Question

Find the area of the sector of a circle with a central angle of 5 radians and a radius of 5 feet. please help!

Answer 1

Do you know the sector area formula?

\[\Large A=\frac{ \alpha }{ 2 }\cdot r^2\]

Answer 2

so the answer is 62.5?

Answer 3

That's what I get. :)

do you want an easy way to remember the formula?

Answer 4

yay! and sure!

Answer 5

Think of it as:

\(\Large A=\dfrac{ \alpha }{ 2\pi }\cdot \pi r^2\)

Because really, all you're doing is taking a PROPORTION of the TOTAL AREA of the circle,right?

So look at my "version" of the formula:

\(\Large \dfrac{ \alpha }{ 2\pi }\) this is the PROPORTION that the angle \(\alpha\) "cuts out" from the WHOLE circle (since the "angle" of the whole circle is \(2\pi\)).

\(\Large \pi r^2\) is the total area of the circle (plain ol' area of a circle formula)

so you get proportion x whole:

\(\Large A=\dfrac{ \alpha }{ 2\pi }\cdot \pi r^2=\dfrac{ \alpha }{ 2 }\cdot r^2\) because the pi's cancel... but it's an easy way to remember it (and to have an intuitive grasp of the formula).

:)

Answer 6

And if you are also doing arc length, that's just a PROPORTION of the TOTAL CIRCUMFERENCE so you can use the same idea:

\(\Large s=\dfrac{ \alpha }{ 2\pi }\cdot 2\pi r=\alpha r\)

The \(2\pi 's\) cancel, but it is intuitively the same thing (portion of circumference).

Answer 7

thank you!!! but what if it asks a question like..what is the exact radian measure of the acute angle formed by the hands of a clock at 7:00? which formula do you use

Answer 8

Are you sure it says "ACUTE" angle?? You just need to think about what the face of a clock looks like at 7:00.....



The angle is 7/12 of the whole circle, so what is that in radians? (that's a proportion of 2pi)

But it isn't acute!

Answer 9

it says acute, haha that's why it confused me

Answer 10

OOPS, it's isn't 7/12 of the circle, it is 5/12 of the circle - that's what I meant to say.

That's stil not an acute angle... but I think it is trying to say, the angle that is rotated in the "shortest" direction... e.g.:

Answer 11

so do you put 5/12 in radians?

Answer 12

Well, kind of... it's 5/12 of the WHOLE circle. the WHOLE circle is 2pi radians.

So you want 5/12 of 2pi. :)

Answer 13

thank you :) but how do you find that?

Answer 14

\[\Large \dfrac{ 5 }{12 }\cdot 2\pi\]

Answer 15

"a proportion of the whole" means to multiply the proportion x the whole

Answer 16

is it 2.61?

Answer 17

You are using a calculator, which is not an "exact" radian measure... :)

Just simplify that product, your answer will be a fraction and will include pi.

Answer 18

10pi/12? sorry, im not the best at math