Help finding the area.?

Question

Here is the picture
http://i40.photobucket.com/albums/e215/xxs3xp1stolxx/11.jpg

the book says to use 3.14 for pi, cause you help me find what is the area of the shaded reigion.

Thanks!

Answer 1

Arc AC is one-quarter of the full circle. Thus the circumference of the full circle is 4 * (5/2)π = 10π

The formula for circumference is:
C = 2π r

Using your value of 10π, you can easily figure out the radius is 5:
10π = 2π r
r = 10π / 2π
r = 5

The radius is 5, so AO is 5, BO is 5 and CO is 5.

Because AE is 2, that means EO is 3.

Now you have a right triangle OEB, one leg is 3 and the hypotenuse is 5. You can prove it with the Pythagorean theorem but EB (the other leg) will be 4 (remember a 3-4-5 right triangle?)

3² + (EB)² = 5²
9 + (EB)² = 25
EB² = 16
EB = 4

So you have a rectangle with sides 3 and 4. This is an area of 12 sq. units.

But the question asks for the shaded area which is the quarter circle minus this rectangle.

The area of the quarter circle is:
1/4 π r²
1/4 π 5² = 25/4 * π

Thus the area of the shaded region will be 25/4 π - 12

Using the estimate of 3.14 for π:

Area of the shaded region = 6.25 * 3.14 - 12
= 7.625 sq. units

Answer 2

Perimeter of a circle = Pi (2r)
Perimeter of 1 quarter of a circle = Pi(2r)/4 = Pi r/2
we know that as arc AC = 5Pi/2 = Pi r/2 ~~~>
r =5 units

Now let's find the area of one quadrant
it is the area of a circle/4

area of a circle is Pi *r^2 = Pi *25 =25 Pi
area of of a quadrant = 25 Pi/4

Find the area of the rectangle ODBE?

AE =2 and OA =r =5 ~~~> OE =5-2 =3

Note that OB = radius of the circle =5
Note also the triangle OEB is right, need to find EB,
OB^2 =OE^2+EB^2
25 = 3^2+EB^2 = 9+EB^2
EB^2 = 16 ~~~> EB =4
area of the rectangle is 4*3 = 12
Hence the area of the shaded reigion.
is just 25Pi/4 -12 =25*3.14/4 -12 =7.625 units^2

Answer 3

Circumference of circle = 4 x 5pi/2, so

2pir = 10pi, so

r = 5, and EO = 3.

By Pythagoras` in triangle OEB,

(EB)^2 = 5^2 - 3^2, so

EB = 4.

Required area = (pi r^2)/4 - 3 x 4

I assume that you can calculate that.

Hope this helps, Twiggy.