Practice Maths

Area of a Circle

Key Ideas

Key Terms

area of a circle
the amount of space enclosed inside a circle; A = πr², where r is the radius.
radius (r)
the distance from the centre to the edge; always use the radius (not the diameter) in A = πr². If given d, find r = d ÷ 2 first.
π (pi)
the mathematical constant ≈ 3.14159…; used in both area and circumference formulas for circles.
square units
the units used for area; if the radius is in cm, the area is in cm². Always write the squared unit in your answer.
inverse — finding radius from area
rearrange A = πr² to get r = √(A ÷ π); take the square root after dividing by π.
Hot Tip In A = πr², you square the radius first, then multiply by π. A common mistake is to multiply by π before squaring. Always follow the order of operations: r² is calculated before multiplying by π.

Worked Example

Question A: Find the area of a circle with radius 6 cm. Give your answer to 2 decimal places.

Step 1 — Write the formula: A = πr²

Step 2 — Substitute r = 6: A = π × 6² = π × 36 = 36π

Step 3 — Evaluate: A ≈ 36 × 3.14159 ≈ 113.10 cm²

Question B: A circle has area 200 m². Find the radius to 2 decimal places.

Step 1 — A = πr², so r² = A ÷ π = 200 ÷ π ≈ 63.662

Step 2 — r = √63.662 ≈ 7.98 m

The Area Formula: A = πr2

The area of a circle tells you how much space is enclosed inside the circle. The formula is A = πr2, where r is the radius. Notice that you always need the radius — not the diameter — for this formula. If you are given the diameter, halve it first to get r.

Example 1: Find the area of a circle with radius 6 cm.
A = π × 62 = π × 36 = 36π ≈ 113.10 cm2

Example 2: Find the area of a circle with diameter 10 m.
First, r = 10 ÷ 2 = 5 m.
A = π × 52 = 25π ≈ 78.54 m2

Why Does the Formula Work? An Intuitive Explanation

Imagine cutting a circle into many thin "pie slices" (sectors). Now flip every second slice upside down and line them up side by side. The resulting shape looks almost like a rectangle. As you cut thinner and thinner slices, the shape gets closer and closer to a perfect rectangle.

The rectangle's length is approximately half the circumference = (12) × 2πr = πr. The rectangle's height is the radius r. So the area = length × height = πr × r = πr2. This is where the formula comes from — it is not just a rule to memorise, it has a beautiful geometric reason behind it.

Finding the Radius When the Area is Known

Sometimes a question gives you the area and asks for the radius. To reverse the formula, rearrange A = πr2:

  • Divide both sides by π: r2 = A ÷ π
  • Take the square root: r = √(A ÷ π)

Example: A circle has an area of 200 cm2. Find the radius.
r2 = 200 ÷ π ≈ 63.66
r = √63.66 ≈ 7.98 cm

Key tip: The most common mistake is squaring the diameter instead of the radius. Always confirm: "I have the radius." If the question gives you d = 8, write r = 4 as your very first step, then substitute r into A = πr2. This one habit eliminates the most frequent error in circle area questions.

Comparing a Circle and a Square with the Same Perimeter

Here is an interesting result: if a circle and a square have the same perimeter, the circle always encloses a larger area. For example, suppose both shapes have a perimeter of 40 cm.

  • Square: side = 40 ÷ 4 = 10 cm. Area = 102 = 100 cm2.
  • Circle: C = 40, so r = 40 ÷ (2π) ≈ 6.366 cm. Area = π × 6.3662 ≈ 127.3 cm2.

The circle's area (127.3 cm2) is larger than the square's (100 cm2). This is why circular storage tanks and silos are used in agriculture — for the same amount of fencing, a circular enclosure holds more area than any other shape.

Units Matter — Areas vs. Lengths

Always write the correct units in your answer. Lengths are in cm, m, mm, etc. Areas are in cm2, m2, mm2, etc. If the radius is in centimetres, the area will be in square centimetres. Forgetting the squared unit is a common way to lose marks in exams.

Mastery Practice

  1. Find the area of each circle. Give answers to 2 decimal places. Fluency

    1. r = 3 cm
    2. r = 7 m
    3. r = 5 mm
    4. r = 10 cm
    5. r = 2.5 m
    6. r = 12 cm
    7. r = 0.8 m
    8. r = 20 mm

  2. Find the area. You are given the diameter. Round to 2 decimal places. Fluency

    1. d = 8 cm
    2. d = 14 m
    3. d = 6 mm
    4. d = 20 cm
    5. d = 3 m
    6. d = 9.4 cm
    7. d = 50 mm
    8. d = 1 m

  3. Find the radius. Round to 2 decimal places. Fluency

    1. A = 50 cm²
    2. A = 78.54 m²
    3. A = 200 mm²
    4. A = 314 cm²
    5. A = 1 m²
    6. A = 100 cm²
    7. A = 400π mm²
    8. A = 9π cm²

  4. Find the area. Leave your answer in terms of π. Understanding

    1. r = 4 cm
    2. d = 12 m
    3. r = 9 mm
    4. d = 1 cm
    5. r = 7 m
    6. d = 6 km

  5. Answer these comparison questions. Understanding

    1. Circle A has radius 3 cm and Circle B has radius 6 cm. How many times larger is the area of Circle B than Circle A?
    2. A circle and a square both have a perimeter/circumference of 20 cm. Which shape has the larger area? Show your working.
    3. If the radius of a circle is doubled, what happens to the area? Explain using the formula.
    4. A circle has diameter 10 cm. Find both the circumference and the area. Which value is larger?

  6. Area of a circle in real-world contexts. Problem Solving

    1. A circular pizza has a diameter of 30 cm. Find its area to 2 decimal places. If a second pizza has diameter 45 cm, how much more area does it have?
    2. A circular pool has an area of 50.27 m². Find the radius of the pool (to 2 dp) and then the circumference (to 2 dp).
    3. A circular coaster has a radius of 5 cm. A square coaster has a side length of 9 cm. How much greater is the area of the square coaster? Round to 2 decimal places.
    4. A homeowner wants to plant a circular garden with radius 4 m. Fertiliser costs $3.50 per m². Find the cost to fertilise the entire garden (to the nearest dollar).

  7. Find the area of each circle shown. Give answers to 2 decimal places. Understanding

    1. Find the area of this circle. r = 5 cm
    2. Find the area of this circle. (Hint: find r first.) d = 12 m
    3. Find the area of this circle. r = 8.5 mm

  8. Complete the table. Round to 2 d.p. where needed. Understanding

    Radius (r)Diameter (d)Area (exact, in terms of π)Area (decimal)
    6 cm_________
    ___10 m______
    4.5 mm_________
    ______49π cm²___
    ___3 km______

  9. Each calculation below contains an error. Find it and give the correct answer. Understanding

    1. A student calculated the area of a circle with d = 10 cm as: A = π × 10² = 100π ≈ 314.16 cm². What did they do wrong?
    2. Another student found the area of a circle with r = 5 m as: A = 2πr = 2 × π × 5 = 31.42 m². Identify and correct the error.
    3. A student said: “If I double the radius, the area doubles.” Use r = 4 cm and r = 8 cm to show this is incorrect. What actually happens?
    4. A student used A = π × d² to find the area. What is the correct formula? Find the area of a circle with d = 8 cm using the correct method.

  10. Extended response. Show all steps and include units. Problem Solving

    1. A sprinkler waters a circular area with radius 4.5 m.
      1. Find the area watered by one sprinkler to 2 dp.
      2. A lawn is rectangular: 30 m × 20 m. Find the area of the lawn.
      3. How many sprinklers (minimum) are needed to water the whole lawn if each sprinkler covers its circular area without overlap? (You may assume the sprinklers are positioned optimally.)
    2. A circular dartboard has diameter 45 cm. The bullseye is a circle of diameter 6 cm.
      1. Find the area of the entire dartboard.
      2. Find the area of the bullseye.
      3. Find the area of the dartboard that is NOT the bullseye.
      4. What percentage of the dartboard is the bullseye? Round to 2 dp.
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