Problems

The triangle visible in the picture is equilateral. The hexagon inside is a regular hexagon. If the area of the whole big triangle is \(18\), find the area of the small blue triangle.

On the left there is a circle inscribed in a square of side 1. On the right there are 16 smaller, identical circles, which all together fit inside a square of side 1. Which area is greater, the yellow or the blue one?

In a pentagon \(ABCDE\), diagonal \(AD\) is parallel to the side \(BC\) and the diagonal \(CE\) is parallel to the side \(AB\). Show that the areas of the triangles \(\triangle ABE\) and \(\triangle BCD\) are the same.

The area of the red triangle is \(25\) and the area of the orange triangle is \(49\). What is the area of the trapezium \(ABCD\)?

Prove that the relation between areas of two similar polygons equals to the square of their similarity coefficient.