what is the relationship between the side length and the apothem in a regular hexagon?
If you draw a diagram, you can see that the apothem is just the altitude of a 30-60-90 right triangle with one side s/2.
Recall that the sides of such a triangle are in the ratio 1:√3:2
In a regular hexagon, the relationship between the side length and the apothem can be explained as follows:
First, let's define the terms:
- Side length: The length of any one side of the hexagon.
- Apothem: The perpendicular distance from the center of the hexagon to any side.
Now, to understand the relationship, we can use the properties of a regular hexagon. In a regular hexagon, all the sides are equal in length, and all the angles are equal. The apothem is the shortest distance from the center to a side, and it intersects the side at a right angle.
To find the relationship, we need to consider the properties of a right triangle formed by the apothem, half the side length (also known as the radius), and a line segment from the center to one of the vertices of the hexagon. This right triangle has the following relationships:
- The apothem is the shortest side of the right triangle.
- The radius of the hexagon is the hypotenuse.
- The line segment from the center to one of the vertices is the longer side of the right triangle.
Using the Pythagorean theorem, we can express the relationship between the side length (s), apothem (a), and radius (r) as:
s^2 = a^2 + r^2
Since the radius (r) is equivalent to half the side length, we can rewrite the equation as:
s^2 = a^2 + (s/2)^2
Simplifying further, we have:
s^2 = a^2 + s^2/4
Multiplying through by 4 to eliminate the fraction, we get:
4s^2 = 4a^2 + s^2
Combining like terms, we have:
3s^2 = 4a^2
Taking the square root of both sides, we find:
s = 2sqrt(3) * a
Therefore, the relationship between the side length (s) and the apothem (a) in a regular hexagon is:
s = 2sqrt(3) * a
This equation tells us that the side length is equal to 2 times the square root of 3, multiplied by the apothem length. Knowing the value of the apothem, we can easily calculate the side length, and vice versa.
The relationship between the side length and the apothem in a regular hexagon can be found using the Pythagorean theorem. In a regular hexagon, the apothem is the distance from the center to the midpoint of any side, and the side length is the length of one side.
To find the relationship, we can start by drawing a line from the center of the hexagon to one of its vertices, creating a right triangle. The length of this line is the apothem, the length of one side is the adjacent side, and half the length of one side is the opposite side of the triangle.
Using the Pythagorean theorem, the relationship can be expressed as:
(side length)^2 = (apothem)^2 + (opposite side)^2
In a regular hexagon, the opposite side is equal to half the side length, so the relationship simplifies to:
(side length)^2 = (apothem)^2 + (0.5 * side length)^2
We can further simplify this relationship by expanding the terms:
(side length)^2 = (apothem)^2 + (0.25 * side length^2)
By rearranging the equation, we can isolate the apothem:
(apothem)^2 = (side length)^2 - (0.25 * side length^2)
(apothem)^2 = (0.75 * side length^2)
Taking the square root of both sides of the equation gives us the final relationship:
apothem = √(0.75 * side length^2)
Therefore, the apothem of a regular hexagon is equal to the square root of 0.75 times the square of the side length.