Why is the apothem of an equilateral triangle one third its height?
The apothem of a regular polygon is the distance from the centre to one of its sides.
Let the equilateral triangle (regular polygon of three sides) be ABC, centred at O. Drop a perpendicular from O to D on side AB.
Consider triangle OAB, ∠OAB and ∠OBA are each 30°.
Since sin(30°)=1/2, it is clear that OD = OA/2 = OC/2
Therefore
Apothem OD=(OD+OA)/3=CD/3=height/3.
The apothem of an equilateral triangle is one third of its height because of the special properties of equilateral triangles. To understand why this is the case, let's first define what the apothem and height of a triangle are.
The apothem of a triangle is the distance from the center of the triangle to any of its sides. In the case of an equilateral triangle, the apothem is the perpendicular distance from the center to one of the sides.
The height of an equilateral triangle is the length of a line segment drawn from one of the vertices to the opposite side, perpendicular to that side.
Now let's look at how to determine the relationship between the apothem and the height of an equilateral triangle. We can start by drawing an equilateral triangle and its height, as shown below:
/\
/__\
/____\
height
Let's call the length of the height "h" and the length of the apothem "a". Since an equilateral triangle has all three sides equal in length, we can also label the length of each side as "s".
Now, let's consider dividing the equilateral triangle into two congruent right triangles by drawing a line segment from the center of the triangle to one of the vertices, as shown below:
/\
/__\
/____\
/ \ / \
height | a | height
Now we have a right triangle with the apothem "a" as one of its legs, the height "h" as the hypotenuse, and the side of the equilateral triangle as the other leg, which is also equal to "s/2".
Using the Pythagorean theorem, we can write the relationship as:
(s/2)^2 + a^2 = h^2
Since the equilateral triangle has all sides equal, s = s/2 + s/2 = h. By substituting h for s in the equation above, we get:
(s/2)^2 + a^2 = s^2
Now, let's simplify the equation:
s^2/4 + a^2 = s^2
Multiply both sides of the equation by 4 to get rid of the fraction:
s^2 + 4a^2 = 4s^2
Rearranging the equation:
3s^2 = 4a^2
Now, we can solve for the apothem "a":
a^2 = (3s^2)/4
Taking the square root of both sides:
a = sqrt((3s^2)/4)
Since s is the length of a side of the equilateral triangle, and all sides are equal, we can substitute s with h:
a = sqrt((3h^2)/4)
Simplifying further:
a = (sqrt(3)h)/2
From the equation above, we can see that the apothem "a" is equal to (sqrt(3)h)/2. If we divide both sides of the equation by the height "h", we have:
a/h = (sqrt(3)h)/2h
Simplifying further:
a/h = sqrt(3)/2
Therefore, we can conclude that the apothem of an equilateral triangle is equal to one third of its height, since sqrt(3)/2 is equal to 1/3.
So, the reason why the apothem of an equilateral triangle is one-third of its height is due to the geometry and properties of equilateral triangles.