Area of a Circle

The space inside a circle equals pi multiplied by the radius, times the radius again.

A = π r²

The radius r goes from the center to the edge. The formula fills in everything shaded.

What each symbol means

Every circle has one measurement that controls its size: the radius. The radius is the distance from the middle of the circle out to the edge. Once you know the radius, this formula tells you exactly how much space fits inside.

The formula is A = π r². In plain words: take the radius, multiply it by itself, then multiply by pi.

What each part means

The little "2" on top of the r means "squared", which just means the radius times itself. If the radius is 5, then r² is 5 × 5 = 25. It does not mean 5 times 2.

Pi (written π) is a number that never changes. It is about 3.14. It shows up any time circles are involved, because it is the link between how far it is across a circle and how far it is around it.

Radius or diameter?: The formula needs the radius. If a problem gives you the diameter (the full distance across), just cut it in half first. A diameter of 10 means a radius of 5.

A worked example

Find the area of a circle with radius 5 cm

  1. Start with the formula: A = π r².
  2. Put in the radius: A = π × 5².
  3. Square the radius: 5² = 25, so A = π × 25.
  4. Multiply by pi (about 3.14): A ≈ 3.14 × 25 = 78.5.
  5. The area is about 78.5 square centimeters.

Why "square" units?: Area is always measured in square units (like square centimeters), because you are counting how many little 1-by-1 squares would cover the shape.

A second example, working from the diameter

A pizza is 12 inches across. How much pizza is that?

  1. The 12 inches is the diameter (all the way across), so the radius is half: r = 6 inches.
  2. Use the formula: A = π × 6².
  3. Square it: 6² = 36, so A = π × 36.
  4. Multiply: A ≈ 3.14 × 36 ≈ 113 square inches of pizza.

Why it works (a picture you can imagine)

  1. Imagine slicing the circle into many thin wedges, like a pizza cut into lots of skinny slices.
  2. Now lay the slices side by side, pointing up, then down, then up, and so on. They fit together into a shape that looks almost like a rectangle.
  3. The bottom of that rectangle is made of the rounded edges of half the slices, so its length is half the distance around the circle. The distance around is 2π r, so half of it is π r.
  4. The height of the rectangle is just the radius, r (the length of one slice from tip to crust).
  5. Area of a rectangle is length times height: π r × r = π r². The thinner the slices, the closer this gets to exact, and that is the formula.

Practice questions

Question 1

Find the area of a circle with radius 3 cm. Use π ≈ 3.14.

  1. Write the formula: A = π r².
  2. Substitute the radius r = 3: A = 3.14 × 3².
  3. Square the radius first (exponents before multiplying): 3² = 3 × 3 = 9.
  4. Multiply by pi: A = 3.14 × 9 = 28.26.
  5. Attach the units. Area is a surface, so it is measured in square cm.

Answer: 28.26 square cm

Question 2

Find the area of a circle with radius 10 m. Use π ≈ 3.14.

  1. Start from A = π r².
  2. Substitute r = 10: A = 3.14 × 10².
  3. Square the radius: 10² = 100.
  4. Multiply: A = 3.14 × 100 = 314 square m.

Answer: 314 square m

Question 3

A circle has a diameter of 8 cm. What is its area? Use π ≈ 3.14.

  1. The formula needs the radius, but we were given the diameter. The radius is half the diameter.
  2. Halve it: r = 8 ÷ 2 = 4 cm.
  3. Now use A = π r² = 3.14 × 4².
  4. Square the radius: 4² = 16.
  5. Multiply: A = 3.14 × 16 = 50.24 square cm.

Answer: 50.24 square cm

Question 4

Find the area of a circle with radius 7 cm. Use π ≈ 227.

  1. Use A = π r² with π ≈ 227 (a good choice when the radius is a multiple of 7).
  2. Substitute r = 7: A = 227 × 7² = 227 × 49.
  3. The 7 in the denominator cancels into 49: 49 ÷ 7 = 7.
  4. Multiply the rest: 22 × 7 = 154 square cm.

Answer: 154 square cm

Question 5

A round table has a radius of 2 feet. How much area does the top cover? Use π ≈ 3.14.

  1. The tabletop is a circle, so its surface area is A = π r².
  2. Substitute r = 2: A = 3.14 × 2².
  3. Square the radius: 2² = 4.
  4. Multiply: A = 3.14 × 4 = 12.56 square feet.

Answer: 12.56 square feet