Question

A circle is divided into 6 sectors so that the angles of the sectors form an arithmetic progression. The largest sector angle is 4 times the smallest sector angle. Given that the radius of the circle is 5cm find the perimeter of smallest sector?

Answer 1

`=12.09 cm`

Explanation:

First you need to find the size of the smallest angle.

Let the smallest angle be `x°`
Therefore the largest angle is `4x°`

You know the size of the first and sixth (last) sectors and that the sum of all the sectors in the circle is `360°`

`S_n = n/2(a+l) " "larr` sum of the terms in an AP

`S_6 = 6/2(x+4x)= 360°`

`3(5x)=360°`

`5x=120°`

`x=24°`

The smallest sector is therefore `24/360 = 1/15` of the circle.

The perimeter is made up of `2` radii and an arc.

The length of the arc is `"fraction" xx "circumference"`

Arc length = `1/15 xx 2pi(5)" "larr (C=2pi r)`

`=2/3pi`

Perimeter `= 5+5+2/3pi`

`=12.09 cm`

This is a really nice question which includes different topics in Maths!!