Find the area of the segment shaded in blue. The radius of the circle is 5 units and the base of the ....?

Question

Find the area of the segment shaded in blue. The radius of the circle is 5 units and the base of the triangle is 8 units. Use pi = 3.14 and round your answers to the nearest hundredth.

http://tinypic.com/view.php?pic=m803nc&s=4

please work out the problem step by step, using the formula:

A=a/3603.14r^2-1/2bh

im a little confused on this particular problem, because of the height. Normally, i can find it, but this time....I JUST NEED TO SEE IT WORKED OUT! :)

thanks alot<3

Answer 1

a=112 degrees

r = 5

The area of the segment is therefore

(112/360) X 3.142 X r^2

=(112/360) X 3.142 X 25

=(112/360) X 78.55

=24.44

Since the two sides of the triangle are equal (the radii) this is an isoceles triangle

If a line is dropped from the centre of the circle to the base it will therefore divide the base into 2 equal parts and each part will become a right angled triangle

Then apply the formula for a right angled triangle

height^2 + base^2^2 = hypotenuse ^2

or rearranging

hypotenuse^2 - base^2 = height^2

= 5^2 - 4 ^2 = height^2

= 25 - 16 = 9 = height^2

height^2 = 9

therefore height = 3

the area of the triangle is 1/2 X base X height

= 1/2 X 4 X 3

= 6

since there are two triangles multiply by 2

6 X 2 = 12

subtract this from the area of the segment of the circle

24.44 - 12

= 12.44

Therefore the area of the segment shaded in blue is 12.44 units

Answer 2

The area is the difference between the sector and the triangle, both of which can be computed from the data given.  For the triangle, drop a perpendicular bisector from the circle center to the secant line, which will give two right triangles with 56 degree angles at the center.  Elementary trig gives the length of the bisector, so we have the area of the triangle.

The area of the sector is even easier.  It is 112/360 of the area of the entire circle, so with that in hand and one subtraction, you are done.