You find the area of a circle using this formula:

Area=πr2\text{Area}=\pi r^2Area=πr2

where rrr is the radius of the circle (the distance from the center to the edge).

Quick Scoop

1. The basic idea

  • A circle’s area is how much flat space it covers inside its boundary.
  • The key measurement you need is the radius rrr, then you plug it into A=πr2A=\pi r^2A=πr2.
  • π\pi π (pi) is about 3.14 or 22/722/722/7, and it’s the same for every circle.

2. Step‑by‑step: using the radius

Suppose the radius is 3 cm.

  1. Write the formula: A=πr2A=\pi r^2A=πr2.
  1. Square the radius: 32=93^2=932=9.
  2. Multiply by π\pi π: A=π×9≈3.14×9=28.26textcm2A=\pi \times 9\approx 3.14\times 9=28.26\\text{cm}^2A=π×9≈3.14×9=28.26textcm2.

So the area is about 28.3textcm228.3\\text{cm}^228.3textcm2.

3. If you know the diameter instead

  • The diameter ddd is the distance across the circle through the center, and d=2rd=2rd=2r.
  • First find the radius: r=d/2r=d/2r=d/2, then use A=πr2A=\pi r^2A=πr2.

Or you can go directly with this equivalent formula:

A=πd24A=\frac{\pi d^2}{4}A=4πd2​

This comes from substituting r=d/2r=d/2r=d/2 into A=πr2A=\pi r^2A=πr2.

Example (diameter 10 cm):

  • Radius r=10/2=5textcmr=10/2=5\\text{cm}r=10/2=5textcm.
  • Area A=π×52=π×25≈78.5textcm2A=\pi \times 5^2=\pi \times 25\approx 78.5\\text{cm}^2A=π×52=π×25≈78.5textcm2.

4. If you know the circumference

Sometimes you’re given the circumference CCC (the distance all the way around the circle).

  • The relationship is C=2πrC=2\pi rC=2πr.
  • Solve for rrr: r=C/(2π)r=C/(2\pi)r=C/(2π).
  • Then plug into A=πr2A=\pi r^2A=πr2, which simplifies to:

A=C24πA=\frac{C^2}{4\pi}A=4πC2​

5. Quick reference table (HTML)

Here’s a small HTML table summarizing the key formulas:

html

<table>
  <thead>
    <tr>
      <th>What you know</th>
      <th>Formula for area</th>
      <th>Notes</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Radius r</td>
      <td>A = π r²</td>
      <td>Most common formula; r is center to edge.</td>
    </tr>
    <tr>
      <td>Diameter d</td>
      <td>A = (π d²) / 4</td>
      <td>Use when distance across the circle is given.</td>
    </tr>
    <tr>
      <td>Circumference C</td>
      <td>A = C² / (4π)</td>
      <td>Use when you only know the distance around.</td>
    </tr>
  </tbody>
</table>

6. A tiny story to remember it

Imagine slicing a pizza into many skinny slices and rearranging them into a shape that almost looks like a rectangle. Each “row” of crust becomes one long side, and the height is the radius. Put enough slices together and that “rectangle” ends up with area very close to πr2\pi r^2πr2, which is why this formula neatly captures how much pizza you get inside the circle.

TL;DR:

  • Use A=πr2A=\pi r^2A=πr2 if you know the radius.
  • Convert diameter or circumference to radius (or use the alternative formulas) and then apply the same idea.

Information gathered from public forums or data available on the internet and portrayed here.