Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Compare Areas

Or search by topic

Age 14 to 16

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Many got the correct answer that the circle which is embedded inside the isosceles triangle is the largest of the three shapes.

Some students measured the diagrams given which luckily had been drawn accurately to scale, while others observed the symmetry of the isosceles triangle and the consequences of joining B to D the mid point of AC. To find the radius of the circle it helps to draw perpendiculars from the centre to the sides of the triangle.

Taking the lengths of the short sides of the triangle as $2$ units:

the radius of the circle is found to be $2-\sqrt{2}$ and the area to be $\pi (6 - 4\sqrt{2})$.

the side length of square one to be $1$ and area $1$;

and the side length of square two to be $\frac{2\sqrt{2}}{3}$ and area $\frac{8}{9}$.

Many students evaluated the ratio of the areas as $1.078 : 1: 0.889$ (to $3$ dec. places)

Ben and Adrian from the Simon Langton Grammar School for Boys used trig ratios, as did Hannah of Maidstone Girls Grammar School, to arrive at the same conclusions.