What is the relationship between the side length and the apothem of an equilateral triangle?
L = Length of a side.
A = Apothem = Altitude.
L/A = L/L*sin60 = 1/sin60 = 1.1547.
So the length of a side is approximately 1.15 times the apothem.
In an equilateral triangle, the apothem is the distance from the center of the triangle to the midpoint of any side, and the side length is the length of any side of the triangle.
The relationship between the side length and the apothem of an equilateral triangle is as follows:
The apothem is equal to half the height of the equilateral triangle, which can be calculated using the Pythagorean theorem.
If "s" represents the side length of the equilateral triangle, the height (h) can be found by applying the Pythagorean theorem to a right triangle formed by the apothem (a) and half of one side (s/2).
Using the formula:
h^2 = s^2 - (s/2)^2
Simplifying the equation, we get:
h^2 = s^2 - s^2/4
h^2 = 3s^2/4
Taking the square root of both sides, we find:
h = (s√3)/2
Thus, the apothem of an equilateral triangle is equal to (s√3)/2, or s multiplied by the square root of 3, divided by 2.
The relationship between the side length and the apothem of an equilateral triangle can be explained as follows:
First, let's define what a side length and an apothem are in relation to an equilateral triangle.
- Side length: The side length refers to the length of any one side of an equilateral triangle. In an equilateral triangle, all three sides are equal in length. Let's denote the side length as "s".
- Apothem: The apothem is the perpendicular distance from the center of the equilateral triangle to any one of its sides. It is denoted by "a".
Now, to understand the relationship between the side length and the apothem, we can use the following formula:
a = (s√3)/2
Here's how you can derive this formula:
1. Draw an equilateral triangle and its height (apothem) as shown:
/\
/ \
s/ \s
/______\
2. Divide the equilateral triangle into two right-angled triangles by drawing a line from the center of the triangle to one of its vertices.
/\
/ |\
/ | \
a/ | \a
/____|___\
3. Each of the resulting right-angled triangles is a 30-60-90 triangle, where the sides have a specific ratio. In this case, the ratio is 1:√3:2.
4. From the properties of a 30-60-90 triangle, the apothem (a) is equal to half the length of one side (s) multiplied by the square root of 3.
a = (s√3)/2
Therefore, the relationship between the side length and the apothem of an equilateral triangle is given by a = (s√3)/2.