Given:
-Square ABCD with side length 2.
-Two quarter circles, one with center A and the other at center B. Call their intersection E. What is the area enclosed by Arc CE, Arc DE, and side CD?
I've already done the problem, but I want to verify my answer.
Also, my solution does not use calculus, trigonometry, and any other form of high-powered math. The whole solution only required geometry.

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- amistre64

ack!!...picture would help

- anonymous

No problem. Give me a minute.

- anonymous

This ought to help. Essentially, finding the area of the shaded.

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- amistre64

thats easier :)

- amistre64

we essentially got 4 areas, lets call them a,b,c, and d

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- anonymous

When I did that, I formuated the equation a=4-b-c-d

- amistre64

thats good... we also need to realize that:
b+d = c+d

- amistre64

b = c

- amistre64

a = 4-2b-d right?

- amistre64

lets try to fill in some values here:
lets say b = 5, c = 5, and d = 10
we know that:
(5+10) + (5+10)
but the area here is exagerated by "an extra 10"

- amistre64

we have: the true area for these overlapping circles is:
2b + 2d - d right?

- amistre64

2b + d jsut gets us right back to where we were .... can we use the circumference of a circle in our solution?

- amistre64

2pi(2) = 4pi for a whole circle; which means that pi is the length for a quarter of it......I think I got it.
unless you already know the answer :)

- amistre64

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- amistre64

do we know that area of an equilateral triangle?

- amistre64

that middle triangle has an area of: sqrt(3)

- anonymous

Thats what I did too!

- amistre64

each side section has an area of 1 so the total unshaded area is 2sqrt(3)

- amistre64

pi(r^2) is the area of a complete circle
the r here =2
4pi is the total area of our representative circle...
we only want 30 out of 360 of that circle

- amistre64

30 degrees is 1/6 of a circle right?

- amistre64

so the area of each side should be 4pi/6 = 2pi/3

- amistre64

got something turned around...hold on.

- amistre64

360/30 = 12, so we have 1/12 of a circles area....
4pi(1/12) = 4pi/12 = pi/3... am i right?

- amistre64

it should be 2pi/3 for the other stuff

- amistre64

a =4-sqrt(3)-(2pi/3)
a = 12 - 3sqrt(3) - 2pi
----------------
3
is my answer

- amistre64

a = .1735 if I did it right :)

- anonymous

I didn't evaluate for that, but the formula for a is what I got too! Awesome, thanks! You just earned a fan.

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