Definition of a Circle

A circle is the set of all points in a plane that are the same distance from a fixed point.

• Center: The fixed point, usually labeled O.
• Radius (r): The distance from the center to any point on the circle.
• Diameter (d): The distance across the circle through the center, and d=2r.

📌 A circle is a perfectly symmetrical shape — the radius is the same in all directions.

Circle Area Formula: Introduction & Derivation

Formula

To calculate the area of a circle, the formula is:

Area = π × r²

Where:

• r = radius
• π ≈ 3.1416 (an irrational number that represents the ratio of a circle’s circumference to its diameter)

Derivation Method: Rearranging Sectors

1. Divide the circle into many small sectors (like pizza slices).
2. Rearrange these sectors alternately to form a shape that approximates a parallelogram or rectangle.
3. The base of this shape is approximately half the circumference: base = ½ × (2πr) = πr
4. The height = radius = r
5. So, area = base × height = πr × r = πr²

Example: Applying the Area Formula

Problem

A circle has a radius of 7 cm. Find its area (in terms of π and also as a decimal)?

Solution

Using the formula – Area = π × r², substitute r=7:

Area = π × r² = π × 7² = π × 49 = 49π cm²

Answer

The area of the circle is 49π cm².

If you need an approximate value:

Area = 49π ≈ 49 × 3.14 = 153.86 cm²

About Think Academy

Think Academy, a leading K–12 math education provider wholly owned by TAL Education Group, is dedicated to helping students build strong mathematical foundations and critical thinking. Our structured curriculum provides multiple course levels designed to accommodate students with diverse academic goals and proficiency levels, ensuring targeted and effective learning experiences. Supported by advanced teaching methods, expert instructors, and innovative AI technology, Think Academy consistently demonstrates excellence, trustworthiness, and proven expertise in mathematics education.

Want more insights on math education and parenting tips? Subscribe to our newsletter for weekly expert advice and updates on the latest learning tools.