Pythagoras, hexagons and more

Triangles and hexagons and circles, oh my! Try your hand at these geometry questions from St John’s Maths lecturer Dr Devinder Sivia

Question 1

With reference to right-angled triangles, express Pythagoras’ theorem in three different ways using sines, cosines and tangents.

Question 2

(2a) The closed curve that encloses the largest area for a given length of perimeter is a circle. Show that for rectangles with a fixed perimeter, a square encloses the largest area.

(2b) Work out the areas of an equilateral triangle, a square and a regular hexagon in terms of their perimeter.

(2c) What will be the area for the case of an n-sided regular polygon in the limit n → ∞?

Question 3

Calculate the perimeters of the regular hexagons that inscribe and circumscribe a circle, and hence place a bound on the value of π.