Circles
The perimeter of a circle has a special name, the circumference.
The circumference of a circle can be calculated with this formula:
$\text{Circumference}=\pi\times\text{diameter}$Circumference=π×diameter
The diameter is twice the length of the radius, so we can also write the formula as:
$\text{Circumference}=2\times\pi\times\text{radius}$Circumference=2×π×radius
Arc length and perimeter of sectors
Sectors are pieces of a circle that have been sliced out from the centre, like the ones pictured below:
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| examples of sectors | ||
The length of the the circular edge is called the arc length of the sector. The arc length of a quarter circle is a quarter of the circumference of the whole circle: $\frac{1}{4}\times2\pi r$14×2πr. A quarter circle has a contained angle of $90^\circ$90° out of a total possible $360^\circ$360° and the corresponding proportion is indeed a quarter:$\frac{90}{360}=\frac{1}{4}$90360=14. If instead we have a sector with a contained angle of $30^\circ$30°, then we have $30^\circ$30° out of a total possible $360^\circ$360°, so the fraction of the circle that we have is $\frac{30}{360}=\frac{1}{12}$30360=112 and the arc length will be $\frac{1}{12}\times2\pi r$112×2πr. This is the idea behind the arc length formula.
A sector with contained angle $\theta$θ corresponds to a fraction $\frac{\theta}{360}$θ360 of a full circle and so its arc length $l$l is given by the formula:
$l=\frac{\theta}{360}\times2\pi r$l=θ360×2πr
where $r$r is the radius of the circle.
Remember that the perimeter of a 2D shape is the total distance around the boundary of the shape. For a sector the boundary consists of a curved part and two straight edges each with length equal to the radius of the circle.
When finding the perimeter of a sector, don't forget to add the lengths of the straight edges to the arc length!
| $\text{Perimeter of a sector}=\text{arc length}+2r$Perimeter of a sector=arc length+2r |
Practice questions
QUESTION 1
Find the circumference of the circle shown, correct to two decimal places.
QUESTION 2
If the radius of a circle is equal to $27$27 cm, find its circumference correct to one decimal place.
Question 3
A circle with radius $4$4 cm has been drawn with a dashed line.
A sector is outlined with a filled line.
Find the exact circumference of the whole circle.
Find the exact length of the arc of the sector.
Find the perimeter of the sector.
Round your answer to two decimal places.
Question 4
A sector has radius $6$6 cm and an angle of $55^\circ$55°.
Find the exact length of the arc.
Find the perimeter of the sector.
Round your answer to two decimal places.
Composite shapes with curves
Now that we also know how to find the circumference of a circle, we can find the perimeter of composite shapes that involve parts of a circle The concept is the same, that all perimeters can be found by adding up one side at a time as we travel around the shape (even if the side is circular).
This is a composite shape, made up of a semicircle and a rectangle. Although, we are missing one side of the rectangle and the base of the semicircle as that common line lies inside the shape.
We have the following:
- $3$3 straight sides: two of length $4$4 cm and one of length $8.4$8.4 cm.
The sum of these lengths $=2\times4+8.4=16.4$=2×4+8.4=16.4 cm
- A semicircle has radius $r=\frac{8.4}{2}=4.2$r=8.42=4.2 cm.
Arc length $=\frac{2\pi\times4.2}{2}=4.2\pi$=2π×4.22=4.2π cm.
Thus, the total perimeter is:
| $\text{Perimeter}$Perimeter | $=$= | $2\times4+8.4+\frac{2\pi\times4.2}{2}$2×4+8.4+2π×4.22 |
| $=$= | $16.4+4.2\pi$16.4+4.2π cm | |
| $=$= | $29.6$29.6 cm (rounded to $1$1 decimal place) |
Practice questions
QUESTION 5
Find the perimeter of the arch shown (shaded).
(Give your answer in terms of $\pi$π or correct to two decimal places.)
Question 6
Find the perimeter of the shape (shaded) shown.
Give your answer correct to $2$2 decimal places.