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Finding areas of sectors using radians

There is a question on the area of a sector of a circle on which I am stuck and was wondering whether anyone could give me a pointer.

I have attached an image of the diagram to which the question refers...

The diagram shows an arc of ABC of a circle centre o and radius 10cm.

Angle AOB = $\displaystyle \theta$ radians and angle AOC is a right angle.

a. Write down, in terms of theta, the area of sector AOB.

I can do this easily enough: $\displaystyle area = \frac{1}{2} r^2 * \theta$

area = $\displaystyle 50 \theta $

b. Show that the area of the shaded segment is $\displaystyle 25(\pi -2\theta - 2cos \theta)cm^2 $

This is where I get stuck.

I can find the area of the sector OBC using the formula and get $\displaystyle 50(\frac{\pi }{2} - \theta)$ = $ 25\pi - 50 \theta $

But I can't see how to get the area of the triangle.

Having said that, multiplying out $\displaystyle 25(\pi - 2\theta - 2 cos \theta)$ gives:

$\displaystyle 25\pi - 50\theta -50 cos\theta $, so the area of the triangle must be $\displaystyle 50 cos \theta $.

I just can't see how to get at this area for the triangle, though. Any advice would be greatly appreciated!

Re: Finding areas of sectors using radians

Quote:

Originally Posted by

**pturo1** The diagram shows an arc of ABC of a circle centre o and radius 10cm. Angle AOB = $\theta$ radians and angle AOC is a right angle.

a. Write down, in terms of theta, the area of sector AOB.

I can do this easily enough: $\displaystyle area = \frac{1}{2} r^2 * \theta$

area = $\displaystyle 50 \theta $

b. Show that the area of the shaded segment is $\displaystyle 25(\pi -2\theta - 2cos \theta)cm^2 $

You can use this webpage.

Re: Finding areas of sectors using radians

That's a isosceles triangle having two sides of length 10 and one angle of measure $\displaystyle \theta$. If you drop a perpendicular, you divide it into two right triangles having hypotenuse 10 and angle $\displaystyle \theta/2$. So the two legs of the right triangle have length $\displaystyle 10cos(\theta)$ and $\displaystyle 10sin(\theta)$. Since the legs are the "base" and "altitude" of a right triangle, the area of each is $\displaystyle (1/2)(10 cos(\theta/2))(10 sin(\theta/2)$ and the area of the entire triangle is $\displaystyle 100 sin(\theta/2)cos(\theta/2)$.

Re: Finding areas of sectors using radians

Thank you for your replies. Plato, thanks for the link; it looks interesting but involves cotangents which I have not yet got to in my mathematical education.

Thanks for your explanation Halls of Ivy. I can follow your working but there is just one thing I don't understand (I am probably failing to see something obvious):

How do you know one of the angles in the isocoles triangle is $\displaystyle \theta$? The only conclusion I could draw was that the angle would be $\displaystyle \frac{\pi}{2} - \theta$ radians.

Thanks again for your help.