Some special sectors we should make note of are semicircles which make half a circle's area and quadrants which make up a quarter of a circle's area.
We can think of an arc as a fraction of the circle. Looking at it like this, we can see that an arc length is simply a fraction of the circumference.
As such, we can calculate arc lengths by finding the circumference of the circle they are a part of and then taking the appropriate fraction.
The angle \angle AXB is equal to 120\degree.
What fraction of the whole circle lies on the arc AB?
Find the exact length of the arc AB.
To find the fraction of a circle taken up by an arc length, put the angle of the arc over the angle of a circle which is 360. \text{Fraction}=\dfrac{\text{Angle of arc}}{360} We can then use this fraction to find the arc length by multiplying it by the circumference: \text{Arc length}=\dfrac{\text{Angle of arc}}{360} \times 2\pi r
We can find the perimeter of a sector in the same way that we find any perimeter, by adding up the lengths of the sides. A sector has three sides, two straight and one curved. So how do we find the length of each side?
We know from the definition that a sector is the region between two radii and the circle, meaning that the two straight sides must be radii and the curved side is an arc.
Find the perimeter of the sector shown, correct to two decimal places.
To find the perimeter of a sector, add the two radii and the arc length that the sector is bounded by.
Similar to how the arc length is simply a fraction of the circumference, the area of a sector is simply a fraction of the circle's area.
We can calculate the area of a sector by finding the area of the circle they are a part of and then taking the appropriate fraction.
The sector in the diagram has an angle of 30\degree and a radius of 6\,cm.
What fraction of the circle's area is covered by this sector?
Find the exact area of the sector.
To find the area of a sector we can use the formula:\text{Area of sector}=\dfrac{\text{Angle of sector}}{360} \times \pi r^2
An annulus is a composite shape formed by subtracting the area of a smaller disc from a larger one, where the centre of the two discs is the same.
As such, these are all annuli.
And these are not annuli.
Since annuli are composed of an inner and outer circle, we can also say that they have an inner and outer radii, which are the distances from the central point to inside and outside edges respectively.
The inner radius is the distance from the central point to the inside edge of the annulus.
The outer radius is the distance from the central point to the outside edge of the annulus.
We can see that the perimeter of an annulus will be the sum of the circumferences of the inner and outer circles.
We can also see that the area of an annulus will be the difference between the area of the outer circle and the area of the inner circle.
The annnulus has an inner diameter of 10\, \text{cm} and an outer diameter of 18\,cm.
Find its exact area.