A way to find this shaded area without calculus? This is a popular problem spreading around. Solve for the shaded reddish/orange area. (more precisely: the area in hex color #FF5600)



$ABCD$ is a square with a side of $10$, $APD$ and $CPD$ are semicircles, and $ADQB$ is a quarter circle. The problem is to find the shaded area $DPQ$.

I was able to solve it with coordinate geometry and calculus, and I verified the exact answer against a numerical calculation on Desmos.

Ultimately the result is 4 terms and not very complicated. So I was wondering: Is there was a way to solve this using trigonometry? Perhaps there is a way to decompose the shapes I am not seeing.

A couple of years ago there was a similar "Find the shaded area" problem for Chinese students. I was able to solve that without calculus, even though it was quite an involved calculation.

Disclosure: I run the YouTube channel MindYourDecisions. I plan to post a video on this topic. I'm okay posting only the calculus solution, but it would be nice to post one using only trigonometry as many have not taken calculus. I will give proper credit to anyone that helps, thanks!

Update: Thanks for everyone's help! I prepared a video for this and presented 3 methods of solving it (the short way like Achille Hui's answer, a slightly longer way like David K and Seyed's answer, and a third way using calculus). I thanked those people in the video on screen, see around 1:30 in this link: https://youtu.be/cPNdvdYn05c.
The area is equal to difference between the area of two lenses. 

It is easy to find the area of lenses like the one I did in this question before: How to find the shaded area
Let's assume we have 4 semi circles inside ABCD. There will be 4 intersections and all will be equal to DP.So we can calculate DP by 
4 times area of DP + the area of square = 2 times the area of circle (r=5)
so DP= 50/4 (pi=3)
İF we assume one more semi circle in ABCD (APB)
and one more quarter circle in ABCD (CDHB) 
so area of 2 quarter circles - the area of the square will give us the area of intersection 
DB 
10*10*3/2= 150  10*10=100 150-100= 50
In half the area of intersection there are 2 red areas and DP so
50/2= 25 and (25-(25/2))/2=7.5 will be our answer