Circular sector

Do you want to calculate the area of a circular sector? Use our online calculator and you will get the area of the sector automatically as well as its arc length and perimeter.

All you need is to know the radius of the circle (r) and the number of degrees (α) that the circular sector occupies. When you have it, press the calculate button and you will get the result immediately.

What is a circular sector?

Circular sector

When we want to calculate the area of a circleIn this case, we do it taking into account its entire surface. However, we may only be interested in a portion of the circle Therefore, we refer to this portion as the circular sector.

If we look at the figure above these lines, the circular sector would correspond to the part colored in turquoise. It is formed by the rotation of the radius a given number of degrees or radians.

Area of a circular sector

Now that we know what a circular sector is, we are going to learn how to calculate its area. To do so, we will use the following mathematical formula:

Formula for calculating the area of a circular sector

In this formula we have two unknowns that we must know:

  • r is the radius of the circle
  • α is the angle in degrees of the sector

If we were given the sector angle expressed in radians we have two options. The first is to switch from radians to degrees while the second option is to apply this other formula:

Formula for calculating the area of the circular sector in radians

Rememberthis second one formula for calculating the area of the circular sector we will use it if we are given the angle expressed in radians.

Arc length

Circumferential arch

If we are asked to calculate the arc length of the circular sectorrefer to the distance from point A to point B. We will have to calculate the length of that arc by applying the following formula:

Formula for calculating the length of an arc

Calculate the perimeter of a circular sector

In the event that we want to remove the perimeter of the entire circular sectorwe must add to the length of the arc 2 times the radius, that is to say:

Circular sector perimeter = arc length + 2r

If we express it mathematically, the formula to calculate the perimeter of the circular sector is:

Formula for calculating the perimeter of a circular sector


Circular sector exercises

Circular sector exercise

To put into practice everything we have learned so far, we are going to do a solved exercise in which we are asked for calculate the area, arc length and perimeter of a circular sector whose radius is 6cm and α=90º.

We will start with the area since it is the first formula we have seen in the theory:

Example of an exercise to calculate the area of a circular sector

Remember that area units are squared. If you do not leave it indicated, you may be deducted points in the grade of the exercise.

Next we will calculate the arc of the circular sector:

Exercise of calculating the arc of a circular sector

With the arc already calculated, the perimeter is obtained in a very simple way since we only have to add twice the length of the radius. The complete process to calculate it would be as follows:

Exercise solved of perimeter of circular sector

If you have any questions or have any problems solving an exercise, leave us a comment and we will help you solve it.

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