Circles • Topic 3 of 3

Area and Applications of Circles

What is the area of a circle? The area of a circle is the total amount of flat two-dimensional space contained inside its curved boundary line. Think of the amount of cheese needed to cover a pizza base, or the amount of paint required to fill a painted circular target ring.

The Formula for Area: To calculate interior flat space, we multiply Pi ($\pi$) by the square of the radius ($r \times r$, written as $r^2$).

\[ \text{Area} = \pi r^2 \]

If you are only given the diameter, you must always divide it by $2$ first to find the radius before squaring it!

Real-World Applications: Circles are common in architecture, landscaping, and mechanical engineering. Common application scenarios include:

Circular Paths and Borders: Finding the space of a running track or a brick ring path around a circular garden lawn. This involves subtracting a smaller inner circle area from a larger outer circle area.
Rotational Distance: Calculating how far a car, bicycle, or unicycle travels. Every time a circular wheel completes one full turn ($1$ rotation), the vehicle rolls forward a linear distance exactly equal to the wheel's circumference.

Solved Examples

Worked Example

Calculate the surface area of a circular grass lawn that has a radius of $10\text{ meters}$. (Use $\pi = 3.14$)

Solution*Step 1: Identify the given radius value:* $r = 10\text{ m}$ *Step 2: Recall the mathematical area formula:* $\text{Area} = \pi r^2$ *Step 3: Substitute the parameters into the equation:* $\text{Area} = 3.14 \times 10^2$ *Step 4: Evaluate the square term first:* $10^2 = 10 \times 10 = 100$ *Step 5: Multiply the product by $3.14$:* $\text{Area} = 3.14 \times 100 = 314\text{ m}^2$

Key Points

The area formula is $\text{Area} = \pi r^2$, and the result is always expressed in square units (e.g., $\text{cm}^2, \text{m}^2$).
Always check if you are given the diameter; if so, halve it to get the radius before calculating area.
A wheel covers a linear distance equal to its circumference in one full rotation.
To find the area of a circular ring path, subtract the inner circle area from the outer circle area: $\text{Area} = \pi R^2 - \pi r^2$.
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Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.The area of a circle is:

Explanation: $\pi r^2$.

Q2.For $r=7$ ($\pi=\tfrac{22}{7}$), the area is:

Explanation: $\tfrac{22}{7}\cdot49=154$.

Q3.The area of a semicircle is:

Explanation: Half a circle.

Q4.If the radius is doubled, the area becomes:

Explanation: Area $\propto r^2$.

Practice more MCQs → 📝 Assignment · 35 marks