Triangle Puzzle Explanation

( Where did that Black Square come from? )

The Red-Dotted Line shows the Hypotenuse of a true '8 x 5' Right-Angled Triangle

Quick Observations:

  1. The two large '8 x 5' right-angled triangles shown above are not true triangles; look closely, they are actually 4-sided figures (quadrilaterals).

  2. The hypotenuse (side opposite the right-angle) of the Green Triangle slopes '2 in 3' and the hypotenuse of the Yellow Triangle slopes '3 in 5'.

  3. In addition, the top image has an area slightly less than a true '8 x 5' right-angle triangle and the bottom image has an area slightly greater.

Areas of Individual Coloured Pieces: (assuming inches as units)

  4a. Red: 4 square inches.   4b. Blue: 5 sq. inches.   4c. Green: (3 x 2) / 2 = 3 sq. inches.   4d. Yellow: (5 x 3) / 2 = 7½ sq. inches.

Areas of the Constructed Shapes:

  5a. Top Shape: 3 + 4 + 5 + 7½ = 19½ square inches.   5b. Bottom Shape: 3 + 4 + 5 + 7½ + The Black Square = 20½ square inches.

  6. A True '8 x 5' right-angled Triangle has an area of (8 x 5) / 2 = 20 sq. ins.   Compare its red-dotted hypotenuse with the Constructed Shapes.

  7. The two (seemingly identical) constructed triangles differ in area by 1 square inch - The Black Square - and now you know why.  I cheated!