Working backwards: find r when you know A → r = √(A/π)
Area of a semicircle and quarter circle
Area of composite shapes involving circles
Comparing circle area to squares and rectangles
📖 Learn: Area of a Circle
Part 1: The Formula A = πr²
The area of a circle is the amount of space inside it. The formula uses π (pi) ≈ 3.14159… and the radius (r) — the distance from the centre to the edge.
📌 Formula: A = π × r² (pi times radius squared)
The radius is half the diameter: r = d ÷ 2
Always square the radius first, then multiply by π.
💡 Use the π button on your calculator for full precision, or use π ≈ 3.142 when calculating by hand.
Part 2: Why A = πr² ? (The Sector Proof)
Imagine slicing a circle into many thin sectors (like pizza slices). Rearrange them alternating up/down — they form a rough rectangle!
Height of the "rectangle" = r (the radius)
Width of the "rectangle" = πr (half the circumference, because C = 2πr)
Area of the rectangle = height × width = r × πr = πr² ✅
🍕 The more slices, the more rectangular it becomes. See the Visualizer tab for the animation!
Part 3: Area Given the Radius
When given the radius, substitute directly into A = πr².
Example: r = 5 cm → A = π × 5² = π × 25 ≈ 78.5 cm²
Example: r = 3.5 m → A = π × 3.5² = π × 12.25 ≈ 38.5 m²
Example: r = 12 mm → A = π × 144 ≈ 452.4 mm²
⚠️ The most common mistake is forgetting to square r before multiplying. r² ≠ 2r!
Part 4: Area Given the Diameter
When given the diameter, always halve it to find the radius first!
📌 r = d ÷ 2, then use A = πr²
Example: d = 14 cm → r = 7 cm → A = π × 49 ≈ 153.9 cm²
Example: d = 9 m → r = 4.5 m → A = π × 20.25 ≈ 63.6 m²
💡 Never use d directly in the formula! A = π(d/2)² is correct, but it's cleaner to halve first.
Part 5: Working Backwards — Find r Given Area
If you know the area and need to find the radius, rearrange the formula:
📌 A = πr² → r² = A ÷ π → r = √(A ÷ π)
Example: A = 200 cm² → r² = 200 ÷ π ≈ 63.66 → r = √63.66 ≈ 7.98 cm
Example: A = 50.3 m² → r² = 50.3 ÷ π ≈ 16.0 → r = √16 = 4.0 m
🔁 Steps: (1) Divide area by π. (2) Take the square root. That's your radius!
Part 5b: Semicircle and Quarter Circle
A semicircle is half a circle; a quarter circle is one quarter.
Semicircle area: A = πr² ÷ 2
Quarter circle area: A = πr² ÷ 4
Example (semi): r = 6 cm → A = π × 36 ÷ 2 ≈ 56.5 cm²
Example (quarter): r = 10 m → A = π × 100 ÷ 4 ≈ 78.5 m²
💡 Calculate the full circle area first, then divide by 2 or 4. Same radius, different fraction!
💡 Worked Examples
Example 1: Area Given Radius
Find the area of a circle with radius 7 cm. Give your answer to 1 decimal place.
Step 1: Write the formula: A = πr²
Step 2: Substitute: A = π × 7²
Step 3: Square: A = π × 49
Step 4: Multiply: A = 153.938… ≈ 153.9 cm²
📝 Always include the unit squared (cm², m², etc.). Area is always in square units.
Example 2: Area Given Diameter
A circular pond has a diameter of 18 m. Find its area to 1 d.p.
Step 1: Find radius: r = 18 ÷ 2 = 9 m
Step 2: Apply formula: A = π × 9²
Step 3: Square: A = π × 81
Step 4: Calculate: A = 254.469… ≈ 254.5 m²
⚠️ Common error: using d = 18 directly → π × 18² = 1017.9 (wrong — 4× too large!)
Example 3: Find Radius Given Area
A circle has area 314.2 cm². Find its radius to 1 d.p.
💡 For composite shapes: break into simpler parts, find each area, then add (or subtract if there's a hole).
⭕ Circle Area Visualizer
🍕 Sector Rearrangement — Why A = πr²
Watch the circle slice into sectors and rearrange into a rectangle. Height = r, Width = πr, so Area = πr²!
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📐 Live Circle Calculator
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A = π × 6² = 113.1 cm²
🔷 Composite Shape Builder
Click shapes to add them to your composite figure (using r = 5 cm as the base unit). Click a placed shape to remove it.
⭕ Full Circle (r=5)
🌙 Semicircle (r=5)
🍕 Quarter (r=5)
🟦 5×5 Square
🟦 10×5 Rectangle
Click shapes above to add them here…
Total Area: 0.0 cm²
🤔 Is the Circle Bigger?
A circle and a square both have the same "size". The circle's radius = the square's side. Which has greater area?
7 cm
⭕ Circle (r = 7 cm)
🟦 Square (side = 7 cm)
Your prediction: which has greater area?
✏️ Exercise 1: Area Given Radius
Calculate the area of each circle. Give answers to 1 decimal place. Use A = πr²
✏️ Exercise 2: Area Given Diameter
Find the radius first (r = d ÷ 2), then calculate the area to 1 decimal place.
✏️ Exercise 3: Find the Radius
Use r = √(A ÷ π) to find the radius. Give answers to 1 decimal place.
🍕 Exercise 4: Semicircles & Quarter Circles
Calculate the shaded area to 1 decimal place. Remember: semi = πr²÷2, quarter = πr²÷4.
🔷 Exercise 5: Composite Shapes
Break each shape into parts, calculate each area, then combine. Give answers to 1 decimal place.
📝 Practice Questions
Find the area of a circle with radius 4 cm. (1 d.p.)
Find the area of a circle with radius 9 m. (1 d.p.)
Find the area of a circle with radius 2.5 cm. (1 d.p.)
Find the area of a circle with diameter 10 cm. (1 d.p.)
Find the area of a circle with diameter 7 m. (1 d.p.)
Find the area of a circle with diameter 13 cm. (1 d.p.)
A circle has area 78.5 cm². Find its radius. (1 d.p.)
A circle has area 200.96 cm². Find its radius. (1 d.p.)
Find the area of a semicircle with radius 5 cm. (1 d.p.)
Find the area of a semicircle with diameter 12 m. (1 d.p.)
Find the area of a quarter circle with radius 8 cm. (1 d.p.)
A rectangle (12 cm × 8 cm) has a semicircle of diameter 8 cm on one short end. Find the total area. (1 d.p.)
A square of side 10 cm has a quarter circle of radius 10 cm cut from one corner. Find the remaining area. (1 d.p.)
A circular pizza has diameter 30 cm. Find its area in cm². (1 d.p.)
A circular flower bed has radius 3.5 m. Find the area in m². (1 d.p.)
Two circles: Circle A has r = 6 cm, Circle B has r = 3 cm. By how much is Circle A's area greater than Circle B's? (1 d.p.)
A circle and a square share the same "radius" — the circle has r = 5 cm and the square has side 5 cm. How much bigger is the circle's area than the square's area? (1 d.p.)
A semicircle sits on top of a rectangle. The rectangle is 8 cm × 5 cm and the semicircle has diameter 8 cm. What is the total area? (1 d.p.)
A dartboard has a bullseye circle of radius 6 mm inside a larger circle of radius 18 mm. Find the area of the ring between them. (1 d.p.)
A running track has two semicircular ends each with radius 40 m and two straight sections 80 m long and 5 m wide. Find the area of the track surface only (not the inner field). (1 d.p.)
Two semicircle track rings (outer r=45, inner r=40). Ring area = π(45²−40²) = π×425 ≈ 1335.2. Two straight sections: 2×(80×5) = 800. Total ≈ 2135.2 m²
🏆 Challenge: Multi-Step Problems
A garden contains a rectangular lawn (15 m × 10 m) with a circular pond of diameter 4 m cut into it. Find the remaining lawn area to 1 d.p.
A semicircle is placed on top of a triangle. The triangle has base 10 cm and height 8 cm. The semicircle has diameter 10 cm. Find the total area to 1 d.p.
Circle A has radius r. Circle B has radius 2r. Show that Circle B's area is exactly 4 times Circle A's area.
A target consists of 3 concentric circles with radii 2 cm, 5 cm, and 9 cm. Find the area of each ring (between consecutive circles) to 1 d.p.
A logo is made of a full circle of radius 8 cm with two quarter circles of radius 4 cm cut from opposite corners of an 8 cm × 8 cm square inside it. Find the logo's shaded area to 1 d.p. (Hint: find full circle area, subtract square, add back the two cut quarters' inverse.)
A wheel has diameter 60 cm. It rolls 10 full rotations. The road surface it rolls over is a rectangle 10 cm wide and as long as the distance covered. Find the area of the road in m². (Hint: circumference × rotations = length.)
A farmer has 100 m of fencing. He can make a circular pen or a square pen. Which encloses more area, and by how much? Give answers to 1 d.p.
The ratio of two circles' radii is 3:5. The smaller circle has area 113.1 cm². Find the larger circle's area to 1 d.p. without measuring the larger radius.
Lawn: 150 m². Pond (r=2): π×4 ≈ 12.6 m². Remaining ≈ 137.4 m²