# Semicircles, Quadrants and Composite Shapes

> Find the perimeter of half circles, quarter circles and shapes built from circles, rectangles and triangles.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/semicircles-and-quadrants/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The running track
The school oval has a running track shaped like a rectangle with a half circle on each end. Coach Lee wants to paint a white line around the inside edge.

The straight parts are easy: two lines of 80 m each. The curved ends are two half circles, each 42 m across. Two half circles make one whole circle, so the curved parts together are $\frac{22}{7} \times 42 = 132$ m.

The line is $80 + 80 + 132 = 292$ m long. Coach Lee split the shape into pieces she knew. That is the secret to every composite shape.

**In short:** The perimeter of a semicircle is half the circle's curve plus the straight diameter: $\frac{1}{2}\pi d + d$. A quadrant is a quarter circle, so its perimeter is a quarter of the curve plus two radii. Always remember to add the straight edges.

## Before you start

- Circumference of a circle: $C = \pi d$, with $\pi \approx \frac{22}{7}$ or $3.14$ (see Circles and Circumference).
- The diameter is twice the radius.
- Perimeter means the total distance around the **outside** of a shape.

## Semicircles

A **semicircle** is half a circle. Its perimeter has two parts: the curved arc and the straight diameter.

*Diagram: A semicircle with its curved arc and straight diameter, and a quadrant with its curved arc and two straight radii*
*Do not forget the straight edges: one diameter for a semicircle, two radii for a quadrant.*

### Key idea: Perimeter of a semicircle
$$P = \frac{1}{2} \times \pi \times d + d$$

### Worked example: Semicircles
Diameter 14 cm, $\pi = \frac{22}{7}$: arc $= \frac{1}{2} \times 44 = 22$ cm. Perimeter $= 22 + 14 = 36$ cm.

Diameter 20 cm, $\pi = 3.14$: arc $= \frac{1}{2} \times 62.8 = 31.4$ cm. Perimeter $= 31.4 + 20 = 51.4$ cm.

## Quadrants

A **quadrant** is a quarter of a circle. Its edge is a curved arc plus **two radii**.

### Key idea: Perimeter of a quadrant
$$P = \frac{1}{4} \times \pi \times d + 2r$$

### Worked example: Quadrants
Radius 14 cm, $\pi = \frac{22}{7}$: the full circle would be $\frac{22}{7} \times 28 = 88$ cm. A quarter is 22 cm. Perimeter $= 22 + 14 + 14 = 50$ cm.

Radius 10 cm, $\pi = 3.14$: full circle $62.8$ cm, quarter $15.7$ cm. Perimeter $= 15.7 + 20 = 35.7$ cm.

### Watch out: The straight edges count
The most common slip is to give only the curved part. A semicircle's perimeter includes its diameter. A quadrant's includes both radii.

## Composite shapes

A **composite shape** is made of simpler shapes joined together.

### Key idea: Trace the outside
1. Run your finger around the **outside** edge only.
2. List every piece you pass: straight lines and curved arcs.
3. Work out each piece, then add.

Edges **inside** the shape, where two pieces join, are not part of the perimeter.

### Worked example: Rectangle with a semicircle on top
A rectangle 14 cm wide and 3 cm tall has a semicircle on its top edge. Use $\pi = \frac{22}{7}$.

Outside edge: the arc, which is half of 44, so 22. Then the right side 3, the bottom 14 and the left side 3. The top of the rectangle is inside, so it does not count.

$P = 22 + 3 + 14 + 3 = 42$ cm.

### Worked example: Triangle and semicircle
An equilateral triangle with perimeter 42 cm sits on top of a semicircle. The triangle's sides are each 14 cm, and its base is the semicircle's diameter.

Outside edge: two triangle sides, 28 cm, and the semicircle arc, 22 cm. $P = 50$ cm.

### Exam tip: Two halves make a whole
Two equal semicircles have the same curved length as one full circle. Four equal quadrants also make one full circle. Spotting this saves time.

### Challenge: Selective level: a cut-out
A rectangle 36 cm wide and 18 cm tall has a semicircle of diameter 10 cm cut out of the middle of its bottom edge. Use $\pi = 3.14$.

Outside edge: top 36, two sides of 18, two bottom pieces of $\frac{36 - 10}{2} = 13$ each, and the arc of the cut-out, $\frac{1}{2} \times 3.14 \times 10 = 15.7$.

$P = 36 + 18 + 18 + 13 + 13 + 15.7 = 113.7$ cm. Cutting a hole into the edge makes the perimeter **longer**, not shorter.

**Mind map**

```text
Curved perimeters
  Semicircle
    Half of πd
    Plus the diameter
  Quadrant
    Quarter of πd
    Plus two radii
  Composite shapes
    Trace the outside only
    Inside joins do not count
  Shortcuts
    2 semicircles = 1 circle
    4 quadrants = 1 circle
```

## Quick summary

- Semicircle perimeter $= \frac{1}{2}\pi d + d$.
- Quadrant perimeter $= \frac{1}{4}\pi d + 2r$.
- Always add the straight edges.
- For a composite shape, trace the outside edge and add every piece. Edges inside the shape do not count.
- Two equal semicircles make a full circle of curve, and so do four equal quadrants.

## Quiz

Try these questions from the chapter quiz.

1. Using $\pi = \frac{22}{7}$, what is the curved length of a semicircle with diameter 14 cm?
   - A. 44 cm
   - B. 22 cm
   - C. 11 cm
   - D. 36 cm
   - Answer: B. 22 cm. A full circle is 44 cm, so half is 22 cm.

2. Using $\pi = \frac{22}{7}$, what is the full perimeter of a semicircle with diameter 14 cm?
   - A. 22 cm
   - B. 36 cm
   - C. 58 cm
   - D. 44 cm
   - Answer: B. 36 cm. Arc 22 cm plus the diameter 14 cm.

3. How many straight edges does a quadrant have?
   - A. 0
   - B. 1
   - C. 2
   - D. 4
   - Answer: C. 2. A quadrant has two radii as straight edges.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/semicircles-and-quadrants/#quiz