He **circle perimeter **is the set of points that form the contour of a circle and is also known as *length* *of the circumference*. It depends on the radius, since a larger circumference will obviously have a larger contour.

Be *P* the perimeter of a circle and *R.* the radius of it, then we can calculate *P* with the following equation:

*P = 2π.R*

Where π is a real number (read “pi”) with a value of approximately 3.1416… The ellipses are due to the fact that π has infinite decimal places. Therefore, when making the calculations, it is necessary to round its value.

However, for most applications, it is enough to take the amount indicated here, or use all the decimal places that the calculator with which you work.

If instead of having the radius, you prefer to use the diameter D, which we know is twice the radius, the perimeter is expressed as follows:

*P = π.2R= π.D*

Since perimeter is a length, it should always be expressed in units such as meters, centimeters, feet, inches, and more, depending on which system is preferred.

[toc]

**circumferences and circles**

They are often terms that are used interchangeably, that is, as synonyms. But it happens that there are differences between them.

The word «perimeter» comes from the Greek «peri» which means contour and «meter» or measure. The circumference is the outline or perimeter of the circle. It is formally defined like this:

*A circle is the set of points with the same distance from a point called the center, this distance being the radius of the circle.*

For its part, the circle is defined as follows:

*A circle is the set of points whose distance from a point called the center is less than or equal to a fixed distance called the radius.*

The reader can notice the subtle difference between both concepts. The circumference only refers to the set of points on the edge, while the circle is the set of points from the edge to the interior, of which the circumference is the border.

**exercises ****d****demonstration of the calculation of the perimeter of the circle**

Through the following exercises, the concepts described above will be put into practice, as well as some others that will be explained as they appear. We will start from the simplest and the degree of difficulty will gradually increase.

**– Exercise 1**

Find the perimeter and area of the circle of radius 5 cm.

**Solution**

The equation given at the beginning is directly applied:

*P** = 2π.R**= 2π.5cm** =10πcm** = 31.416 cm*

To calculate the area *TO* the following formula is used:

*TO* = *π.R2* = *π. (5cm)2= 25π cm2= 78.534 cm2*

**– Exercise 2**

a) Find the perimeter and area of the blank region in the following figure. The center of the shaded circle is the red point, while the center of the white circle is the green point.

b) Repeat the previous section for the shaded region.

**Solution**

a) The radius of the white circle is 3 cm, therefore we apply the same equations as in exercise 1:

*P** = 2π.R**= 2π.3cm** =6πcm** = 18.85cm*

*TO *= *π.R2* = *π. (3cm)2= 9π cm2= 28.27 cm2*

b) For the shaded circle, the radius is 6 cm, its perimeter is twice that calculated in part a):

*P** = 2π.R**= 2π.6cm** =12πcm** = 37.70cm*

And finally the area of the shaded region is calculated as follows:

– First, find the area of the shaded circle as if it were complete, which we will call A´, like this:

*TO *= *π.R2= π.(6 cm)2 =36π cm2= 113.10 cm2*

*– *Then to the area *TO* the area of the white circle, previously calculated in section a), is subtracted, in this way the requested area is obtained, which will be denoted simply as A:

*A = A´ – 28.27 cm2 = 113.10-28.27 cm2 = 84.83 cm2*

**– Exercise 3**

Find the area and perimeter of the shaded region in the following figure:

**Solution**

**Calculation of the area of the shaded region**

We first calculate the area of the *circular sector* or wedge, between the straight segments OA and OB and the circular segment AB, as shown in the following figure:

For this, the following equation is used, which gives us the area of a circular sector, knowing the radius R and the central angle between the segments OA and OB, that is, two of the radii of the circumference:

*A circular sector = π.R2. (αº/360º)*

Where αº is the central angle –it is central because its vertex is the center of the circle- included between two radii.

**Step 1: Calculate the area of the circular sector**

Thus, the area of the sector shown in the figure is:

*A circular sector = π.R2. **(αº/360º) = π. (8cm)**)2. (60º/360º) = (64/6) **πcm2**= 33.51 cm2*

**Step 2: Calculating the area of the triangle**

Next we will calculate the area of the white triangle in figure 3. This triangle is equilateral and its area is:

*A triangle = (1/2) base x height *

The height is the dotted red line seen in figure 4. To find it you can use the Pythagorean theorem, for example. But it is not the only way.

The observant reader will have noticed that the equilateral triangle is divided into two identical right triangles, whose base is 4 cm:

In a right triangle the Pythagorean theorem is fulfilled, therefore:

Since we have the height of the triangle, both the rectangle and the equilateral, its area is calculated:

*A triangle = (1/2) base x height = (1/2) 8 cm x 6.93 cm = 27.71 cm2.*

**Step 3: Calculating the shaded area**

It is enough to subtract the greater area (that of the circular sector) from the smaller area (that of the equilateral triangle): A shaded region = *33.51 cm2 – 27.71 cm2 = 5.80 cm2.*

**Calculation of the perimeter of the shaded region**

The perimeter sought is the sum of the rectilinear side of 8 cm and the arc of circumference AB. Now, the complete circumference subtends 360º, therefore an arc that subtends 60º is one sixth of the complete length, which we know is 2.π.R:

*AB = 2.π.R / 6 = 2.π.8 cm / 6 = 8.38 cm*

Substituting, the perimeter of the shaded region is:

*P = 8cm + 8.38cm = 16.38cm.*

**Applications**

The perimeter, like the area, is a very important concept in geometry and with many applications in daily life.

Artists, designers, architects, engineers and many other people make use of the perimeter while carrying out their work, especially that of a circle, since the round shape is everywhere: from advertising to food to machinery.

To directly know the length of a circumference, it is enough to wrap it with a thread or string, then extend this thread and measure it with a tape measure. The other alternative is to measure the radius or diameter of the circle and use one of the formulas described above.

In daily tasks, the concept of perimeter is used when:

-The right mold is chosen for a certain size of pizza or cake.

-An urban road is going to be designed, by calculating the size of a vial where cars can turn to change direction.

-We know that the Earth revolves around the Sun in an approximately circular orbit –in fact, planetary orbits are elliptical, according to Kepler’s laws-, but the circumference is a very good approximation for most of the planets.

-The appropriate size of a ring or ring that is going to be bought in an online store is chosen.

-We choose a key of the right size to loosen a nut.

And many more.

https://youtu.be/Cr8xJRYL5Tk

**References**

Free Mathematics Tutorials. Area and Perimeter of a Circle – Geometry Calculator. Retrieved from: analyzemath.com.

Math Open Reference. Circumference, Perimeter of a circle. Retrieved from: mathopenref.com.

Monterey Institute. Perimeter and Area. Retrieved from: montereyinstitute.org.

Science. How to find the Perimeter of a Circle. Retrieved from: science.com.

Wikipedia. circumference. Retrieved from: en.wikipedia.org.