Circle Formulas: Area, Circumference, and Radius

Circles are a core part of geometry. To solve circle problems, you need to be comfortable with the constant Pi ($\pi \approx 3.14$) and the difference between radius and diameter.

Key Definitions

Radius ($r$): The distance from the center to any point on the edge.
Diameter ($d$): The distance across the circle through the center.
$$d = 2r$$
Circumference ($C$): The distance around the circle (perimeter).
Area ($A$): The space inside the circle.

The Formulas

1. Circumference:

$$C = 2\pi r \quad \text{or} \quad C = \pi d$$

2. Area:

$$A = \pi r^2$$

Step-by-Step Example

Find the Area and Circumference of a circle with a diameter of $10$ cm. Use $\pi \approx 3.14$.

Step 1: Find the Radius
The problem gives the diameter ($d = 10$).

$$r = \frac{d}{2} = \frac{10}{2} = 5$$

Step 2: Calculate Circumference

$$C = 2\pi r = 2 \cdot 3.14 \cdot 5$$

$$C = 10 \cdot 3.14 = 31.4 \text{ cm}$$

Step 3: Calculate Area

$$A = \pi r^2 = 3.14 \cdot (5)^2$$

$$A = 3.14 \cdot 25$$

$$A = 78.5 \text{ cm}^2$$

Solution: $C = 31.4 \text{ cm}$, $A = 78.5 \text{ cm}^2$

Why reading isn’t enough

Confusing radius and diameter is the #1 mistake students make with circles. Also, remembering to square the radius before multiplying by $\pi$ for Area is crucial.

Weekzen’s adaptive exercises constantly switch between giving you the radius or the diameter, training you to always check which one you have before applying the formula.