Product of the areas of three shapes in a circle.

Geometry Level 3

We have an Equilateral Triangle, Rectangle and a Square embedded in the

Circle of radius 2 as shown:

One of the vertices of the Equilateral triangle lies on the Centre of this Circle.

One side of the Rectangle has a length of 1.

What is the prouct of the areas of the Equilateral Triangle, Square and Rectangle ?

The answer is 6.

1 solution

Blan Morrison
Oct 19, 2018

First, the area of the triangle.

The height of the triangle can be calculated by the Pythagorean Theorem: $2^2=h^2+1^2$ $h^2=4-1$ $h=\sqrt{3}$

Since the base is 2, the area of the triangle is $\sqrt{3}$ . By this same logic, the dimensions of the rectangle are $1\times \sqrt{3}=\sqrt{3}$ .

Now, we know the diagonal of the square is 2. Therefore, the side length of the square is $\frac{2}{\sqrt{2}}=\sqrt{2}$ . That means the area is simply $\sqrt{2}^2=2$ . $2\times \sqrt{3}^2=6$ $\beta_{\lceil \mid \rceil}$

Product of the areas of three shapes in a circle.

The answer is 6.

1 solution

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