What is the relationship between the side length and the apothem of an equilateral triangle?
let the side be 2x
let the height , (your apothem) be h
then
x^2 + h^2 = (2x)^2
h^2 = 3x^2
h = √3 x
ratio of side : apothem
= 2x : √3 x
= 2 : √3
The relationship between the side length and the apothem of an equilateral triangle can be derived using its properties.
An equilateral triangle is a special type of triangle in which all three sides are equal in length and all three angles are 60 degrees.
The apothem of a regular polygon is defined as the perpendicular distance from the center of the polygon to any side. In an equilateral triangle, the apothem is also the distance from the center to any vertex.
To find the relationship between the side length (s) and the apothem (a), we can divide the triangle into two congruent right triangles by drawing a line from the center to one of the vertices. This line segment is the apothem.
In the right triangle, the apothem (a) serves as the height, and one-half of the side length (s/2) is the base.
Applying the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can write:
(a^2) = (s/2)^2 + (s^2)
a^2 = s^2/4 + s^2
a^2 = (1/4) * s^2 + s^2
a^2 = (1/4 + 1) * s^2
a^2 = (5/4) * s^2
Taking the square root of both sides, we find:
a = √(5/4) * s
Hence, the relationship between the side length (s) and the apothem (a) of an equilateral triangle is that the apothem is equal to the square root of 5 divided by 4, multiplied by the side length.