# Math Help - Finding the Area (parameter)

1. ## Finding the Area (parameter)

Given is a square, ABCD, the lenght of a side is equal to "a". A and D are centers of circles, BD and AC are arcs of circles and they intersect each other in K. Find the area of brightened figure on the picture

2. ## Re: Finding the Area (parameter)

Hello, Telo!

Given square ABCD with side $a.$
A and D are centers of quarter-circles.
The arcs intersect at K.
Find the area of shaded region.

Code:
B *-*-------*-* C
|    K*:::::|
|   */..*:::|
| *./.....*:|
|*./.......*| a
|./.........|
*/..........*
A *-----------* D
a
Draw chord $AK.$

The area of quarter-circle $AKCD$ is:- $\tfrac{1}{4}\pi a^2$

The area of sector $AKD$ is:- $\tfrac{1}{6}\pi a^2$

The area of segment $AK$ is:- $\tfrac{1}{6}\pi a^2 - \tfrac{\sqrt{3}}{4}a^2$

$\text{Shaded region} \:=\:\text{(quarter-circle)} - \text{(sector + segment)}$

. . . . . . . . . . $=\;\tfrac{1}{4}\pi a^2 - \left(\tfrac{1}{6}\pi a^2 + \left[\tfrac{1}{6}\pi a^2 - \tfrac{\sqrt{3}}{4}a^2\right]\right)$

. . . . . . . . . . $=\;\left(\tfrac{\sqrt{3}}{4} - \tfrac{\pi}{12}\right)a^2$

3. ## Re: Finding the Area (parameter)

Originally Posted by Telo
Given is a square, ABCD, the lenght of a side is equal to "a". A and D are centers of circles, BD and AC are arcs of circles and they intersect each other in K. Find the area of brightened figure on the picture

area of square = a^2 area of quarter circle ADB = area of quarter circle ADC = 1/4 ( pi x a^2)
Area of square ABCD - area of quarter circle ADB - area of quarter circle AD= areaBKC
Now I am sure you can compute the area of the shaded region.

4. ## Re: Finding the Area (parameter)

Hi,
Soroban beat me to the same solution. I was busy constructing the attached drawing; I enjoy drawing pictures. I'll post my version of his solution anyway.