Hello, classicstrings!

Careful . . . 5-5-8 is **not** a right triangle.

. . But your game plan is correct.

The area of the intersection of two circles of radius 5 cm

which have their centres 8 cm apart is? Code:

C
*
* : *
5* :3 *5
* α : *
* * * * * * * * *
A 4 4 B

I get a triangle with sides 5, 5, and 8.

We also get a couple of 3-4-5- right triangles.

I work out the angle between the 5 side and 8 side.

$\displaystyle \sin\alpha \:=\:\frac{3}{5}\quad\Rightarrow\quad \alpha \:= \:\arcsin(0.6) \:\approx \:0.6435$ radians.

then times that by 2 to get the full angle.

$\displaystyle \theta \:=\:2\alpha \:=\:1.287$ radians

Then use the formula: .$\displaystyle \text{ segment} \:= \:\frac{1}{2}r^2(\theta - \sin\theta)$ $\displaystyle \text{segment} \:= \:\frac{1}{2}\left(5^2\right)\left[1,287 - \sin(1.287)\right] \:\approx\:4.0875$

then times that answer by two to get the final answer (the interection)

$\displaystyle \text{Area} \:=\:2 \times 4.0875 \:=\:8.175$ cm².