Arc length, sector area and sector perimeter from the angle, and working back to the angle or the radius.
What you will learn
- Find an arc length and a sector area
- Find the perimeter of a sector
- Find the angle or radius of a sector from its arc or area (Higher)
A sector is a fraction of a circle
A sector is a slice of a circle between two radii, like a slice of pizza.
Its curved edge is an arc. A sector with angle θ at the centre is θ/360 of the whole circle. So take that fraction of the circumference, or of the area.
arc length = θ/360 × 2πr
sector area = θ/360 × πr²
Not drawn accurately
- arc = 80/360 × 2 × π × 6 = 8.3775... = 8.38 cm (3 s.f.)
- area = 80/360 × π × 6² = 25.132... = 25.1 cm² (3 s.f.)
- perimeter = 8.3775... + 6 + 6 = 20.4 cm (3 s.f.)
The fraction also works backwards. A sector of 90° is 90/360 = 1/4 of the circle; if its area is 20 cm², the whole circle is 20 ÷ 1/4 = 80 cm². Cutting a circle into 8 equal slices gives each slice an angle of 360 ÷ 8 = 45°, and each slice is 1/8 of the area and 1/8 of the circumference.
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- A sector is a fraction of a circle
- Working backwards (Higher)
- Try it: Fractions of a circle
7 practice questions, marked as you go; about 25 minutes in all.
Key terms
Sector, Arc
In this unit: Area, perimeter and volume
Perimeter and area of rectangles, triangles, parallelograms, trapeziums and compound shapes, circles, arcs and sectors, and the volume and surface area of prisms, cylinders, pyramids, cones and spheres.