Grade by Grade
E. Spatial Sense · Spatial Sense: 3D Objects, Transformations, Circles, Surface Area and Volume

The Area of a Circle

E2.5 A1About 50 minutes
Learning goals
  • I can explain where the formula A = π × r × r comes from by cutting a circle into sectors and rearranging them.
  • I can find the area of a circle when I know its radius or diameter.
  • I can work backwards from an area to find the radius.
  • I can find the areas of half circles, quarter circles and rings.

Words to know

area
The amount of surface a 2D shape covers, measured in square units like cm² and m².
sector
A slice of a circle shaped like a piece of pizza, between two radii.
semicircle
Half of a circle, cut along a diameter.
r² (r squared)
r × r. For example, if r = 5, then r² = 5 × 5 = 25.
π (pi)
About 3.14. The number of diameters that fit around a circle.

Remember from earlier grades

In Grade 5 you found the area of a parallelogram: base × height. In Grade 6 you found areas of trapezoids, rhombuses, kites and shapes made of polygons. A circle has no straight sides, so we need a clever idea.

Let's start with a pizza

One pizza has a diameter of 40 cm. Another has a diameter of 20 cm. Is the big one twice as much pizza? It's much more than that! To find out how much more, we need the area of a circle.

A large cheese pizza and a small cheese pizza about half as wide, side by side on a wooden table, seen from above.
The small pizza is about half as wide, but it has only about one quarter of the area. Doubling the diameter makes the area four times as big. AI-generated image

Where the formula comes from

  1. Cut a circle into equal sectors, like pizza slices.
  2. Lay them in a row, pointing up and down, taking turns.
  3. The new shape looks almost like a parallelogram. The more slices you cut, the straighter its sides get.
  • Its height is the radius, r.
  • Its base is made of the crust of half the slices, so it's half the circumference: half of 2 × π × r = π × r.

Area of a parallelogram = base × height = π × r × r.

Area of a circle: A = π × r × r = π × r²

→rabout π × r(half the circumference)
A circle cut into 8 sectors and rearranged into a shape that is close to a parallelogram (not to scale). Its height is about the radius r, and its base is about half the circumference, π × r. So the area is about π × r × r.
Worked example

Example 1: Given the radius

Find the area of a circle with a radius of 5 cm.

  1. A = π × r × r ≈ 3.14 × 5 × 5.
  2. 5 × 5 = 25, and 3.14 × 25 = 78.5.

About 78.5 cm².

Worked example

Example 2: Given the diameter

A pizza has a diameter of 30 cm. What is its area?

  1. Find the radius first: r = 30 ÷ 2 = 15 cm.
  2. A ≈ 3.14 × 15 × 15 = 3.14 × 225 = 706.5.

About 706.5 cm².

Worked example

Example 3: Working backwards

A lawn sprinkler waters a circle with an area of 314 m². How far does the water reach (the radius)?

  1. A = π × r × r, so r × r = A ÷ π ≈ 314 ÷ 3.14 = 100.
  2. Which number times itself makes 100? 10 × 10 = 100, so r = √100 = 10 (the square root of 100, from the Number Sense lesson on square roots).

The radius is about 10 m.

Worked example

Example 4: A ring

A round flower bed has a radius of 3 m. A path 2 m wide goes all the way around it, so the outer edge of the path has a radius of 5 m. What is the area of the path?

  1. Area of the big circle: 3.14 × 5 × 5 = 78.5 m².
  2. Area of the flower bed: 3.14 × 3 × 3 = 28.26 m².
  3. The path is the big circle minus the flower bed: 78.5 − 28.26 = 50.24 m².

About 50.24 m².

Worked example

Example 5: A semicircle

Find the area of a semicircle with a diameter of 8 cm.

  1. r = 8 ÷ 2 = 4 cm.
  2. Whole circle: 3.14 × 4 × 4 = 50.24 cm².
  3. Half: 50.24 ÷ 2 = 25.12 cm².

About 25.12 cm².

Don't mix up the formulas

  • Circumference (distance around, in cm): C = π × d = 2 × π × r.
  • Area (surface covered, in cm²): A = π × r × r.

Area always uses the radius, and its unit is always squared. And r² means r × r, not r × 2!

r = 20r = 10d = 40 cm: about 1256 cm²d = 20 cm: about 314 cm²1256 ÷ 314 = 4Double the diameter,get 4 times the pizza!
Back to the pizzas: 3.14 × 20 × 20 ≈ 1256 cm² and 3.14 × 10 × 10 = 314 cm². The big pizza is 4 times as much, not 2 times, because the radius is multiplied by itself.
Hands-on activity

Slice and rearrange

You need: a paper plate (or a circle cut from paper), a ruler, scissors, glue or tape, and paper. Ask a grown-up to help with scissors.

  1. Fold the circle in half 3 times, then open it. You now have 8 equal sectors. Colour the crust edge of 4 of them, taking turns, then cut along the folds.
  2. Arrange the sectors in a row, alternating up and down, so they make a shape like a parallelogram. Glue them down.
  3. Measure the radius of your circle and the base and height of your parallelogram shape. Is the height close to r? Is the base close to half the circumference?
  4. Bonus: trace a circle on 1 cm grid paper and count the squares inside (count squares that are more than half inside as 1). Compare with π × r × r.

What to look for: the base is close to 3.14 × r, and the area from counting is close to the formula.

Just for fun

Why did the circle keep its area formula a secret?

It was too pi-vate.

Your turn

1

Find the area of a circle with a radius of 3 cm.

2

Find the area of a circle with a radius of 10 m.

3

Find the area of a circle with a diameter of 8 cm.

4

Find the area of a circle with a radius of 1.5 m.

5

A circle has an area of 12.56 cm². What is its radius?

6

Find the area of a semicircle with a radius of 6 cm.

7

Is the area of a circle measured in cm or cm²? Why?

More practice: warm-up

1

Area of a circle with r = 1 cm?

2

Area of a circle with r = 2 cm?

3

Area of a circle with d = 2 cm?

4

What is r² when r = 7?

5

Area of a circle with r = 4 cm?

More practice: on your own

1

A round rug has a diameter of 2 m. What is its area?

2

Which is more pizza: one pizza with a diameter of 40 cm, or two pizzas with diameters of 20 cm? Explain.

3

Find the area of a quarter circle with a radius of 10 cm.

4

A ring has an outer radius of 6 m and an inner radius of 4 m. Find its area.

5

A circle has an area of 78.5 m². Find its radius and diameter.

Challenge

1

A circle fits exactly inside a square with 10 cm sides. What is the area of the square that is not covered by the circle?

2

If you double the radius of a circle, what happens to its area? Test it with r = 3 and r = 6.

3

A goat is tied with a 5 m rope to a post in the middle of a long, straight fence. It can only graze on one side of the fence. How much area can it graze?

Check your understanding

Quiz

1.

Which formula gives the area of a circle?

2.

What is the area of a circle with a radius of 10 cm?

3.

What is the area of a circle with a diameter of 6 cm?

4.

When the sectors of a circle are rearranged into a parallelogram shape, why is the base about π × r?

5.

A circle has an area of 50.24 cm². What is its radius?

6.

A ring has an outer radius of 2 m and an inner radius of 1 m. What is its area?

For parents and teachers

Big idea: E2.5 asks students to use the relationships between the radius, diameter and area of a circle to explain the area formula and to solve problems. The sector rearrangement shows why A = π × r × r, connecting it to the parallelogram formula from Grade 5.

How to help at home: Do the paper-plate activity. Compare pizza sizes using area (a great real-life surprise). Ask your child to explain the formula in their own words.

Common difficulties:

  • Using the diameter in π × r × r. Halve it first.
  • Thinking r² means r × 2. It means r × r.
  • Mixing up area and circumference, or writing cm instead of cm².
  • Working backwards: after dividing by π, students need the number that times itself makes r × r. That is a square root, which they learned in the Grade 7 Number Sense lesson Square Roots of Perfect Squares (B1.2): √100 = 10, √25 = 5. Our problems use perfect squares; estimating the square root of a number that is not a perfect square comes in Grade 8.

This lesson builds A1 skills: reasoning and proving, and representing an idea in more than one way.