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?

A circle is a round flat shape where every point on the boundary is the exact same distance from the center.
• Radius (r): The distance from the center of the circle to any point on its outer boundary.
• Diameter (d): The distance straight across the circle, passing through the center. The diameter is exactly twice the length of the radius: d=2rd = 2r.
rd
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.
A=πr2A = \pi r^2

3. Fractions of a Circle (Sectors)

A sector is a fraction of a circle, like a slice of pizza. If we know the sector's central angle θ\theta (in degrees), we can calculate its area. Since a full circle is 360∘360^\circ, the sector represents the fraction θ360\frac{\theta}{360} of the entire circle's area:
A=θ360πr2A = \frac{\theta}{360} \pi r^2

• 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 solving the sector area formula for θ\theta:
θ=360×Aπr2\theta = \frac{360 \times A}{\pi r^2}

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 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.

What are the main types of special triangles?

The three main types are: isosceles (two equal sides and two equal base angles), equilateral (all sides and angles equal — each angle is 60°), and the 30-60-90 right triangle (fixed side ratios of 1 : √3 : 2).

Why does 1 cm² equal 100 mm² and not just 10 mm²?

Because area is two-dimensional (length × length), the conversion factor must be applied twice. Since 1 cm = 10 mm, we have 1 cm² = 1 cm × 1 cm = 10 mm × 10 mm = 100 mm². In general, if 1 unit = k sub-units, then 1 unit² = k² sub-units². The same logic applies to all other area unit conversions.

Area: Circle and Sectors | SealMath