Find circumference and area of a circle

What you will learn

• Find circumference and area of a circle

Models used: number sentences, stories, diagrams.

Before you start

• Find areas of composite rectangles, squares and triangles

From the radius to the area

The radius runs from the centre to the edge. Here it is 7 cm.

1. The centre is the middle point. A radius joins it to the edge; a diameter passes through it.

   The radius runs from the centre to the edge. Here it is 7 cm.

   A diameter contains two radii. A partial-circle perimeter may also contain straight edges; its area concerns the enclosed surface.

2. The centre is the middle point. A radius joins it to the edge; a diameter passes through it.

   The diameter crosses the centre from edge to edge. It is two radii long.

   diameter = 2 × radius

3. The centre is the middle point. A radius joins it to the edge; a diameter passes through it.

   A string wrapped around a circle is about 3.14 times its diameter. We call that constant ratio π.

   circumference = π × diameter = 2πr

4. The blue and orange pieces all came from the same circle. Rearranging them does not add or remove area.

   Cut a circle into equal sectors, like pizza slices. None of its area is lost.

   Area counts all the surface inside the circle. Moving its pieces without gaps or overlaps keeps that surface unchanged.

5. The blue and orange pieces all came from the same circle. Rearranging them does not add or remove area.

   Alternate the slices pointing up and down. They begin to look like a rectangle.

   Area counts all the surface inside the circle. Moving its pieces without gaps or overlaps keeps that surface unchanged.

6. The blue and orange pieces all came from the same circle. Rearranging them does not add or remove area.

   Its height approaches one radius. Its width approaches half the circumference, π times the radius.

   Area counts all the surface inside the circle. Moving its pieces without gaps or overlaps keeps that surface unchanged.

7. The blue and orange pieces all came from the same circle. Rearranging them does not add or remove area.

   Finer slices make the fit closer. Width times height gives πr × r.

   circle area = πr²

   Area counts all the surface inside the circle. Moving its pieces without gaps or overlaps keeps that surface unchanged.

Worked example: A new example, together

Find the area of a circle with radius 7 cm. Use π = 22/7.

1. Mark the centre, radius and required outer boundary or surface before calculating.

   Circular area scales with the square of the radius and the fraction of the circle used.

2. The radius is 7 cm; diameter is 14 cm.

   Find the area of a circle with radius 7 cm. Use π = 22/7.

   154 cm²

3. The radius is 7 cm; diameter is 14 cm.

4. Area = (22/7) × 7² ÷ 1.

5. The result is 154 cm².

6. Check the result against the question we started with.

   The area is 1 of the corresponding full-circle area.

   We found 154 cm². Circular area scales with the square of the radius and the fraction of the circle used.

Common mistakes to discuss

• Using the diameter in the area. Circumference uses the diameter; area uses the radius. For r = 7 cm, area is 22/7 × 7 × 7 = 154 cm².

Check your understanding

• Find circumference and area of a circle
• justify circle calculations using the correct radius and region
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.