CIRCLE – PERIMETER, AREA OF A SECTOR OF A CIRCLE

CIRCLE – PERIMETER, AREA OF A SECTOR OF A CIRCLE

Area of a sector of a circle and perimeter. In the previous article, we discussed terms related to circles and their definitions. We also defined the sector of a circle as a plane figure bounded by two radii and an arc. In this article, we will explain in detail, the sector of a circle – how to calculate the perimeter and area of a sector of a circle.

PERIMETER OF A SECTOR OF A CIRCLE

The word perimeter simply means the distance round an object. So the perimeter of a sector of a circle is the distance round the sector. That is the sum of the two radii and the arc which forms the sector. The diagram below shows the perimeter of a sector AOB

perimeter of a sector

So the perimeter of the sector AOB is the sum of two radii (2r) and the length of the arc l, where r is the radius and l, length of the arc.

Therefore, the Perimeter of a sector, AOB=2r+l units.

Example 1: Find the perimeter of the sector of radius 3.5cm which subtends an angle of 45^{0}. {Take π=\frac{22}{7}}.

The perimeter of a sector =2r+l

r=3.5cm, l=?, θ=45^{0}

\frac{45}{360}\times 2\times \frac{22}{7}\times 3.5

=\frac{1}{8}\times 2\times \frac{22}{7}\times \frac{7}{2}

\frac{22}{8}=\frac{11}{4}=2.75cm

∴ Length of arc, l=2.75cm

So the perimeter of the sector is 2r+l.

=(2\times 3.5)+2.75

7+2.75=9.75cm

Example 2: Find the perimeter of the major arc of the circle below correct to the nearest cm. {Take π=\frac{22}{7}}.

perimeter of a major arc jpg

From the diagram above, ACB is the major arc

∴ \angle ACB=360^{0}-105^{0}=255^{0}

So r=6cm, θ = 255^{0}, and

So the Length of  arc AOB is \frac{255}{360}\times 2\times \frac{22}{7}\times 6cm

=\frac{85}{120}\times 2\times \frac{22}{7}\times 6cm

=\frac{17}{24}\times 2\times \frac{22}{7}\times 6cm

l=26.71cm

∴ The perimeter of the sector =2r+l

=(2\times 6)+26.71

=12+26.71

=38.71cm

Read also: Circle – length of an arc, Standard form of a number, Quadratic Equation.

We will consider one final example on the perimeter of the sector of a circle. In this example, we are given the radius and the angle subtended by the arc and required to find the the perimeter of the sector. If you are a student preparing for NECO, WAEC (SSCE or GCE), this are the kind of tricky questions you should expect in your examination.

Example 3: A sector of a circle with radius 6cm subtends an angle of 60^{0} at the center. Calculate its perimeter in terms of π. (WAEC 2020). 

Firstly, list all the parameters you are given and the one you are asked to find.

Given the parameters, r=6cm, θ = 60^{0} and Perimeter = ?

So Perimeter =l+2r, where l= the length of an arc

∴ Perimeter =\frac{60}{360}\times 2\times \barwedge \times 6+2\times 6

=\frac{6}{36}\times 2\times \barwedge \times 6+12

\frac{1}{6}\times 2\times \barwedge \times 6+12

=2\barwedge +12

=2(\barwedge +6)cm

AREA OF SECTOR OF A CIRCLE

The sector of a circle is a portion of the whole circle. Recall that the area of a circle is πr^{2}, so the area of a sector is,

area of a sector

Therefore, the area of a sector is

area of a sector

Where θ = Angle formed at the center by the arc of the circle.

Note: If the angle subtended at the center is in radian, then the area of the sector is given by:

area of a sector in radian

∴ The area of the sector of a circle (with the angle in radian) is

=\frac{1}{2}\times (radius^{2})\times angle (in radian) subtended by the arc at the center.

Example 1: Find the area of the sector of a circle of radius 4.8cm which subtends an angle of 135^{0} at the centre. { Take π = 3.142}.

Given the following parameters: r=4.8cm, θ = 135^{0} and π = 3.142

∴ Area of sector = \frac{135}{360}\times3.142\times (4.8)^{2}

=\frac{3}{8}\times 3.142\times 4.8\times 4.8

=0.375\times 3.142\times 4.8\times 4.8

=27.1468cm^{2}

≅ 27.15cm^{2} ( to 2 decimal place).

AREA OF THE SECTOR OF A CIRCLE GIVEN THE LENGTH OF THE ARC OF THE CIRCLE

The area of the sector of a circle can also be found, if the length of the arc of the circle is known. For instance, if AB is an arc of a circle, center 0, radius rcm and AB=lcm, it can be shown that the area of this sector AOB is  \frac{rl}{2}cm^{2}.

perimeter of a sector

Proof: Let θ be the angle subtended at the center by arc AB and l be the length of arc AB, as shown in the diagram above.

Simplifying further, we have

When we substitute θ in (i) into (ii), we will have

Simplifying the above expression, we have

This is the formula for the area of a sector when the length of arc and radius of the circle are known.

Example 1: AB is an arc of a circle of length 10.2cm  with center o and the radius is 4.8cm . Find the area of the sector AOB. 

Given the parameters, length of arc, l=10.2cm and radius, r=4.8cm

∴ the area of the sector, AOB=\frac{rl}{2}cm^{2}

=\frac{10.2\times 4.8}{2}

=\frac{48.96}{2}cm^{2}

=24.48cm^{2}

Take quizzes on this topic. Click HERE to download the worksheet. The worksheet contains some West African Examination Council (WAEC) SSCE and GCE questions. Solving questions on perimeter and area of a sector, can be very easy and interesting. Follow the examples above, and solve further related questions to consolidate your knowledge on the topic.

In our next article, we will discuss the perimeter and area of segments of a circle. We urge you to look forward to that article.

For insightful Mathematics videos, click HERE.

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October 1, 2021
MATH ARTICLES
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