The perimeter of a cardioid can be found by integrating the length of its curve. The equation for the cardioid is r = a(1 - cosθ), where r is the distance from the origin to a point on the curve, θ is the angle between the polar axis and the line connecting the origin to the point, and a is a constant.

To find the length of the curve, we need to find the derivative of r with respect to θ, which gives us dr/dθ = a sinθ. Then, we can use the arc length formula to find the length of the curve:

L = ∫√(r^2 + (dr/dθ)^2) dθ
= ∫√(a^2(1 - cosθ)^2 + a^2 sin^2θ) dθ
= ∫√(a^2(1 - 2cosθ + cos^2θ + sin^2θ)) dθ
= ∫√(a^2(2 - 2cosθ)) dθ
= ∫a√(2 - 2cosθ) dθ.

This integral can be evaluated using trigonometric identities. Let's use the identity cos(θ/2) = ±√((1 + cosθ)/2) to simplify the integral:

L = ∫a√(2 - 2cosθ) dθ
= a∫√(2 - 2cosθ) dθ
= a∫√(4sin^2(θ/2)) dθ
= 2a∫|sin(θ/2)| dθ
= 4a∫sin(θ/2) dθ
= -4a cos(θ/2) + C.

To find the total perimeter of the cardioid, we need to evaluate this expression for the range of θ values that cover the entire curve. The range of θ for a cardioid is typically from 0 to 2π, so:

L = -4a cos(2π/2) + 4a cos(0/2)
= -4a cos(π) + 4a cos(0)
= -4a(-1) + 4a(1)
= 8a.

Therefore, the perimeter of the cardioid r = a(1 - cosθ) is 8a.