Apothem

Do you have to calculate the apothem of a regular polygon? Use our online calculator and get its value automatically. We just need you to enter the number of sides the polygon has and the length of one of them to get the result.

If you want to know more about how to calculate the apothem of a regular polygonRead on because we will tell you all about it.




What is the apothem?

Calculate apothem and area of a pentagon

First of all, let's solve the most common doubt: What is the apothem of a polygon? There are several definitions:

  • It is the segment with the shortest distance between the center of the regular polygon and one of its sides.
  • It is the segment that joins the center of the regular figure with the midpoint of one of its sides, so it is perpendicular to it.

Apothema of a pyramid

In the case of a regular pyramid, the apothem is the segment that starts from the top vertex and goes to the center of any of the sides of the base. In this case, the length of the apothem coincides with the height (h) of each triangular face that we find on the sides.

How to calculate the apothem

Apothem Calculator

We can calculate the apothem of any regular polygon by the following general formula:

Formula for calculating apothem

Being:

  • L = Length of one of the sides of the figure
  • α = central angle of the polygon

How to calculate the value of the central angle in order to solve the formula of the apothem? With the following formula:

Angle formula

In it we simply divide 360º by the number of sides of the polygon (N).

There are specific formulas for calculating the value of the apothem in each geometric figure but we believe that it is much more useful to learn the general formula since you can apply it to any case as we will see in the following example.

Exercise solved

Apothem of a hexagon

We will calculate the apothem of a hexagon whose side measures 20 centimeters.

To solve the exercise, the first thing we will do is to calculate the value of the central angle. As we know that a hexagon has 6 sides, the formula will be:

α = 360º / 6 sides = 60º.

Now we have all the necessary data to be able to apply the general formula:

ap = 20cm / 2tan(60º/2) = 20cm /2tan(30º) = 17.26 centimeters

As you can see, it is very easy to solve this type of exercise.

Formulas for calculating the apothem of regular polygons

If the general formula does not work for you, we will explain below how to calculate the apothem of the main polygons regular:

Triangle

Apothema triangle

Like we can only calculate the apothem in a regular polygonIf we have a scalene or isosceles triangle, it will not be possible to find its value in a scalene or isosceles triangle because we need all sides to be equal, so it will only be possible to calculate the value of the apothem if we have an equilateral triangle.

If you meet this requirement, the formula to use is as follows:

Formula for calculating the apothem of a triangle

L being the length of one of the sides. For example, if we want to calculate how much is the ap in a triangle of side 7 centimeters, we have that:

ap = √3 x 7cm / 6 = 2.03 cm

Square

Square apothem

The apothem of a square is very simple to calculate since its value coincides with half the length of one of the sides of the figure. That is to say:

ap = L / 2

For example, if we have a square whose sides measure 8cm, the value of its ap will be:

ap = 8cm / 2 = 4cm

Pentagon

Calculate apothem and area of a pentagon

To calculate the apothem of a pentagon (in this link you have more info), you have to use this mathematical formula:

Formula to find the apothem of a pentagon

Being:

  • rthe distance from the center of the polygon to one of the vertices of the polygon.
  • lthe length of one of the sides of the pentagon

Hexagon

Apothem of a hexagon

Calculate the apothem in a hexagon knowing the side is very easy since there is an equivalence between the length of its sides and the radius of the circumscribed circle. This makes that we simply have to solve the following formula:

Formula for calculating the apothem of a regular hexagon

R being the length of one of the sides of the figure. In this case we will not extend much more since you have solved examples with more detail in the link that we have left in the previous paragraph.

Heptagon

heptagon

The case of the heptagon is special in that it there is no specific formula to get the apothem of this 7-sided regular polygon, so we are forced to resort to the general formula that we remember was the following:

Formula for calculating apothem

In this case, the heptagon has a central angle of 51.43º so if we substitute in the previous expression and simplify the formula as much as possible, we get this way:

ap heptagon = L / 0,9631

L being the length of one of its sides.

For example, if we want to calculate the value of ap in a heptagon of side equal to 10 centimeters, we have that:

ap = 10 cm / 0.96 = 10.35 cm

Octagon

Apothema octagon

For the octagon something similar happens to the heptagon and we don't have a concrete formula to get the apothem, that is why we resort to the general formula which we will simplify since we know that the central angle of an octagon is 45º.

Therefore, the value of ap will be:

ap = L / 0.8284

Pyramid

Apothema of a pyramid

Find the apothem of a pyramid is a more complex process based on the Pythagorean Theorem and it requires more steps that we will detail so that you do not have any doubt.

The formula to calculate it that we have to use is the following:

Pyramid apothem formula

Being:

  • h: the height of the pyramid
  • apb the apothem of the base

That is, the first thing we have to do is to calculate the apothem of the base of the pyramid. You already know how to do it depending on whether it is triangular, square, hexagonal or any other shape with the explanations that we have been giving you up to this point.

Once you have calculated apjust substitute in the formula and solve for the square root.

If you have an exercise about the apothem that you do not know how to solve, leave us a comment and we will try to help you as soon as possible.

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