Math Is Fun Forum

AT&T

Member

Registered: 2022-07-21

Posts: 4

the area of a sector of one radian

Hi,

given:

a circle, radius = r;
a sector, angle = 1 radian;

What is the area?

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(1) Area of the circle = pi . r^2

(2) Area of a sector of one radian = 1 Rad

(3) If a circle has 2.pi.rad
then
Area of the circle = 2.pi.Rad

From (1) and (3) => pi . r^2 = 2 . pi . Rad <=> r^2 = 2 . Rad <=>

<=> Rad = 1/2 . r^2

It turns out that Area of a sector of one radian is half a square.

I do not know what to think.

AT&T

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: the area of a sector of one radian

hi AT&T

Although degrees have been used since the time of the ancient Babylonians, radians are the angle measure of choice for advanced mathematics.

For example the gradient function for sin is given by d(sin x)/dx = cos x only when the angle x is measured in radians.

What you have discovered is  the result of the way radians are defined.

A radian is just under 60 degrees (one sixth of the circle), so your result does not seem unreasonable.  If pi= 3 then the whole circle is 3r^2 so one sixth is 3r^2/6 =(r^2)/2  So an approximate calculation confirms the result.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

AT&T

Member

Registered: 2022-07-21

Posts: 4

Re: the area of a sector of one radian

Thank you, Bob.

AT&T