Question

a square is inscribed in an equilateral triangle that is inscribed in a circle.

which represents the area of the shaded region?

• area of the circle - area of the square - area of the triangle
• area of the triangle - area of the square + area of the circle
• area of the triangle + area of the square + area of the circle
• area of the circle - area of the triangle + area of the square

Explanation:

Identify the nested geometric figures

The problem describes three nested figures:

1. A circle (outermost figure).
2. An equilateral triangle inscribed in the circle.
3. A square inscribed in the equilateral triangle.

Analyze the shaded region

The shaded region consists of:

1. The region inside the circle but outside the equilateral triangle.
2. The region inside the square.

Formulate the area equation

Let \(A_{\text{circle}}\) be the area of the circle, \(A_{\text{triangle}}\) be the area of the equilateral triangle, and \(A_{\text{square}}\) be the area of the square.

• The area of the region inside the circle but outside the triangle is:

$$ A_{\text{circle}} - A_{\text{triangle}} $$

• Adding the area of the square inside the triangle:

$$ \text{Area of shaded region} = (A_{\text{circle}} - A_{\text{triangle}}) + A_{\text{square}} = A_{\text{circle}} - A_{\text{triangle}} + A_{\text{square}} $$

Answer:

• area of the circle – area of the square – area of the triangle
• area of the triangle – area of the square + area of the circle
• area of the triangle + area of the square + area of the circle
• area of the circle – area of the triangle + area of the square (Correct answer)