Question

Find the area \(A\) and circumference \(C\) of a circle of radius 2 feet.

Short answer

The area is approximately 12.56 square feet and the circumference is approximately 12.56 feet.

Step-by-step solution

- Recall the formulas

To find the area and circumference of a circle, use the following formulas: Area: \[\begin{equation} A = \pi r^2 \end{equation}\]Circumference: \[\begin{equation} C = 2 \pi r \end{equation}\]where \( r \) is the radius of the circle.

- Substitute the radius

Given the radius \( r = 2 \) feet, substitute this value into the formulas. For the area: \[\begin{equation} A = \pi (2)^2 \end{equation}\]For the circumference: \[\begin{equation} C = 2 \pi (2) \end{equation}\]

- Calculate the area

Calculate the area by simplifying the expression. \[\begin{equation} A = \pi (4) = 4 \pi \end{equation}\]Since \( \pi \approx 3.14 \), \[\begin{equation} A \approx 4 \times 3.14 = 12.56 \text{ square feet} \end{equation}\]

- Calculate the circumference

Calculate the circumference by simplifying the expression. \[\begin{equation} C = 2 \times 2 \pi = 4 \pi \end{equation}\]Since \( \pi \approx 3.14 \), \[\begin{equation} C \approx 4 \times 3.14 = 12.56 \text{ feet} \end{equation}\]

Key concepts

area of a circle

The area of a circle is the space contained within its circumference. This is calculated using the formula \(A = \pi r^2\). Here, \(A\) stands for area, \(\pi\) (pi) is a constant approximately equal to 3.14, and \(r\) is the radius of the circle. To find the area, you square the radius (multiply it by itself) and then multiply the result by \(\pi\).
For example, if the radius is 2 feet, the calculation for the area would be: \[A = \pi (2)^2 = \pi (4) = 4\pi\approx 12.56 \text{ square feet}\]
This means that the area of a circle with a radius of 2 feet is approximately 12.56 square feet.

circumference of a circle

The circumference of a circle is the distance around the edge of the circle. It’s akin to the perimeter in polygons. To find the circumference, you use the formula \(C = 2\pi r\), where \(C\) represents the circumference, \(\pi\) is pi, and \(r\) is the radius.
If we have a circle with a radius of 2 feet, then its circumference would be calculated as follows: \[ C = 2 \pi (2) = 4 \pi \approx 12.56 \text{ feet}\]
This circumference calculation tells us that the length around the outside of this circle is approximately 12.56 feet.

radius and diameter

Understanding the radius and diameter is key to mastering circle measurements. The radius \(r\) is the distance from the center of the circle to any point on its edge. This distance remains constant for any point on the circle. The diameter \(d\), on the other hand, is twice the length of the radius, stretching from one edge of the circle to the opposite edge passing through the center.
The relationship between the radius and the diameter can be shown by the formula: \[ d = 2r\]
For a circle with a radius of 2 feet, the diameter would be: \[ d = 2(2) = 4 \text{ feet}\]
This means that if you know either the radius or the diameter, you can always find the other by using simple multiplication or division by 2.