Help your students prepare for their Maths GCSE with this free arcs and sectors worksheet of 40 questions and answers
An arc of a circle is a portion of the circumference of a circle. To find the length of an arc, we adapt the formula for the circumference of a circle πD, where D is the diameter of the circle. We need to divide the angle of the sector by 360, and multiply this by πD to find the length of the arc. Essentially we are finding the required proportion of the circumference of the circle.
A sector of a circle is a portion of a circle enclosed by two radii and an arc. To find the area of a sector we need to adapt the formula for the area of a circle r², where r is the radius of the circle. A simple example is calculating the area of a semicircle. Here we need to find the area of the whole circle and then divide by 2; this is because a semicircle is half of a circle. For other sectors, we need to divide the angle of the sector by 360 and then multiply this by πr² to find the area of the sector.
The algebraic formula for arc length and area of a sector frequently use Θ for the angle at the centre of the circle so it is important that students recognise this notation.
The perimeter of the sector can be found by summing the arc length and two lots of the length of the radius. As problems involving arcs and sectors use π, answers will often require rounding to a given number of decimal places or significant figures. This topic can be extended further to involve finding the area of a segment which often also requires applications of trigonometry.
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