Question

Find the area \(A\) and circumference \(C\) of a circle of radius 5 meters.

Short answer

Area = 25π square meters, Circumference = 10π meters.

Step-by-step solution

Write Down the Formulas

The area of a circle is given by the formula \[ A = \pi r^2 \] The circumference of a circle is given by the formula \[ C = 2 \pi r \]

Identify the Radius

For this problem, the radius \( r \) of the circle is given as 5 meters.

Compute the Area

Substitute \( r = 5 \) meters into the area formula: \[ A = \pi (5)^2 \] This simplifies to \[ A = 25 \pi \text{ square meters} \]

Compute the Circumference

Substitute \( r = 5 \) meters into the circumference formula: \[ C = 2 \pi (5) \] This simplifies to \[ C = 10 \pi \text{ meters} \]

Key concepts

Understanding the Area of a Circle

The area of a circle represents the total space enclosed within its boundary. To find the area, we use the formula \( A = \pi r^2 \).
In this formula, \( \pi \) (pi) is a mathematical constant approximately equal to 3.14159, and \( r \) is the radius of the circle.
The radius is the distance from the center of the circle to any point on its edge.
In our exercise, we have a circle with a radius of 5 meters.
Substituting this value into the formula, we get: \[ A = \pi (5)^2 \]
This simplifies to: \[ A = 25 \pi \text{ square meters} \]
Therefore, the area of the circle is \ 25 \pi \text{ square meters} \.

Calculating the Circumference of a Circle

The circumference of a circle is the total distance around its edge.
To find the circumference, we use the formula \( C = 2 \pi r \).
Similar to the area formula, \( \pi \) is a constant approximately equal to 3.14159, and \( r \) is the radius of the circle.
For our problem where the radius is 5 meters, we substitute this value into the formula: \[ C = 2 \pi (5) \]
This simplifies to: \[ C = 10 \pi \text{ meters} \]
Therefore, the circumference of the circle is \ 10 \pi \text{ meters} \.

Key Circle Formulas You Need to Know

Circles are fundamental shapes in geometry and come with some key formulas:

• The formula for the area of a circle is \( A = \pi r^2 \).
• The formula for the circumference of a circle is \( C = 2 \pi r \).

Whether you are finding how much space a circular garden takes up or the length of a fence needed to go around it, these formulas are essential.
They rely on knowing the radius, or distance from the center to the edge.
For most problems, once you know the radius, you can easily find both the area and the circumference by simply plugging the radius into these formulas.
Remembering these fundamental formulas can help solve many practical and theoretical problems involving circles!