Inscribed hexagon
Weekly Problem 1 - 2014
The diagram shows a regular hexagon inside a rectangle. What is the sum of the four marked angles?
The diagram shows a regular hexagon inside a rectangle. What is the sum of the four marked angles?
Age
11 to 14
Challenge level
Problem
Image
The diagram shows a regular hexagon inside a rectangle.
What is the sum of the four marked angles?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
As the sum of the angles in a triangle is 180 ° and all four angles in a rectangle are 90 °, then sum of the two marked angles in the triangle 180 ° - 90 ° = 90 °.
Each interior angle of a regular hexagon is 120 ° and the sum of the angles in a quadrilateral is 360 °; hence the sum of the two marked angles in the quadrilateral is 360 ° - 90 ° - (360 ° - 120 °) = 30 °.
Hence the sum of the four marked angles is 90 ° + 30 ° = 120 °.