Radius

In this article we will learn what the radius is and we will see how we can use it to calculate the perimeter of the circle and the area of the circle.

The radius is a segment that connects the center of the circle with any point located on the circle itself. We will illustrate it with a graph

Every circle has a center point. In the following illustration it is marked with the letter O. Now we will draw a line from the center point to any other point on the circle.

This line is the radius of the circle, usually denoted by the letter $R$ uppercase or $r$ lowercase. We can draw an infinite number of radii on each circle and they will all be of identical length.

For example, on this circle we have drawn three radii. All the radii of the circle have the same length. That is, the radius of a circle has a fixed length.

The diameter of the circumference is the chord that passes exactly through the center and is usually denoted by the letter D.

For example:

The length of the diameter is equal to twice the length of the radius. Can you understand why? We can imagine that the diameter is composed of two radii.

With the length of the radius we can calculate the perimeter of the circle and the area of the circle. Just for that we have the formulas that will help us to do it.

We will mark the perimeter of the circle with the letter $P$. The formula to calculate the perimeter of the circle is:

$C=2πr$

Let's explain it in words: the circumference equals $2$ multiplied by the number PI, multiplied by the radius. Recall that the value of PI(which is detailed in other articles), is approximately equivalent to $3.14$

Given a circle, knowing that its radius measures $3$ cm.

What is the perimeter of the circumference?

Solution:

Let's write it down

$R=3$

Now let's remember the formula we just learned to calculate the perimeter:

$P=2πr$

Let's put in the formula the parameters and we will get:

$P=2\times3.14\times3$

$P=18.84$ cm

Thus, we have based on the length of the radius to find the perimeter.

Area of the circle

With the radius we can also calculate the area of the circle, usually denoted by the letter $A$. Just for that we have the following formula:

$A=πr^2$

Let's explain it in words: the area of the circle equals PI times the radius squared.

Given a circle with a radius of $4$ cm.

What is the area of the circle?

Solution:

Let's mark the data:
$R = 4$ cm.

Now let's remember the formula to calculate the area of a circle:

$A=πr^2$

and let's set the parameters.

$A=3.14\times4^2$

$A=3.14\times16$

$A=50.24$

That is, we have arrived at that the area of the circle is $50.24$ cm².

Pay attention to the units of measurement. The length of the radius is given in cm, but it is raised to the power of two and, therefore, the area is measured in cm² (cm squared).

If this article interests you, you may also be interested in the following articles:

• The circumference
• The center of the circumference
• Circle
• Diameter
• Pi
• Perimeter of the circle
• Circular area
• Arcs in a circle
• Chords in a circle
• Central angle in a circle
• Perpendicular to the chord from the center of the circle
• Inscribed angle in a circle

On the Tutorela blog you will find a variety of articles about mathematics.

Request

Given the circle in the figure

The radius of the circle is equal to: $9.5$

What is its circumference?

The radius of the circle is

$r=9\frac{1}{2}$

We use the formula of the circumference

$2\pi r$

We replace accordingly and get

$2\cdot\pi\cdot9\frac{1}{2}=19\pi$

Answer

$19\pi$

Problem

Given that the circumference is equal to: $8$

What is the length of the radius of the circle?

Solution

According to the data $2\pi r=8$

We divide on both sides by $2\pi$

$\frac{2\pi r}{2\pi}=\frac{8}{2\pi}$

Simplify and get

$r=\frac{4}{\pi}$

Answer

$r=\frac{4}{\pi}$

Question

The radius of the circle is $4cm$ centimeters.

The length of the side of the square is $8cm$ centimeters.

In which shape is there a larger perimeter?

Solution

The circumference is: $2\pi r$

We replace the data accordingly.

$2\cdot\pi\cdot4=8\pi$

$8\pi=8\cdot3.14=25.12$

The perimeter of the square is equal to $4a$

$4\cdot8=32$

Answer

Square

Question

Given that the circumference of the circle is equal to $16$

What is the length of the radius of the circle?

Solution

$2\pi r=16$

We divide the two sides by: $2\pi$

We obtain

$\frac{2\pi r}{2\pi}=\frac{16}{2\pi}$

Reduce $2\pi$

$r=\frac{8}{r}$

Answer

$r=\frac{8}{r}$

Question

Given the circle in the figure

The radius of the circle is equal to: $\frac{1}{4}$

What is the circumference?

Solution

The radius of the circle is equal to $r=\frac{1}{4}$

We use the formula of the circumference of the circle $2\pi r$

Replace accordingly

$2\cdot\pi\cdot\frac{1}{4}=\frac{\pi}{2}$

Answer

$\frac{\pi}{2}$

Recall that the radius of a circle is a line segment that passes from the center of the circle and touches one of the points of the circumference, and is half the diameter. Let's look at the following image to see the radius of the circle.

The radius is half of the diameter or we can say that the diameter is twice the radius, therefore if the diameter is equal to $10\text{ cm}$, then half will be the radius and therefore

$r=5\text{ cm}$

The formula for finding the perimeter of a circle (the circumference) is the following:

$P=2\pi r$

We can express it as two times the number pi times the radius.

Since the diameter is twice the radius we can also write this formula as:

$P=\pi D$

We can read the above formula as the perimeter is equal to pi times the diameter.

Now to calculate the area of a circle which is also known as the surface area we have the following formula:

$A=\pi r^2$

If we want to know the radius of a circle and we know the circumference or area we can use the formulas mentioned above,

For example if we know the circumference then we use the formula

$P=2\pi r$

And from this formula we isolate the radius, dividing it all by $2\pi$, leaving the following form:

$\frac{P}{2\pi}=\frac{2\pi r}{2\pi}$

Simplifying the formula:

$\frac{P}{2\pi}=r$

In a general way for any circumference:

$r=\frac{P}{2\pi}$

Now if we know the surface and we want to obtain the radius we use the formula:

$A=\pi r^2$

We isolate the radius in the equation, dividing all by pi:

$\frac{A}{\pi}=\frac{\pi r^2}{\pi}$

$\frac{A}{\pi}=r^2$

Now, take the square root of both sides.

$\sqrt{\frac{A}{\pi}}=\sqrt{r^2}$

Simplifying and rearranging, we get the general way to find the radius of any circle if we are given the surface area

$r=\sqrt{\frac{A}{\pi}}$

In a circle with circumference equal to $12\text{ cm}$, calculate the radius:

From our formula

$r=\frac{P}{2\pi}$

We just substitute the value of the circumference

$r=\frac{12}{2\pi}$

$r=\frac{6}{\pi}$

Answer

$r=\frac{6}{\pi}$

Question

A circle has an area of $64\operatorname{cm}^2$, calculate the radius of the circumference:

Solution

In this case, the data we know is the area therefore from our formulas we have:

$r=\sqrt{\frac{A}{\pi}}$

Substituting the area:

$r=\sqrt{\frac{64\operatorname{cm}^2}{\pi}}$

Find the square root:

$r=\frac{\sqrt{64\operatorname{cm}^2}}{\sqrt{\pi}}$

$r=\frac{8\text{ cm}}{\sqrt{\pi}}$

Result:

$r=\frac{8\text{ cm}}{\sqrt{\pi}}$