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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Euler Line and Nagel Point [07/20/1998]

Can you provide more information on the Euler line and the Nagel point, including proofs?

Euler Line Proof [11/13/2001]

Prove that if the Euler line of a triangle passes through a vertex, then the triangle is either right or isosceles.

Euler's Line Theorem [04/08/2001]

Prove that the circumcenter, orthocenter, and centroid of any triangle lie on the same line using analytical geometry.

Euler's Nine-point Circle [02/21/1999]

What is the "nine-point circle" problem?

Evaluating a Trigonometric Expression [07/31/1999]

Can you help me evaluate tan (arcsin (3/5))?

Existence of the Brocard Point [06/06/2002]

Demonstrate that the Brocard point exists in any triangle.

Explanation and Informal Proof of Pick's Theorem [04/27/2004]

We just learned Pick's Theorem, A = b/2 + I - 1, where b is the boundary pegs, I is the interior pegs, and A is the area. I don't get why it works. Why do you divide the boundary by 2 and subtract 1?

Explanation and Test Case for Pick's Theorem [06/13/2006]

I don't really understand Pick's Theorem and its formula. Can you explain the formula and show how it works for a polygon?

Exploring the Distance from (0,0) to (1,1) with Limits [10/15/2006]

Any route traveling from (0,0) to (1,1) going only north and east will cover a total distance of 2 units. But the straight line distance from (0,0) to (1,1) is sqrt(2) units. It seems that if I think of a staircase connecting the two points and let the stairs become infinitely small, the limit of the north/east route distance should converge to sqrt(2). But it doesn't! What's going on here?

Exterior Angles in Triangles [09/11/1999]

How can you prove that in any triangle, each exterior angle is equal to the sum of the two nonadjacent interior angles?

Familiar Triangles [01/19/1999]

How do you get the lengths of the sides of a 45-45-90 triangle and a 30- 60-90 triangle?

A Fibonacci Jigsaw Puzzle [04/29/1999]

Why is the area of our rectangle, formed from a square, 65 when the square's area was 64?

Fibonacci Riddle [11/21/2001]

We can cut an 8x8 square with an area of 64 into four pieces and reassemble to get a 5x13 rectangle with an area of 65. Where does the extra 1x1 square come from?

The Figure of Maximum Area and Given Perimeter [06/02/1998]

Can you help me show, with and without calculus, that the geometric figure of a maximum area and given perimeter is a circle? What are the dimensions of a triangle with perimeter p that encloses the maximum area?

Filling a Garden with Topsoil [2/5/1996]

I have a garden that is 10' x 10' (100 square feet). I want to add 6" of topsoil to my garden. Topsoil is sold by the cubic yard. How many cubic yards of topsoil will I need for my project?

Find Angle ACB [04/20/2002]

Let A', B' and C' be points on triangle ABC such that AA' BB' CC' are angle bisectors. Suppose angle A'C'B' = 90 degrees. Find angle ACB.

Find angle DEB [5/27/1996]

Given an isosceles triangle ABC...

Find Angles, Area, Perimeter of a Parallelogram [03/23/2001]

I can't understand how to find indicated measures when I am given little information to begin with.

Find a Point 3/8 along the Line [12/04/2002]

Find a point 3/8 from A to B. Given: two endpoints X,Y coordinates. Point A (-2,7) point B (6,-5).

Finding a Missing Angle [01/23/2002]

Using trigonometry, calculate the measure of angles ABC and ACB.

Finding Angles without Using Trigonometry [08/27/2001]

Given the lengths of three sides of a triangle, determine the measures of the three angles using only geometry and algebra.

Finding Area and Volume [04/12/2001]

When working with area and volume of triangular shapes, how do I know when to divide the base by 2 and when to divide it by 3?

Finding Areas of Different Polygons [09/02/1997]

Could you please tell me how to work out the area for an equilateral heptagon, octagon, nonagon, decagon, unedecagon, and dodecagon?

Finding Polygon Areas [03/20/1997]

How do I find the area of polygons?

Finding Rhombi in a Rhombus [9/25/1995]

How can I work out a formula for finding how many rhombi there are in a rhombus? (say 2cm*2cm or 3cm*3cm and so on, etc.)

Finding Side Lengths of a Scalene Triangle [6/2/1996]

Two observers on points A and B of a national park see a beginning fire on point C. Knowing that the angles CAB = 45 degrees, ABC = 105 degrees and that the distance between points A and B is of 15 kilometers, determine the distances between B and C, and between A and C.

Finding the Area of an Irregular Polygon [02/23/2008]

What is the formula for finding the area of an irregular polygon?

Finding the Area of an Irregular Shape [01/03/2007]

I need the area of a parcel of land with 5 sides. I know the lengths of the sides and the angles at the corners, but am not sure how to calculate the area.

Finding the Area of a Regular Pentagon [04/15/1998]

How can you find the area of a regular pentagon given only the length of one side?

Finding the Base of Parts of a Triangle [05/22/2000]

Can you derive an expression for L1 in terms of L2 and L3 such that the area of a triangle with base A1 and the area of a triangle with base A2 are each 10% of the total area?

Finding the Center of the Research Triangle [9/5/1995]

We live in an area known as the Research Triangle, with the triangle's points at the University of North Carolina, North Carolina State University and Duke University. We are interested in finding the center point of our triangle home and whether there is a unique term (or several terms) for the center point of a triangle.

Finding the Coordinates of a Triangle Vertex [10/26/1999]

How can I find the coordinates of the point A of triangle ABC if B lies on the line 3y = 4x, C lies on the line y = 0, the line BC passes through (2/3,2/3) and AOBC forms a rhombus (where O is the origin)?

Finding the Dimensions of a Box [10/21/2001]

You want to construct a cardboard box from a cardboard strip that is 8 inches wide. The dimensions of the box are 8"x8"x4". How long does the strip need to be?

Finding the Incenter of a Triangle [10/08/2006]

What is the equation or method to find the incenter of a triangle? I'm having trouble with the Cartesian coordinates.

Finding Total Area of Several Rectangles [07/08/2005]

I need to find the total square footage of a lot of rectangular lawns. Do I have to find the area of each lawn and add up all the areas, or can I just add all the lengths and all the widths and make one area calculation based on those two totals?

Finding Triangle Vertices from Midpoints [09/18/1999]

If you know the coordinates of the midpoints of the sides of a triangle, how can you find the coordinates of its vertices?

Find Lengths of Sides of Triangle [04/20/2002]

Let ABC be a right-angled triangle with angle C = 90 degrees. Let the bisectors of angle A and angle B intersect BC and CA at D and E respectively. Given that CD = 9 and CE = 8, find the lengths of the sides of ABC.

Find the Area of the Quadrilateral [05/20/2003]

Find the area of an irregular quadrilateral formed by the given intersection of two squares.

Find the Diagonal of a Rectangle [07/26/1997]

We use a tape measure to square different things on the job site by measuring opposite corners...

Find the Fourth Side [12/03/2001]

The successive sides of a quadrilateral are 2, 6, 9, and x. If the diagonals of the quadrilateral are perpendicular, compute x.

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