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Area of Circles

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A circle is one of the most common of shapes. Whether you look at planets' lines of orbits in the solar system, the simple yet effective functioning of wheels, or even molecules at the molecular level, the circle keeps showing up!

A **circle** is a shape in which all points that comprise the boundary are equidistant from a single point located at the center.

Before we discuss the area of circles, let's review the unique characteristics that define the circle's shape. The figure below depicts a circle with a center *O.* Recall from the definition that all points located on the circle's boundary are equidistant (of equal distance) from this center point *O*. The distance from the center of the circle to its boundary is referred to as the **radius**, *R*.

The **diameter**, *D*, is the distance from one endpoint on a circle to another, passing through the center of the circle*. *The diameter is always twice the length of the radius, so if we know one of these measurements, then we know the other as well! A **chord** is a distance from one endpoint to another on a circle that, unlike the diameter, does **not** have to pass through the center point.

Now that we've reviewed the elements of a circle, let's begin with the discussion of the **area** of a circle. First, we will start with a definition.

The **area of a circle** is the space a circle occupies on a surface or plane. The measurements of area are written using square units, such as ft^{2} and m^{2}.

To calculate the area of a circle, we can use the formula:

For this formula, it is important to know that is pi. What is pi? It is a constant represented by the Greek letter and its value is equal to approximately 3.14159.

**Pi **is** **a mathematical constant that is defined as the ratio of the circumference to the diameter of a circle.

You don't have to memorize the value of pi because most calculators have a key for quick entry, shown as . Let's use the area formula in an example to see how we can apply this calculation in practice.

The radius of a circle is 8 m. Calculate its area.

Solution:

First, we substitute the value of the radius into the circle's area formula.

Then, we square the radius value and multiply it by pi to find the area in square units. Keep in mind that does not equal , but rather is equal to .

We have seen the formula for the area of a circle, which uses the **radius**. However, we can also find the area of a circle by using its **diameter**. To do this, we divide the diameter's length by 2, which gives us the value of the radius to input into our formula. (Recall that a circle's diameter is twice the length of its radius.) Let's work through an example that uses this method.

A circle has a diameter of 12 meters. Find the area of the circle.

Solution:

Let's begin with the formula for the area of a circle:

From the formula, we see that we need the value of the radius. To find the circle's radius, we divide the diameter by 2, like so:

Now, we can input the radius value of 6 meters into the formula to solve for the area:

Apart from the area of a circle, another common and useful measure is its circumference.

The **circumference** of a circle is the perimeter or enclosing boundary of the shape. It is measured in length, which means the units are meters, feet, inches, etc.

Let's look at some formulas that relate the circumference to the circle's radius and diameter:

The formulas above show that we can multiply by the diameter of a circle to calculate its circumference. Since the diameter is twice the length of the radius, we can replace it with if we need to modify the circumference equation.

You may be asked to find the area of a circle using its circumference. Lets work through an example.

The circumference of a circle is 10 m. Calculate the area of the circle.

Solution:

First, let's use the circumference formula to determine the radius of the circle:

Now that we know the radius, we can use it to find the area of the circle:

So, the area of the circle with a circumference of 10 m is 7.95 m^{2}.

We may also analyze the circle's shape in terms of **halves** or **quarters**. In this section, we will discuss the area of semi-circles (circles cut in half) and quarter-circles (circles cut in quarters).

A semi-circle is a half circle. It is formed by dividing a circle into two equal halves, cut along its diameter. The area of a semi-circle can be written as:

Where* r * is the radius of the semi-circle

To find the circumference of a **semi-circle**, we first halve the circumference of the whole circle, then add an additional length which is equal to the diameter *d*. This is because the perimeter or boundary of a semi-circle must include the diameter to close the arc. The formula for the circumference of a semi-circle is:

Calculate the area and circumference of a semi-circle that has a diameter of 8 cm.

Solution:

Since the diameter is 8 cm, the radius is 4 cm. We know this because the diameter of any circle is twice the length of its radius. Using the formula for the area of a semi-circle, we get:

For the circumference, we input the value of the diameter into the formula:

A circle can be divided into four equal quarters, which produces four quarter-circles. To calculate the area of a quarter-circle, the equation is as follows:

To get the circumference of a quarter-circle, we start by dividing the circumference of the full circle by four, but that only gives us the quarter-circle's arc length. We then have to add the length of the radius twice to complete the quarter-circle's boundary. This calculation can be performed using the following equation:

Calculate the area and circumference of a quarter-circle with a radius of 5 cm.

Solution:

For the area, we get:

The circumference can be calculated as:

- In a circle, all points which comprise the shape's boundary are equidistant from a point located at its center.
- The line segment that spans from the center of the circle to a point on its boundary is the radius.
- The diameter of a circle is the distance from one endpoint on a circle to another that passes through the center of the circle.
- The circumference of a circle is the arc length of the circle.
- The area of a circle is .
- The circumference of a circle is .

To find the area of a circle you can use the formula:

Area = π r^{2}

^{2}

^{2}

More about Area of Circles

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