How Do I Calculate the Apothem of a Triangle?

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An apothem of a triangle depends upon either the length of one of its side or the radius of a circle that fits inside it. There are two methods of determining an apothem through known measurements.

An apothem of a regular polygon is any line running from the center of the polygon to the midpoint of a side. Every regular polygon has multiple apothems, depending on the number of sides; a polygon with n sides has n apothems. One method of calculating an apothem uses the length of any side of the polygon. Divide 180 by the number of sides of the polygon, which is three in the case of a triangle, and multiply the tangent function of this value by two. Finally, divide the length of one side of the triangle by the previous value. This calculation yields the apothem from the center to that specific side. Knowing the length of each side of the triangle enables the calculation of all three apothems.

The radius of a polygon also yields a value for the apothem. In this case, begin again by dividing 180 by the number of sides. Use the cosine function on this value, then multiply that by the radius of the triangle. This new value is the apothem.

Apothems only exist for regular polygons. Irregular polygons have no true center, so there is no way to calculate an apothem.