Question

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.

Answer 1

ack!!...picture would help

Answer 2

No problem. Give me a minute.

Answer 3

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

Answer 4

thats easier :)

Answer 5

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

Answer 6

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

Answer 7

thats good... we also need to realize that:

b+d = c+d

Answer 8

b = c

Answer 9

a = 4-2b-d right?

Answer 10

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"

Answer 11

we have: the true area for these overlapping circles is:

2b + 2d - d right?

Answer 12

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

Answer 13

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 :)

Answer 14

do we know that area of an equilateral triangle?

Answer 15

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

Answer 16

Thats what I did too!

Answer 17

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

Answer 18

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

Answer 19

30 degrees is 1/6 of a circle right?

Answer 20

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

Answer 21

got something turned around...hold on.

Answer 22

360/30 = 12, so we have 1/12 of a circles area....

4pi(1/12) = 4pi/12 = pi/3... am i right?

Answer 23

it should be 2pi/3 for the other stuff

Answer 24

a =4-sqrt(3)-(2pi/3)

a = 12 - 3sqrt(3) - 2pi
----------------
3

is my answer

Answer 25

a = .1735 if I did it right :)

Answer 26

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