Area of Circles

Circles show up everywhere — pipes, ducts, wheels, tanks, and round openings. To work with circles, you need to understand a few key parts and one essential formula: $A = \pi r^2$.

Parts of a Circle

• Center: The point in the exact middle of the circle.
• Radius ($r$): The distance from the center to any point on the circle’s edge.
• Diameter ($d$): The distance across the circle through the center. The diameter is always twice the radius.

$d = 2r \quad \text{or equivalently} \quad r = \frac{d}{2}$

• Pi ($\pi$): The ratio of every circle’s circumference to its diameter. It is approximately $3.14159$, and for most calculations you can use $\pi \approx 3.14$ or the $\pi$ button on your calculator.

Area of a Circle

The area of a circle with radius $r$ is:

$A = \pi r^2$

This means: multiply pi by the radius squared. Squaring the radius ($r^2 = r \times r$) comes first, then multiply by $\pi$.

Example 1: Find the area of a circle with radius 5 in.

$A = \pi r^2 = \pi (5)^2 = 25\pi \approx 25 \times 3.14 = 78.5 \text{ in}^2$

Answer: The area is approximately 78.5 square inches.

Example 2: Find the area of a circle with radius 3.5 m.

$A = \pi (3.5)^2 = \pi (12.25) = 12.25\pi \approx 38.48 \text{ m}^2$

Answer: The area is approximately 38.48 square meters.

Working with Diameter Instead of Radius

Many real-world measurements give you the diameter rather than the radius. Before using the area formula, convert diameter to radius by dividing by 2.

Example 3: A circular table top has a diameter of 4 ft. Find its area.

Step 1: Find the radius.

$r = \frac{d}{2} = \frac{4}{2} = 2 \text{ ft}$

Step 2: Apply the area formula.

$A = \pi (2)^2 = 4\pi \approx 12.57 \text{ ft}^2$

Answer: The area is approximately 12.57 square feet.

You can also substitute $r = \frac{d}{2}$ directly into the formula to get an equivalent version:

$A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}$

Both formulas give the same result. Use whichever feels more natural.

Area of a Semicircle

A semicircle is half a circle. Its area is half the area of the full circle:

$A_{\text{semicircle}} = \frac{1}{2}\pi r^2$

Example 4: A semicircular window has a diameter of 30 in. Find its area.

Step 1: Find the radius: $r = \frac{30}{2} = 15 \text{ in}$

Step 2: Calculate the semicircle area.

$A = \frac{1}{2}\pi (15)^2 = \frac{1}{2}\pi (225) = 112.5\pi \approx 353.43 \text{ in}^2$

Answer: The area is approximately 353.43 square inches.

Area of a Sector

A sector is a “pie slice” of a circle — the region between two radii and the arc connecting them. If the central angle of the sector is $\theta$ (in degrees):

$A_{\text{sector}} = \frac{\theta}{360} \times \pi r^2$

This formula takes the fraction of the full circle ($\frac{\theta}{360}$) and multiplies it by the full circle’s area.

Example 5: A sprinkler covers a sector with radius 20 ft and a central angle of $90\degree$. What area does it water?

$A = \frac{90}{360} \times \pi (20)^2 = \frac{1}{4} \times 400\pi = 100\pi \approx 314.16 \text{ ft}^2$

Answer: The sprinkler waters approximately 314.16 square feet.

Notice that $90\degree$ is one-quarter of $360\degree$, so the sector area is one-quarter of the full circle area. This makes intuitive sense.

Circle Area Formula Reference

Real-World Application: HVAC — Sizing Round Ductwork

HVAC technicians use the cross-sectional area of round ducts to determine how much air can flow through them. Airflow capacity is directly related to the duct’s area.

Problem: A technician needs to compare two round ducts: one with a 6-inch diameter and one with an 8-inch diameter. How much more area does the larger duct have?

Step 1: Find the area of the 6-inch duct.

$r_1 = \frac{6}{2} = 3 \text{ in}$

$A_1 = \pi (3)^2 = 9\pi \approx 28.27 \text{ in}^2$

Step 2: Find the area of the 8-inch duct.

$r_2 = \frac{8}{2} = 4 \text{ in}$

$A_2 = \pi (4)^2 = 16\pi \approx 50.27 \text{ in}^2$

Step 3: Find the difference.

$A_2 - A_1 = 16\pi - 9\pi = 7\pi \approx 21.99 \text{ in}^2$

Step 4: Find the percentage increase.

$\frac{A_2 - A_1}{A_1} \times 100 = \frac{7\pi}{9\pi} \times 100 = \frac{7}{9} \times 100 \approx 77.8\%$

Answer: The 8-inch duct has approximately 22 more square inches of cross-sectional area — about 78% more than the 6-inch duct. This is a critical insight for HVAC work: increasing a duct’s diameter by just 2 inches nearly doubles its cross-sectional area (and airflow capacity). That is because area depends on the square of the radius, so small changes in diameter produce large changes in area.

Common Mistakes to Avoid

1. Using diameter instead of radius. The formula is $\pi r^2$, not $\pi d^2$. If you are given the diameter, divide by 2 first.
2. Squaring $\pi$ instead of $r$. Calculate $r^2$ first, then multiply by $\pi$. The formula is $\pi \times (r^2)$, not $(\pi \times r)^2$.
3. Forgetting units are squared. Area is in square units ($\text{in}^2$, $\text{ft}^2$), not linear units.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• The area of a circle is $A = \pi r^2$ — always use the radius, not the diameter
• To convert from diameter to radius: $r = \frac{d}{2}$
• Area is in square units ($\text{ft}^2$, $\text{in}^2$, etc.)
• A semicircle has half the area of a full circle: $A = \frac{1}{2}\pi r^2$
• A sector’s area is a fraction of the full circle: $A = \frac{\theta}{360}\pi r^2$
• Because area depends on $r^2$, doubling the radius quadruples the area — small changes in size have a big impact

Return to Geometry for more topics in this section.