IMC 2015 Q6
6. The equilateral triangle and regular hexagon shown have perimeters of the same length.
What is the ratio of the area of the triangle to the area of the hexagon?
- A 5 : 6
- B 4 : 5
- C 3 : 4
- D 2 : 3
- E 1 : 1
Intermediate Mathematical Challenge 2015, Question 6. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 2 : 3
Let the length of the side of the regular hexagon be \(a\). Then its perimeter is \(6a\). Therefore the perimeter of the equilateral triangle is also \(6a\), so the length of each of its sides is \(2a\). The diagrams show that the equilateral triangle may be divided up into 4 equilateral triangles of side \(a\), whereas the regular hexagon may be divided into 6 such triangles. So the required ratio is 4 : 6 = 2 : 3.
Solution © UK Mathematics Trust, from the IMC 2015 solutions. Longer versions with investigations: Solutions and investigations (PDF).