How to calculate the radius in polar coordinates?

To calculate the radius in polar coordinates, use the formula r = √(x^2 + y^2).

In polar coordinates, a point is represented by an angle θ and a radius r. The angle θ is measured from the positive x-axis, and the radius r is the distance from the origin to the point.

To find the radius r, we can use the Pythagorean theorem. Consider a point (x, y) in the Cartesian plane. The distance from the origin to the point is given by the hypotenuse of a right-angled triangle with sides x and y. Therefore, we have:

r = √(x^2 + y^2)

This formula gives us the radius r in terms of the Cartesian coordinates x and y. To convert to polar coordinates, we need to express x and y in terms of r and θ. This can be done using the following formulas:

x = r cos(θ)
y = r sin(θ)

Substituting these expressions into the formula for r, we get:

r = √(x^2 + y^2)
= √((r cos(θ))^2 + (r sin(θ))^2)
= √(r^2 cos^2(θ) + r^2 sin^2(θ))
= √(r^2 (cos^2(θ) + sin^2(θ)))
= √(r^2)
= r

Therefore, the radius in polar coordinates is simply the distance from the origin to the point, given by the formula r = √(x^2 + y^2).