [07/05/2001]

Label the point of intersection of the angle bisectors of triangle ABC as Q. Let M be the midpoint of side BC. Given that MQ = QA, find the minimum value of angle MQA.

Miquel Circles [09/17/2003]

Given an acute triangle ABC, consider all equilateral triangles XYZ, where points A, B and C lie on segments XZ, XY, YZ. Prove that all centers of gravity of all these triangles XYZ lie on one circle.

Mixtilinear Incircle Proof [12/11/2000]

In triangle ABC, AB = AC. A circle is tangent internally to the circumcircle of triangle ABC and also to sides AB and AC at points P and Q, respectively. How can I prove that the midpoint of the segment PQ is the center of the mixtilinear incircle of triangle ABC?

Nagel Point [04/15/2001]

What relation does the Nagel Point have to the incenter, centroid, and Spieker point of a triangle?

Naming Corresponding Parts of Congruent Figures [04/23/2003]

Given triangle OPS congruent to triangle TQR, name the corresponding sides and angles of the two triangles.

Naming the Isosceles Triangle [09/23/1998]

How did the isosceles triangle receive its name?

The Napoleon Point and More [09/04/1998]

How do you prove that the Napoleon point will always exist, given the proper conditions? Is there a stronger theorem?

Napoleon's Triangle [08/10/1999]

What is Napoleon's triangle?

Nine-Sided Polygon [06/11/1997]

Can you construct a regular 9-sided polygon inside a circle using only a compass and straight-edge?

Nonagon or Enneagon? [02/06/2003]

Is 'enneagon' really the correct name for a 9-sided polygon?

Non-Congruent Triangles [12/12/2001]

Construct and prove that there can be two non-congruent triangles in which five parts of one triangle are equal to five parts of another.

Nonconvex Polygon Angle Measure [02/03/1999]

What is the formula to find the interior angle measurements of a nonconvex polygon?

Number of Lines of Symmetry in a Regular Polygon [03/12/1998]

In a regular polygon, is the number of lines of symmetry the same as the number of lines or angles of that polygon?

Number of Points in a Star [7/16/1996]

Is there a way to predict the number of points in a star given only the internal angle of the corners?

Octagon Construction Using Compass Only [02/22/2002]

Construct the vertices of a regular octagon using just a compass. The only thing you know about the octagon is the circumradius.

Octagon Formula [07/30/1997]

If you're building an octagon on a 12-foot radius, what is the length of each side?

Octagon Side Lengths [08/22/2001]

If I know that the dimension of an octagon from one side to the other is 8 feet, how can I find the lengths of a side?

One- and Two-sided Polygons [12/07/1999]

Can you explain what a monogon and a digon are?

Order of a 3D Triangle [08/29/2001]

If I visit the vertices of a 3D triangle in order going from a to b to c, am I going clockwise or anticlockwise?

Origami Equilateral Triangle [04/26/2001]

How can I create an equilateral triangle from a piece of paper using only origami-like folds?

Orthic Triangle [04/09/1999]

How do you find the angles of the triangle ABC, similar to triangle A1B1C1, where AA1, BB1, and CC1 are the altitudes of triangle ABC?

Overlapping right triangle problem [09/14/1997]

Given right triangles ABC and DCB with rt angles at B and C, triangle ABC's hypotenuse 20 and triangle DCB's hypotenuse 30. The hypotenuses intersect at point E, a distance of 10 from BC. Find the length of BC.

Parallelogram Perimeter [10/01/1997]

The diagonals of a parallelogram are 10 and 24 in length. If one side of the paralellogram is 13, what is the perimeter?

Parallelogram Side Length [08/12/2003]

In a parallelogram ABCD, K is a point on AB such that angle DKC = 90 degrees and AD = AK. If AD = 10cm, find the length of AB.

Pascal's Theorem [10/07/1998]

Can you explain Pascal's Theorem? How does it relate to conic sections?

Pedan Trapezium [05/23/2000]

How can I prove that a isosceles trapezium whose parallel side lengths are 7 and 4 respectively, and whose slant sides have length 6, is a Pedan trapezium?

Pentagon Area Using No Trig [05/14/2001]

Where I am stumped is finding the area of one of the five triangles.

Perimeter Equals Area in a Triangle [4/2/1996]

When will the area and perimeter of a right triangle be numerically equal?

Perimeter of an Inscribed Regular Polygon [12/10/1998]

What is the formula for the perimeter of a regular polygon inscribed inside a circle?

Perimeter of a Right Triangle [08/14/2001]

What is the perimeter of a right triangle with hypotenuse 65 that can be circumscribed about a circle with radius 12?

Perimeter of a Strange Figure [03/23/2002]

I don't understand how to find the area or perimeter of a shape that looks like a rectangle with a small triangle to the left on top...

Perimeter of Octagon [04/02/1997]

What is the perimeter of a an octagonal garden with a diameter of six feet?

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