Question

A circle has a radius with endpoints at $\left(2,5\right)$and $\left(-1,-4\right)$. Find the circumference and area of the circle.

Short answer

The circumference and area of the circle are59.6 unitsand282.9 square units respectively.

Step-by-step solution

Step 1. Given Information.

Given a circle has a radius with endpoints $\left(2,5\right)$and $\left(-1,-4\right)$.

The circumference and area of the circle are to be determined.

Step 2. Explanation.

The distance between two points $\left(x_1,y_1\right),\left(x_2,y_2\right)$ is given by $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$.

Plugging the given values in the equation to find the radius:

$\begin{array}{l}r=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ r=\sqrt{{\left(-1-2\right)}^2+{\left(-4-5\right)}^2}\\ r=\sqrt{{\left(-3\right)}^2+{\left(-9\right)}^2}\\ r=\sqrt{9+81}\\ r=\sqrt{90}\\ r=3\sqrt{10}\end{array}$

The circumference of a circle with radius r is given by $C=2\pi r$

Plugging the values in the equation:

$\begin{array}{l}C=2\pi r\\ C=2\times \frac{22}{7}\times 3\sqrt{10}\\ C=59.6\text{units}\end{array}$

The area of a circle with radius r is given by$A=\pi r^2$

Plugging the values in the equation:

$\begin{array}{l}A=\pi r^2\\ A=\frac{22}{7}{\left(3\sqrt{10}\right)}^2\\ A=\frac{22}{7}\left(90\right)\\ A=282.9\text{squareunits}\end{array}$

Step 3. Conclusion.

Hence, the circumference and area of the circle are 59.6 units and 282.9 square units respectively.