Stacking Shapes
Weekly Problem 28 - 2017
The diagram on the right shows an equilateral triangle, a square and a regular pentagon. What is the sum of the interior angles of the resulting polygon?
Problem
An equilateral triangle, square and regular pentagon all have the same side length. The triangle is drawn on the top edge of the square, and the pentagon is drawn on and below the bottom edge of the square. All of the edges match up exactly.
What is the sum of the interior angles of the resulting polygon?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
The interior angles are shown coloured in the diagram on the right.
The three blue angles are the interior angles of a triangle, so add up to $180^\circ$.
The four red angles are the interior angles of a square, which is a type of quadrilateral, so add up to $360^\circ$.
The five green angles are the interior angles of a pentagon, so add up to $540^\circ$.
Therefore, the total interior angle is $180^\circ + 360^\circ + 540^\circ = 1080^\circ$.
Since the shapes are regular, the angles could be individually counted. The blue angles are all $60^\circ$, the red angles are all $90^\circ$ and the green angles are all $108^\circ$.