Tag Archives: Pythagorean theorem

3 Points on Triangles

There are several easy demonstrations using cut-outs than can help show some of the properties of  triangles.

1. Area of a Triangle


The area of a triangle equals 1/2 base times height. This is true for any type of triangle, but easiest to demonstrate for a right triangle. Simply use 2 identical right triangles, flip one upside down and put them together to form a rectangle (shown below) with Area = a x b). The area of the triangle is simply one half of this (1/2 a x b).



It is slightly more complicated to show this with other triangles because the triangles will need to be rearranged a bit.  For these acute and obtuse triangles, draw a line for the height, which must form right angle with the base. For this example, assume that the longest side on the obtuse triangle is the base. Take one of the triangles and cut along this height line and then arrange the three pieces into a rectangle with Area = a x b (shown below).


There is a second way to look at the obtuse triangle. You can use one of the shorter sides as the base. But, then you have to cut both triangles, as shown in the picture below.


To see this demonstration as a computer animations check out these Animated Triangles.

2. Angles of a Triangle

The angles of a triangle always add to 180o (an angle of 180o is simply a straight line). This can be shown by taking three identical triangles and lining up the 3 angles as shown in the picture. You can also take 1 triangle and cut or tear off 2 of the angles and line them up with the third. See how they add up to a straight line? this works for any type of triangle.


There is an interactive web tool showing angle sums here. There is also a Wolfram demonstration showing how this can be demonstrated with folding (you’ll need the free Wolfram CDF player to view).

3. Pythagorean Theorem

The Pythagorean Theorem applies to only right triangles. It states that the sum of the squares of the 2 sides adjacent to the right angle is equal to the square of the other side.  You may remember the formula a2 + b2 = c2. This can be demonstrated a couple of ways.

For the first demonstration, take 4 identical right triangles, arrange them into a square as shown below, leaving a diagonal square of side c in the center. Draw the outer square on the paper below, then rearrange the triangles to give 2 smaller squares of sides a and b. This shows that the area of the diagonal square (c2)  in the first arrangement is the same as the sum of the 2 squares (a2 +b2) in the second arrangement.


You can see the same demonstration in the animation below.


Source: Wikimedia Commons
Author: JohnBlackburne

You can also arrange the triangles to create a square of side c by putting the hypotenuse (the side opposite the right angle) along the outside edge of the square as shown in this animation:

Copyright © 1998 by Davis Associates, Inc.All Rights Reserved

Copyright © 1998 by Davis Associates, Inc.
All Rights Reserved

For the second demonstration, start with two squares of side a and b. (I’ve included the triangles in the picture below, so you can see that the lengths of the sides are the same.) Then draw diagonal lines to create right triangles of sides a and b – the easiest way to do this is to lay the triangles down on the squares as shown and draw a line along the triangle edge. Next, cut along these diagonal lines and rearrange the pieces to create a single square of side c. So, a2 + b2 becomes c2.


There is also a nice Wolfram Demonstration: An Intuitive Proof of the Pythagorean Theorem