# Does the area of a sector change when the radius of the circle is doubled?

If the radius of a circle is doubled and the central angle of a sector is unchanged, how is the area of the sector changed?

Relevance

it is increased by a factor of 4

a sector is a fraction f of a circle radius r.

Area of the circle is pi * r * r

Area of the sector is f x pi x r^2

Length of radius is doubled... r2 = 2 x r

A2 is now pi x 2 x r x 2 x r

ie. 4 pi r^2

Area2 of the sector is f x 4 x pi x r^2

The area of a sector is a fraction of the area of the circle. So just think if you increase the radius of a circle, the area is 4 times the original area. Sector of new circle will thus be also 4 times the area of the old circle.

Just imagine you have two pizzas one with 8" dia (or 4" radius) and the other with 16" dia (or 8" radius). If you cut each of the pizzas into say 6 slices, in both cases the angle of the sector would be 60 degrees. However you can yourself imagine the slice of 16" dia pizza would be much bigger (infact 4 times) the slice of 8" dia pizza.

So even with the same central angle of a sector the area always depends on the radius (or diamaeter) and is proportional to the square of the radius (or diameter). If you double the radius the area (of circle itself or any of its sectors) becomes 4 times the original.

Area of sector = Area of circle times (angle at the centre divided by 360 degree)

In this case angle is not changed threfore

Area of sector is proportional to area of circle, but the are of circle depends on the square of the radius therefore area of sector will be 4 times if radius is doubled

PI R^2 = Area of a circle (A), with PI a constant, R is the radius. R^2 means R squared.

A "sector" area (S) of any given angle (alpha, which is set constant by your assumption) is simply a portion of the area (k*A). So S = k*A = k*PI*R^2 = K*R^2; where K = constant = k*PI.

Therefore S = K*R^2; so that the area of a sector varies as the square of the radius of the circle in which the sector is embedded. In your case, when R is doubled, the area of the sector increases by four-fold.

Source(s): Any good geometry book.
• Anonymous
4 years ago

Area Of The Sector

• Anonymous

The sector area is quadrupled, since it is proportional to pi*r^2. That is, when you double the radius, the sector area increases by 2^2, or 4.

I don't think that the area will change since the radius is 1/ 2 the diameter and since u would have already used that radius to find the area of the sector then the diameter would be useless.

Yes. If the radius is doubled, the area of the sector will increase four fold.

• Anonymous

it will change by the same ratio that the entire area of a circle will change if you double the radius.

if the radius goes from 1 to 2 then, using the formula

a=pi*r^2 then r goes from 1 to 2 and r squared goes from 2 to 4