Sector Area In Circle | Calculation Concepts

In geometry, there are occassions where you have to compute the area of a sector belonging to a circle.

What is this sector about?

Sector is an area of a part of the complete circle. It is not a length!

To understand more, let us do a maths question on finding the sector area.

Question:



Find the area of the green sector given the ratio of the length of the green outline to the whole circle outline is r /L.

("L" being the radius of the circle)

Before I present the solution and steps in achieveing the answer, I would like you to understand the principles and concept of each steps. This way of studying will resort less on memory but more on long term retention of knowledge.

Solution:
Area of a sector is given by the formula S = (1/2) x (radius)² x q,

where q, is angle in radian (unit).

Click here for more information on sector area.

Since length of the sector outline is given, we need to understand the property of this parameter. This parameter is also term "Arc" length.

Arc length is given by the formula Length of Arc = r q.


Here, you see that the arc length is proportional to the angle compassing the arc or sector width. (Property of Arc Length)

In the question, the ratio of the outlines is given as r / L.

This also meant that r / L = q / 2 p. === (A)

Why?

Because "r" is proportional to the angle q, and "L" is the angle of the whole circle.
(Refer to the Property of Arc Length).

From the relation (A) above, you can re-write the expression as q = (r/L) x 2 p.


Replace this new q expression into the sector area formula.
Sector Area S = (1/2) x (radius)² x q = (1/2) x L² x (r/L) x 2 p

= (1/2) x L x (2pr)

In another words, you can say that the Sector Area is determined by :

half the radius of the large circle multiplied by the circumference of a circle with radius "r", "r" being a hidden parameter that equals the sector's arc length.

How's the going? Tough?

Hope not. But if it is, do not despair.

Review the concept a few times and it will be clearer.

Remember, knowing and mastering the concepts and principles has a long term benefit than that of pure memory of the Sector Area formula. Cheers!

:)