Why is the apothem of an equilateral triangle its one third height?

The relationship between the apothem and the height of an equilateral triangle can be derived from its properties. An equilateral triangle is a special type of triangle where all three sides are equal in length, and all three angles are 60 degrees.

To understand why the apothem is one third of the height, let's break down the problem step by step:

Step 1: Draw an equilateral triangle.
Draw an equilateral triangle with one side labeled as "s" and a height extending from one vertex to the midpoint of the opposite side.

Step 2: Divide the height into three equal parts.
Since an equilateral triangle has all three angles equal to 60 degrees, the height divides the triangle into two congruent right-angled triangles. Divide the height into three equal parts.

Step 3: Label the height segments.
Label the total height of the triangle as "h." Each of the three parts will be "h/3".

Step 4: Consider one of the right-angled triangles.
Focus on one of the right-angled triangles formed by the height. Its height is "h/3", and the base is half the length of one side of the triangle, which is "s/2".

Step 5: Define the apothem.
The apothem of an equilateral triangle is the perpendicular distance from the center of the triangle to one of its sides.

Step 6: Draw a line segment to the center of the triangle.
Draw a line segment from the midpoint of one side of the equilateral triangle to its center, creating a right angle. This segment represents the apothem.

Step 7: Notice the relation between the triangle and the apothem.
The right-angled triangle formed by the apothem, the height, and one-half of a side of the equilateral triangle is similar to the right-angled triangle formed by the height and one-half of a side.

Step 8: Apply the properties of similar triangles.
Since the two right-angled triangles formed are similar, we can set up a proportion: the height divided by one-half of a side is equal to the apothem divided by the base of the right-angled triangle.

(h/3) / (s/2) = apothem / (s/2)

Simplifying the equation:

2(h/3) = apothem

Therefore, the apothem is equal to two-thirds (2/3) of the height of an equilateral triangle.

In conclusion, the apothem of an equilateral triangle is not one-third of its height, but rather two-thirds of its height.