Circle - Examples, Exercises and Solutions

O is the center of the circle in the figure below.

What is its circumference?

We use the formula:$P=2\pi r$

We replace the data in the formula:$P=2\times8\pi$

$P=16\pi$

$16\pi$ cm

Look at the circle in the figure:

Its radius is equal to 4.

What is its circumference?

The formula for the circumference is equal to:

$2\pi r$

8π

Look at the circle in the figure:

The radius is equal to 7.

What is the area of the circle?

Remember that the formula for the area of a circle is

πR²

We replace the data we know:

π7²

π49

49π

Given that the diameter of the circle is 7 cm

What is the area?

First we need the formula for the area of a circle:

 $\pi r^2$

In the question, we are given the diameter of the circle, but we still need the radius.

It is known that the radius is actually half of the diameter, therefore:

$r=7:2=3.5$

We substitute the value into the formula.

$\pi3.5^2=12.25\pi$

$12.25\pi$ cm².

O is the center of the circle in the diagram below.

What is its area?

Remember that the formula for the area of a circle is

πR²

We insert the known data:

π3²

π9

$9\pi$ cm²

Look at the circle in the figure.

What is its circumference if its radius is equal to 6?

Formula of the circumference:

$P=2\pi r$

We insert the given data into the formula:

$P=2\times6\times\pi$

$P=12\pi$

$12\pi$

Look at the circle in the figure:

The diameter of the circle is 13.

What is its area?

First, let's remember what the formula for the area of a circle is:

$S=\pi r^2$

The problem gives us the diameter, and we know that the radius is half of the diameter therefore:

$\frac{13}{2}=6.5$

We replace in the formula and solve:

$S=\pi\times6.5^2$

$S=42.25\pi$

42.25π

Look at the circle in the diagram.

AB is a chord.

Is it possible to calculate the area of the circle?

Since AB is just a chord and we know nothing else about the diameter or the radius, we cannot calculate the area of the circle.

It is not possible.

A circle has an area of 25 cm².

What is its radius?

Area of the circle:

$S=\pi r^2$

We insert the known data:

$25=\pi r^2$

Divide by Pi:$\frac{25}{\pi}=r^2$

Extract the root:$\sqrt{\frac{25}{\pi}}=r$

$\frac{5}{\sqrt{\pi}}=r$

$\frac{5}{\sqrt{\pi}}$ cm

Is it possible that the circumference of a circle is 8 meters and its diameter is 4 meters?

To calculate, we will use the formula:

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

Pi is the ratio between the circumference of the circle and the diameter of the circle.

The diameter is equal to 2 radii.

Let's substitute the given data into the formula:

$\frac{8}{4}=\pi$

$2\ne\pi$

Therefore, this situation is not possible.

Impossible

Given the semicircle:

What is the area?

Formula for the area of a circle:

$S=\pi r^2$

We complete the shape into a full circle and notice that 14 is the diameter.

A diameter is equal to 2 radii, so:$r=7$

We replace in the formula:$S=\pi\times7^2$

$S=49\pi$

24.5π

A circle has a circumference of 31.41.

What is its radius?

To solve the exercise, first we must remember the circumference formula:

$P= 2\pi R$

P is the circumference and Pi has a value of 3.14 (approximately).

We substitute in the known data:

$31.41=2\cdot3.141\cdot R$

Keep in mind that the result can be easily simplified using Pi:

$\frac{31.41}{3.141}=2R$

$10=2R$

Finally, we simplify by 2:

$5=R$

5

A circle has a circumference of 50.25.

What is its radius?

We use the formula:

$P=2\pi r$

We insert the known data into the formula:

$50.25=3.14\times2r$

$50.25=2\times r\times3.14$

$50.25=6.28r$

$\frac{50.25}{6.28}=\frac{6.28r}{6.28}$

$r=8$

8

The circumference of a circle is 14.

How long is the circle's radius?

We begin by using the formula:

$P=2\pi r$

We then insert the given data into the formula:

$14=2\times\pi\times r$

Lastly we divide Pi by 2:

$\frac{14}{2\pi}=\frac{2\pi r}{2\pi}$

$\frac{7}{\pi}=r$

$\frac{7}{\pi}$

Below is a circle bounded by a parallelogram:

All meeting points are tangential to the circle.
The circumference is 25.13.

What is the area of the parallelogram?

First, we add letters as reference points:

Let's observe points A and B.

We know that two tangent lines to a circle that start from the same point are parallel to each other.

Therefore:

$AE=AF=3$
$BG=BF=6$

And from here we can calculate:

$AB=AF+FB=3+6=9$

Now we need the height of the parallelogram.

We know that F is tangent to the circle, so the diameter that comes out of point F will also be the height of the parallelogram.

It is also known that the diameter is equal to two radii.

Since the circumference is 25.13.

Circumference formula:$2\pi R$
We replace and solve:

$2\pi R=25.13$
$\pi R=12.565$
$R\approx4$

The height of the parallelogram is equal to two radii, that is, 8.

And from here you can calculate with a parallelogram area formula:

$AlturaXLado$

$9\times8\approx72$

$\approx72$