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?

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Answer 1 · 2017-12-07T07:56:46

$=12.09 cm$

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 $= 5+5+2/3pi$

$=12.09 cm$

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