The area of the cardioid can be found using polar coordinates. The equation of the cardioid in polar coordinates is r = a(1 - cosθ), where r is the distance from the origin to a point on the curve, θ is the angle made with the positive x-axis, and a is a constant.

To find the area, we can integrate over a range of θ values and then multiply by 1/2 to account for the symmetry of the cardioid.

Let's integrate over the range 0 ≤ θ ≤ 2π.

The area is given by the integral of (1/2)r^2 dθ. Substituting r = a(1 - cosθ), we have:

A = (1/2) ∫[0 to 2π] (a(1 - cosθ))^2 dθ

Expanding and simplifying the expression, we get:

A = (1/2) ∫[0 to 2π] a^2(1 - 2cosθ + cos^2θ) dθ

Using the trigonometric identity cos^2θ = (1 + cos(2θ))/2, the expression becomes:

A = (1/2) ∫[0 to 2π] a^2(1 - 2cosθ + (1 + cos(2θ))/2) dθ

Simplifying further, we have:

A = (1/2) ∫[0 to 2π] (a^2 - 2a^2cosθ + (a^2/2) + (a^2/2)cos(2θ)) dθ

Breaking the integral into parts, we get:

A = (1/2) ∫[0 to 2π] a^2 dθ - ∫[0 to 2π] 2a^2cosθ dθ + (1/2) ∫[0 to 2π] (a^2/2) dθ + (1/2) ∫[0 to 2π] (a^2/2)cos(2θ) dθ

Evaluating each integral, we have:

A = (1/2) (a^2)(2π) - 0 + (1/2) (a^2/2)(2π) + 0

Simplifying further, we get:

A = πa^2 - (πa^2)/4

A = (3πa^2)/4

Therefore, the area of the cardioid r = a(1 - cosθ) is (3πa^2)/4.