Area of a Segment of a Circle, when only the radius is known.

[malam1990]

A circle has Centre 0 and radius 2. A, B and C are points on the circumference of a circle such that AB is the perpindicular bisector of 0C.

Find the area of the segment of the circle bounded by the line segment AB and the minor arc ACB.

Give the area in exact forms in terms of surds and pi.

I have uploaded the circle diagram as well.

I understand how to find the area of a segment of a circle when I know the angle and radius, but am unsure when I only know the radius. Any help will be greatly appreciated!

Cheers

 

[dacruick]

Well show me how you would do it if you knew the angle

 

[malam1990]

I would use the formula:

A = \Pir²\theta/360 - 0.5r²sin\theta



cheers

 

[tiny-tim]

hi malam1990!

(have a theta: θ )

A circle has Centre 0 and radius 2. A, B and C are points on the circumference of a circle such that AB is the perpindicular bisector of 0C.
…
I understand how to find the area of a segment of a circle when I know the angle and radius, but am unsure when I only know the radius.

but you do know the angle …

you know the lengths of two sides of triangle AOD (where D bisects AB)

 

[SammyS]

OC, OA, and OB are all radii of the circle, right?

 

[malam1990]

hi malam1990!

(have a theta: θ )


but you do know the angle …

you know the lengths of two sides of triangle AOD (where D bisects AB)

Hi, thanks for the help, but I'm still not sure how I get to know the angle, the answer is probably really obvious but unfortunately I am just not seeing it. So far I have θ as 90 degress but am unsure whether this is correct.

Cheers for the help again! it is much appreciated!

 

[malam1990]

OC, OA, and OB are all radii of the circle, right?

yes they are.

cheers for any help.

 

[tiny-tim]

hi malam1990!

you know the lengths of two sides of triangle AOD (where D bisects AB)

Hi, thanks for the help, but I'm still not sure how I get to know the angle …

what are the lengths of two sides of triangle AOD ?

 

[malam1990]

hi malam1990!



what are the lengths of two sides of triangle AOD ?

well I know OA is 2 as it is the radius, but I am still unsure how I would get to know either OD or DA?

cheers

 

[tiny-tim]

but you know that D is the bisector of OC

 

[malam1990]

but you know that D is the bisector of OC

I'm really sorry but I'm still not getting it, I can't see how it helps me know definitely the length of one of the other sides of the traingle AOD.

 

[SammyS]

OC, OA, and OB are all radii of the circle, right?

Therefore, AB bisects OC.

That should let you find the angle!

 

[malam1990]

Would I be correct in thinking that OD would be 1 and AD being \sqrt{}3 ?

which would then give me 60deg for the angle AOD? and 120 deg for the angle of the whole sector AOB?

 

[tiny-tim]

yup! … AB bisects OC, so OD = OC/2 = 1, so the right-angled triangle has sides 2 1 and √3, and cosAOD = 1/2, so AOD = 60°

 

[malam1990]

yup! … AB bisects OC, so OD = OC/2 = 1, so the right-angled triangle has sides 2 1 and √3, and cosAOD = 1/2, so AOD = 60°

thank you so much for guiding me to finally seeing how to do it!

thanks again! you really have helped me out!