To find the area lying outside the circle r = 2acosθ, we need to determine the area of the entire circle and then subtract the area enclosed by the circle.

The equation of the circle r = 2acosθ can be rewritten in terms of x and y coordinates using the conversion formulas x = rcosθ and y = rsinθ.

So, r = 2acosθ becomes x² + y² = 4a²cos²θ.

Now, we can find the equation of the circle in terms of x and y coordinates:

x² + y² = 4a²cos²θ
r² = 4a²cos²θ

The area of a circle is given by the formula A = πr².

So, the area of the entire circle is A = π(4a²cos²θ).

Now, we need to find the area enclosed by the circle. This can be done by finding the area of the sector formed by the circle and then subtracting the area of the triangle formed by the center of the circle and the two points on its circumference.

The angle of the sector is θ, and the radius of the circle is 2acosθ.

The area of the sector is given by A_sector = 0.5θr².

So, A_sector = 0.5θ(2acosθ)² = 2a²θcos²θ.

The area of the triangle is given by A_triangle = 0.5bh, where b is the base and h is the height.

In this case, the base is the distance between the two points on the circumference of the circle, which is 2r = 4acosθ.

The height is the distance from the center of the circle to the line connecting the two points on the circumference, which is r = 2acosθ.

So, A_triangle = 0.5(4acosθ)(2acosθ) = 4a²cos²θ.

Therefore, the area enclosed by the circle is A_enclosed = A_sector - A_triangle = 2a²θcos²θ - 4a²cos²θ = 2a²θcos²θ(1 - 2).

Finally, the area lying outside the circle is A_outside = A - A_enclosed = π(4a²cos²θ) - 2a²θcos²θ(1 - 2) = π(4a²cos²θ) - 2a²θcos²θ + 4a²θcos²θ = π(4a²cos²θ) + 2a²θcos²θ.

Therefore, the area lying outside the circle r = 2acosθ is π(4a²cos²θ) + 2a²θcos²θ.