Area: Circle

Learn and practice calculating the area of circles, sectors, composite shapes, and shapes with cutouts.

Circle

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

Learning Topics

📖 Area Study Guide

1. What Are a Circle, Radius, and Diameter?

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Circle: Radius & Diameter

2. Circle Area Formula

The area of a full circle is calculated using its radius and the constant π\pi (pi, which is approximately 3.141593.14159). Note that π\pi is an irrational number (π∈R\pi \in \mathbb{R} and π∉Q\pi \notin \mathbb{Q}), meaning it cannot be written as a simple fraction and has infinite non-repeating decimal digits. To learn more about this classification, refer to the Real & Complex Number Sets topic.

3. Area of a Sector

• Common angles: 180∘180^\circ (semicircle, half of a circle) and 90∘90^\circ (quadrant, quarter of a circle).
180°Semicircle
90°Quadrant

4. Deriving the Angle from Area

If we are given the area AA of a sector and its radius rr, we can work backward to find the angle θ\theta by using the derived formulas:

Mastering SealMath: Entering Pi (π\pi)

To enter the pi symbol (π\pi) in the math input, you have three options:
• Keyboard shortcut: Type pi in the input box. It will instantly convert to π\pi.
• LaTeX input: Type \pi and press Enter (or click on the popup suggestion) to get the symbol π\pi.
• Virtual keyboard: Click the ⌨️ keyboard icon inside the input box to open the on-screen keyboard, switch to the greek tab, and press the π\pi button.

Frequently Asked Questions

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 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: Circle and Sectors | SealMath