10.03 Perimeter

The perimeter is the distance around a two dimensional shape. To find the perimeter of any polygon (straight sided shape) add all the lengths of the sides, ensuring all lengths are written in the same unit.

Perimeter is a measure of length, so different units of measure could be used to measure the perimeter of different sized objects. Millimetres could be used to measure the perimeter of a SIM card, centimetres to measure the perimeter of a wallet, metres to measure the perimeter of a room and kilometres to measure the perimeter of a town.

If no particular unit for the context or question is given it is also good mathematical practice to use the word 'units'.

Here is a scalene triangle. To find the perimeter add the three side lengths. Notice that all sides are measured using the same units.

Here is a rectangle. Recall that rectangles have opposite sides of equal length.

A square has 4 sides of the same length, so the perimeter of a square will be 4 times one of the side lengths.

All perimeters can be found by traveling around the shape, adding up one side at a time.

Here is a trapezium. To find its perimeter add the side lengths.

Also keep in mind the ability to construct simple rules for other shapes as necessary, and to group together sides of the same lengths.

Examples

Example 1

Find the perimeter of the shaded region shown, where edges are 1 unit in length each.

Worked Solution

Create a strategy

Count the unit edges around the shaded region.

Apply the idea

\text{Perimeter}=20\text{ units}

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Example 2

Find the perimeter of the figure shown.

Worked Solution

Create a strategy

The perimeter of a figure is the sum of all of its sides.

Apply the idea

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 10 + 7 + 9 + 11 + 9 + 8	Add the sides
	\displaystyle =	\displaystyle 54 \text{ cm}	Evaluate

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The perimeter of a circle, the circumference, can be used to find the perimeter of composite shapes that involve full circles, or semicircles.

The circumference C of a circle is given by: \\C=2\pi r=\pi d, where r is the radius and d is the diameter of the circle.

This is a composite shape, made up of a semicircle and a rectangle. Although, we are missing one side of the rectangle, we can use the circumference of a circle, and halve this, to find the length of the rounded edge.

We have 3 straight sides, two with length 4 cm and one with length 8.4 cm. The sum of these lengths is equal to:

2 \times 4 + 8.4 = 16.4\text{ cm}

Lastly we have a semicircle of radius \dfrac{8.4}{2}=4.2\text{ cm}. So the length of the rounded edge is equal to:

\dfrac{2\pi \times 4.2}{2} = 4.2\pi\text{ cm}

In total, the perimeter is:

\text{Perimeter}=16.4 + 4.2\pi \approx 29.6 \text{ cm} (1 \text{d.p.})

Examples

Example 3

Find the circumference of the circle shown, correct to two decimal places.

Worked Solution

Create a strategy

The circumference of the circle can be found using the formula: C=2\pi r.

Apply the idea

\displaystyle C	\displaystyle =	\displaystyle 2\times \pi \times 8	Substitute r=8
	\displaystyle =	\displaystyle 50.27\text{ cm}	Evaluate

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Example 4

Find the perimeter of the shape (shaded) shown, correct to two decimal places.

Worked Solution

Create a strategy

We need to add the lengths of the borders for the perimeter.

Apply the idea

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 37.0 + 37.0 + \dfrac{37.0}{2}\pi + \dfrac{37.0}{2}\pi	Substritute r=37.0
	\displaystyle =	\displaystyle 37.0 + 37.0 + 18.5\pi + 18.5\pi	Evaluate the division
	\displaystyle =	\displaystyle 37.0 + 37.0 + 116.24	Add and simplify terms with \pi
	\displaystyle =	\displaystyle 190.24 \text{ units}	Evaluate

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An arc of a circle is a curved line formed from part of the circumference of the circle. The length of an arc is called the arc length, 1.

A sector is a shaped formed from part of a circle, where the sector's boundary or perimeter is formed by two radii and an arc.

Finding the perimeter of a sector involves first calculating the arc length, then adding on the lengths of the two radii.

Note that a quarter of a circle, is sometimes called a quadrant, and half of a circle is usually called a semicircle.

Examples

Example 5

Find the perimeter of the sector equal to \dfrac{1}{4} of a circle with radius 6.5 cm. Round to one decimal place.

Worked Solution

Create a strategy

We need to first find the value of arc lenght (l) and then substitute it to the formula: \\\text{Perimeter}=\text{arc length}+2\times \text{radius}.

Apply the idea

To find the arc length:

\displaystyle l	\displaystyle =	\displaystyle \dfrac{1}{4} \times 2\pi r	Write the formula
	\displaystyle =	\displaystyle \dfrac{1}{4} \times 2\times \pi \times 6.5	Substitute r=6.5
	\displaystyle =	\displaystyle 3.25\pi	Evaluate and leave answer in terms of \pi

To find the perimeter:

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle l + 2 \times r	Write the formula
	\displaystyle =	\displaystyle 3.25\pi + 2 \times 6.5	Susbstitute l=3.25\pi,\,r=6.5
	\displaystyle =	\displaystyle 23.2 \text{ cm}	Evaluate and round to 1 decimal place

Example 6

What is the perimeter of a semicircle with diameter 8 cm, correct to two decimal places?

Worked Solution

Create a strategy

We can use the circumference formula: C = \pi \times D, then multiply it by \dfrac{1}{2} since it is semicircle, and the add the diameter to find the perimeter.

Apply the idea

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle \pi \times 8 \times \dfrac{1}{2} + 8	Substitute D=8
	\displaystyle =	\displaystyle 12.57 + 8	Evaluate the multiplication
	\displaystyle =	\displaystyle 20.57\text{ cm}	Evaluate

Example 7

A circle with radius 4\text{ cm} has been drawn with a dashed line. A sector is outlined with a filled line.

a

Find the exact circumference of the whole circle.

Worked Solution

Create a strategy

We can use the formula for circumference: C=2\pi r

Apply the idea

\displaystyle C	\displaystyle =	\displaystyle 2\pi \times 4	Substitute r=4
	\displaystyle =	\displaystyle 8\pi\text{ cm}	Simplify

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b

Find the exact length of the arc of the sector.

Worked Solution

Create a strategy

The sector is a quarter of the circle, so we can use the formula for the arc length of the sector: l= \dfrac{1}{4} \times 2\pi r .

Apply the idea

We know the sector is a quarter of the circle because of the right-angle marking at the centre.

\displaystyle l	\displaystyle =	\displaystyle \dfrac{1}{4} \times 2\times \pi \times 4	Substitute r=4
	\displaystyle =	\displaystyle 2\pi\text{ cm}	Simplify

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c

Find the perimeter of the sector. Round the answer to two decimal places.

Worked Solution

Create a strategy

The perimeter of a sector made up of the arc length, l, and two radii, 2r. So we add these length together.

Apply the idea

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle l + 2r	Add the arc and the radii
	\displaystyle =	\displaystyle 2\pi + 2 \times 4	Substitute l and r
	\displaystyle =	\displaystyle 14.28 \text{ cm}	Evaluate and round

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