x = radius of circle
y = side of triangle
Total length = 3y + 2πx = 8
total area = y²√3/4 + πx²
3y + 2πx = 8
y = (8–2πx)/3
A = y²√3/4 + πx²
A = [(8–2πx)/3]²√3/4 + πx²
A = √3/36(8–2πx)² + πx²
A = √3/9(4–πx)² + πx²
A = √3/9(16–8πx+π²x²) + πx²
A = (16/9)√3 – (8/9)√3πx + (√3/9)π²x² + πx²
A = (√3/9+1)π²x² – (8/9)√3πx + (16/9)√3
differentiate, set equal to zero and solve for x
A' = 2(√3/9+1)π²x – (8/9)√3π = 0
2(√3/9+1)πx = (8/9)√3
x = (8/9)√3 / 2(√3/9+1)π = 1.5396 / 7.492 = 0.205
check the arithmetic.