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Area: Inscribed Circle & Square

Learn about squares inscribed in circles and circles inscribed in squares. Understand why their area ratios are fixed and solve exercises finding area differences.

Inscribed Circle & Square

Use the workspace below. Write equations like A = 30 to solve for the area.

Learning Topics

📖 Area Study Guide

1. Inscribed Shapes & Ratios

An inscribed shape is a geometric figure that is drawn inside another figure, so that their boundaries touch.
We will explore two fundamental inscribed setups:
• A square inscribed in a circle (its vertices lie on the circle).
• A circle inscribed in a square (it is tangent to all four sides of the square).

2. Square Inscribed in a Circle

rad = 2r
Square inscribed in a circle: Diagonal d=2rd = 2r, side aa. Area ratio is exactly 2/pi.
Let a circle have radius rr. The square inscribed inside has vertices touching the circle boundary.

The diagonal of the square is equal to the diameter of the circle: d=2rd = 2r.
Using the Pythagorean theorem for side length aa of the square:
a2+a2=d2  ⟹  2a2=(2r)2=4r2  ⟹  a2=2r2a^2 + a^2 = d^2 \implies 2a^2 = (2r)^2 = 4r^2 \implies a^2 = 2r^2

• Area of the square: Asquare=a2=2r2A_{\text{square}} = a^2 = 2r^2
• Area of the circle: Acircle=πr2A_{\text{circle}} = \pi r^2

The ratio of the area of the square to the circle is constant:
AsquareAcircle=2r2πr2=2π≈0.637\frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{2r^2}{\pi r^2} = \frac{2}{\pi} \approx 0.637

Notice that the radius r2r^2 cancels out completely! The ratio is always exactly 2π\frac{2}{\pi} regardless of the size.

3. Circle Inscribed in a Square

ra = 2r
Circle inscribed in a square: Side length a=2ra = 2r. Area ratio is exactly pi/4.
Let a circle of radius rr be inscribed in a square of side length aa. The circle fits perfectly inside, tangent to all four sides.

The diameter of the circle is equal to the side length of the square: d=a=2rd = a = 2r.
• Area of the square: Asquare=a2=(2r)2=4r2A_{\text{square}} = a^2 = (2r)^2 = 4r^2
• Area of the circle: Acircle=πr2A_{\text{circle}} = \pi r^2

The ratio of the area of the circle to the square is constant:
AcircleAsquare=πr24r2=π4≈0.785\frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi r^2}{4r^2} = \frac{\pi}{4} \approx 0.785

Again, the radius r2r^2 cancels out! The ratio is always exactly π4\frac{\pi}{4} regardless of the size.

Frequently Asked Questions

How is the area of a shape defined?

The area of a shape is defined by how many unit squares of 1 by 1 fit inside it. For example, if a rectangle can be divided exactly into 30 squares of 1 by 1, its area is 30.

How do you calculate the area of a rectangle and a square?

The area of a rectangle is calculated as width × height (A = w × h). A square is a special type of rectangle where all sides are equal (w = h = s). Thus, the area of a square is side × side, or side squared (A = s²).

How do you calculate the area of a right-angled triangle?

The area of a right-angled triangle is calculated by multiplying its two perpendicular legs and dividing by 2 (A = ab / 2). This is because a right-angled triangle is exactly half of a rectangle with the same width and height.

How do you calculate the area of a general triangle?

The area of any general triangle is calculated as half the base times the height (A = (1/2) × b × h or A = bh/2), where the height is the perpendicular distance from the base to the opposite vertex.

How do you calculate the area of a circle?

The area of a circle is calculated as π times the radius squared (A = πr²). Since π (pi) is an irrational number, the area of a circle with a rational radius will always be an irrational number.

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). It is used to find a missing side length when the other two are known.

What is the area ratio of inscribed circles and squares?

For a square inscribed in a circle, the area ratio of the square to the circle is always exactly 2/π ≈ 0.637. For a circle inscribed in a square, the area ratio of the circle to the square is always exactly π/4 ≈ 0.785. These ratios are constant regardless of the shapes' actual sizes.

Can the area of a shape be an irrational number?

Yes, if the side lengths are irrational numbers (such as √2), the resulting area can be either rational or irrational. You can learn more about these classifications in our Number Sets - Real & Complex topic.

Area of Inscribed Circle & Square | SealMath | SealMath