What's the difference between radius and diameter?

Radius is the distance from the center to the edge; diameter is the distance across the entire circle through the center. Diameter is always twice the radius. Using diameter in the area formula gives the wrong answer, so always convert to radius first.

How do I calculate the area if I only know the circumference?

Use the Circumference of a Circle formula (C = 2\pi r) to find the radius by solving r = \frac{C}{2\pi}, then substitute into the area formula.

Do I need to include π in my answer, or should I round it to a decimal?

Check your assignment instructions. Most geometry problems accept A = 25\pi square units as a final answer, but some ask you to approximate: 25\pi \approx 78.54 square units.

What if the radius is given as a decimal or fraction?

Square it the same way as any number: if r = 0.5, then r^{2} = 0.25; if r = \frac{1}{3}, then r^{2} = \frac{1}{9}. Then multiply by π.

Area of a Circle

The area enclosed by a circle, calculated using its radius; needed for geometry and real-world problems involving circles like pools, wheels, or pizza.

A = \pi r^2

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What each symbol means

What Area of a Circle takes

Symbol	Meaning
$A$	The total area enclosed within the circle, measured in square units (e.g., square inches, square centimeters); if interpreted as diameter or circumference instead, the answer will be completely wrong.
$r$	The radius of the circle, measured from the center to any point on the edge; if confused with diameter (which is twice the radius), your answer will be off by a factor of four.

When to use it

Use this when you know a circle's radius and need to find how much space it covers.

Level

Usually taught in: Geometry · Appears on: SAT, ACT

Worked examples

1. Find the area with a simple radius

Problem

Find the area of a circle with radius 5 cm.

$A = \pi r^{2}$
Write the area formula.
$A = \pi (5)^{2}$
Substitute $r = 5$ into the formula.
$A = 25\pi \text{ cm}^{2}$
Square the radius: $5^2 = 25$ .

Answer: $25\pi \text{ cm}^{2} \text{ or } 78.54 \text{ cm}^{2}$

Straightforward application of the formula; substitute the radius value and simplify.

2. Find the area when given the diameter

Problem

A circular plate has diameter 8 inches. Find its area.

$r = \frac{d}{2} = \frac{8}{2} = 4 \text{ in}$
Convert diameter to radius by dividing by 2.
$A = \pi r^{2}$
Write the area formula.
$A = \pi (4)^{2} = 16\pi \text{ in}^{2}$
Substitute $r = 4$ and square: $4^2 = 16$ .

Answer: $16\pi \text{ in}^{2} \text{ or } 50.27 \text{ in}^{2}$

A common setup where you must first convert diameter to radius by dividing by 2, then apply the area formula.

3. Find total area for two circular regions on a road trip

Problem

During a road trip, you stop at two rest areas. The first circular rest area has radius 30 m, the second has radius 40 m. Find the total area of both rest areas.

$A_{1} = \pi r_{1}^{2} = \pi (30)^{2} = 900\pi \text{ m}^{2}$
Calculate the area of the first rest area by substituting $r_1 = 30$ m.
$A_{2} = \pi r_{2}^{2} = \pi (40)^{2} = 1600\pi \text{ m}^{2}$
Calculate the area of the second rest area by substituting $r_2 = 40$ m.
$A_{\text{total}} = 900\pi + 1600\pi = 2500\pi \text{ m}^{2}$
Add the two areas together to find the combined space.

Answer: $2500\pi \text{ m}^{2} \text{ or } 7853.98 \text{ m}^{2}$

Word problems often require calculating multiple areas and combining them; apply the formula to each circle separately, then add the results.

Common mistakes

Where Area of a Circle usually goes wrong

The mistake	Why it is wrong	The fix
Using the diameter in the formula: $A = \pi d^{2}$ instead of converting to radius first	Diameter is twice the radius, so using diameter directly gives an area that's four times too large.	Always convert diameter to radius first by dividing by 2, then use $A = \pi r^{2}$ .
Using the circumference formula $A = 2\pi r$ or forgetting to square the radius, writing $A = \pi r$	The area formula specifically requires squaring the radius; missing the exponent gives the circumference or something in between.	Remember: circumference is $2\pi r$ , but area is $\pi r^{2}$ —the radius must be squared.
Forgetting to include π and calculating only $r^{2}$ as the answer, e.g., for $r = 5$ , writing $A = 25$ instead of $A = 25\pi$	π is part of the formula and must be included; omitting it underestimates the area by a factor of approximately 3.14.	Always include π in your final answer unless instructed to round to a decimal approximation.

Tips and when to use something else

If you're given the circumference instead, use Circumference of a Circle to find the radius first, then apply this formula.
When the radius is a fraction, remember to square the numerator and denominator separately: $\left(\frac{a}{b}\right)^{2} = \frac{a^{2}}{b^{2}}$ .
For a sector (part of a circle), use Area of a Sector instead; it modifies this formula by multiplying by the angle fraction.
Always double-check whether you've been given the radius or diameter before substituting—it's the most common source of error.

Related formulas

Circumference of a CircleThe circumference of a circle is the total distance around its perimeter, calculated using either the radius or diameter.Area of a SectorCalculates the area of a circular sector enclosed by two radii and an arc when you know the radius and the central angle in radians.Volume of a CylinderCalculates the volume of a cylinder—the three-dimensional space inside—used for any cylindrical container or object in math and engineering.

Frequently asked questions

What's the difference between radius and diameter?: Radius is the distance from the center to the edge; diameter is the distance across the entire circle through the center. Diameter is always twice the radius. Using diameter in the area formula gives the wrong answer, so always convert to radius first.
How do I calculate the area if I only know the circumference?: Use the Circumference of a Circle formula ( $C = 2\pi r$ ) to find the radius by solving $r = \frac{C}{2\pi}$ , then substitute into the area formula.
Do I need to include π in my answer, or should I round it to a decimal?: Check your assignment instructions. Most geometry problems accept $A = 25\pi$ square units as a final answer, but some ask you to approximate: $25\pi \approx 78.54$ square units.
What if the radius is given as a decimal or fraction?: Square it the same way as any number: if $r = 0.5$ , then $r^{2} = 0.25$ ; if $r = \frac{1}{3}$ , then $r^{2} = \frac{1}{9}$ . Then multiply by π.

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Reviewed 2026-09-18