Circles

A circle can fool you twice: the distance around it is a boundary, while the space inside it is an area. Semicircles and quarter circles add straight edges to that curved boundary, so the safest habit is to point to every piece before calculating.

Parents: let your child name the curved and straight pieces first, then reveal the working. This page uses $\pi=\frac{22}{7}$ whenever a numerical circle calculation is shown.

By the end you'll be able to identify radius and diameter, find circumference and area, and handle semicircles, quarter circles and composite figures.

Radius and diameter

The radius runs from the centre to the circumference. The diameter runs from one edge to the other through the centre. A diameter is always two radii, so the two links are:

$d=2r$ and $r=\frac{d}{2}$.

Show the worked solution

For that check, halve the given diameter: 18 cm ÷ 2 = 9 cm.

Write the given length beside the centre before choosing a formula.

Circumference: the distance around

The circumference is a length. Use $C=\pi d$. If the question gives a radius, double it first. The result has a plain length unit such as cm.

For the worked example below, the diameter is 14 cm and the question states the convention. Predict before opening the calculation.

Show the worked solution

Circumference

$$C=\pi d=\frac{22}{7}\times14$$

$$C=44\,\text{cm}$$

Area: the space inside

The area of a circle uses the radius twice: $A=\pi r^2$. If a diameter is given, halve it before squaring. The result has a square unit.

Show the worked solution

Area of a circle

$$A=\pi r^2=\frac{22}{7}\times7^2$$

$$A=\frac{22}{7}\times49=154\,\text{cm}^2$$

Semicircles and quarter circles: trace every edge

A semicircle has a curved half and a straight diameter. Its area is half of a full circle, but its outside boundary includes both pieces. A quarter circle has one curved quarter and two straight radii.

━ curved boundary━ straight boundary┄ shared/internal edge

Show the worked solution

Semicircle boundary

$$\text{curved half}=\pi r=\frac{22}{7}\times7=22\,\text{cm}$$

$$\text{outside boundary}=22+14=36\,\text{cm}$$

━ curved boundary━ straight boundary┄ shared/internal edge

Show the worked solution

Quarter-circle boundary

$$\text{curved quarter}=\frac{1}{4}\times2\pi r=\frac{1}{2}\times\frac{22}{7}\times14=22\,\text{cm}$$

$$\text{outside boundary}=22+14+14=50\,\text{cm}$$

Composite area and perimeter

Composite figures are built from familiar pieces. For area, add attached pieces or subtract a cut-out. For perimeter, trace only the outside; never add the edge where two pieces touch inside the shape.

━ curved boundary━ straight boundary┄ shared/internal edge

Show the worked solution

Rectangle plus semicircle

$$\text{rectangle}=14\times7=98\,\text{cm}^2$$

$$\text{semicircle}=\frac{1}{2}\times\frac{22}{7}\times7^2=77\,\text{cm}^2$$

$$\text{total area}=98+77=175\,\text{cm}^2$$

━ curved boundary━ straight boundary┄ shared/internal edge

Show the worked solution

Square with a curved cut

$$\text{square boundary}=4\times14=56\,\text{cm}$$

$$\text{quarter arc}=\frac{1}{2}\times\frac{22}{7}\times7=11\,\text{cm}$$

$$\text{outside boundary}=56-7-7+11=53\,\text{cm}$$

━ curved boundary━ straight boundary┄ shared/internal edge

Show the worked solution

Rectangle plus triangle boundary

$$\text{outside}=5+6+5+8+10$$

$$\text{outside}=34\,\text{cm}$$

Try the circle parts lab

🔬

Circle parts lab: inside or outside?

🤔 Predict first: For a semicircle, which pieces belong to its outside boundary?

Curved arc and straight diameterCurved arc onlyStraight diameter only

The lab keeps the radius, diameter, radii and shared edges visible. Decide whether you are counting inside area or tracing the full outside before you calculate.

Watch out — easily mixed up

Three graduated practices

1. Area of a quarter circle

━ curved boundary━ straight boundary┄ shared/internal edge

Show the worked solution

Practice 1

$$A=\frac{1}{4}\times\frac{22}{7}\times14^2$$

$$A=\frac{616}{4}=154\,\text{cm}^2$$

━ curved boundary━ straight boundary┄ shared/internal edge

2. Boundary of an attached semicircle

━ curved boundary━ straight boundary┄ shared/internal edge

Show the worked solution

Practice 2

$$\text{curved half}=\pi r=\frac{22}{7}\times7=22\,\text{cm}$$

$$\text{outside}=14+7+7+22=50\,\text{cm}$$

━ curved boundary━ straight boundary┄ shared/internal edge

3. Remaining area after a curved cut

━ curved boundary━ straight boundary┄ shared/internal edge

Show the worked solution

Practice 3

$$\text{square}=14\times14=196\,\text{cm}^2$$

$$\text{cut-out}=\frac{1}{4}\times\frac{22}{7}\times7^2=38.5\,\text{cm}^2$$

$$\text{remaining}=196-38.5=157.5\,\text{cm}^2$$

━ curved boundary━ straight boundary┄ shared/internal edge

Quick recap

🎯 Mastery check

Answer all 8 — your progress is saved on this device.

1. A circle has radius 9 cm. What is its diameter?

   4.5 cm18 cm81 cm

2. Use $\pi=\frac{22}{7}$. A circle has diameter 14 cm. What is its circumference?

   22 cm154 cm44 cm

3. Use $\pi=\frac{22}{7}$. A circle has radius 7 cm. What is its area?

   154 cm²44 cm²77 cm²

4. Use $\pi=\frac{22}{7}$. A semicircle has radius 7 cm and diameter 14 cm. What is its full outside boundary?

   22 cm44 cm36 cm

5. Use $\pi=\frac{22}{7}$. A quarter circle has radius 14 cm. What is its full outside boundary?

   22 cm50 cm154 cm

6. Use $\pi=\frac{22}{7}$. A 14 cm by 7 cm rectangle has a radius-7 semicircle attached along its 14 cm top edge. What is the total area?

   175 cm²50 cm²98 cm²

7. Use $\pi=\frac{22}{7}$. A 14 cm square has a radius-7 quarter-circle cut from one corner. What area remains?

   196 cm²38.5 cm²157.5 cm²

8. A 6 cm by 5 cm rectangle shares its 6 cm side with a right triangle whose other sides are 8 cm and 10 cm. What is the outside boundary?

   40 cm34 cm24 cm

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