The apothem of a regular polygon is the distance from the center to any side.
If the length of the apothem remains constant at 10 inches, the formula for the perimeter of a regular polygon as a function of the number of sides is ( ) ( )( ).
As the regular polygon changes from a pentagon (5 sides) to an octagon (8 sides), what is the approximate average rate of change in the perimeter?
2+2=1
To find the formula for the perimeter of a regular polygon as a function of the number of sides, we can use the formula:
Perimeter = Number of sides * Length of each side
Since the length of the apothem remains constant at 10 inches, the length of each side can be found using trigonometry. The formula for the length of each side in terms of the apothem is:
Length of each side = 2 * Apothem * tan(180 degrees / Number of sides)
Now we can substitute this into the perimeter formula:
Perimeter = Number of sides * (2 * Apothem * tan(180 degrees / Number of sides))
So the formula for the perimeter of a regular polygon as a function of the number of sides is:
Perimeter = 2 * Number of sides * Apothem * tan(180 degrees / Number of sides)
To calculate the approximate average rate of change in the perimeter as the regular polygon changes from a pentagon (5 sides) to an octagon (8 sides), we can find the difference in perimeters and divide by the difference in the number of sides:
Average rate of change = (Perimeter of octagon - Perimeter of pentagon) / (Number of sides in octagon - Number of sides in pentagon)
Let's calculate this:
Perimeter of pentagon = 2 * 5 * 10 * tan(180 degrees / 5) ≈ 75.41 inches
Perimeter of octagon = 2 * 8 * 10 * tan(180 degrees / 8) ≈ 138.56 inches
Number of sides in octagon = 8
Number of sides in pentagon = 5
Average rate of change = (138.56 - 75.41) / (8 - 5) ≈ 21.05 inches
Therefore, the approximate average rate of change in the perimeter as the regular polygon changes from a pentagon to an octagon is approximately 21.05 inches.
To find the formula for the perimeter of a regular polygon as a function of the number of sides, we need to use the known formulas for the perimeter and the apothem of a regular polygon.
The formula for the perimeter of a regular polygon is given by:
Perimeter = Number of sides × Length of each side
And the apothem is the distance from the center of the polygon to any side.
Since the length of the apothem remains constant at 10 inches, we can substitute this value into the formula for the perimeter as follows:
Perimeter = Number of sides × Length of each side
Perimeter = Number of sides × 2 × apothem
Perimeter = Number of sides × 2 × 10
Perimeter = 20 × Number of sides
Therefore, the formula for the perimeter of a regular polygon as a function of the number of sides is 20 times the number of sides.
Now, to calculate the approximate average rate of change in the perimeter as the regular polygon changes from a pentagon to an octagon, we can use the following formula:
Average Rate of Change = (Change in Y) / (Change in X)
In this case, the change in X represents the change in the number of sides, and the change in Y represents the change in the perimeter.
The change in the number of sides is 8 - 5 = 3, and the change in the perimeter is (20 × 8) - (20 × 5) = 160 - 100 = 60.
Plugging these values into the formula, we get:
Average Rate of Change = (60) / (3)
Average Rate of Change = 20
Therefore, the approximate average rate of change in the perimeter as the regular polygon changes from a pentagon to an octagon is 20 inches.
If the perimeter is p, the area is A = 1/2 ap.
So, since p = ns, what is the side s of an n-gon?
The central angle of each of the n isosceles triangles is θ=360/n, so the side is
s = 20 tan(θ/2)
Thus, the area is
A = 1/2 a (ns) = 1/2 (10) (n*20 tan 180/n)
= 100n tan(180/n)
You can probably handle the rest, eh?