The area of a polygon is 48 mi2. What is the formula that shows the length of side a?
unless it is a regular polygon, there is no such formula.
For a regular polygon of n sides and area A, the side length is
a = √(A/n tan(π/n))
so, once you have figured out how many sides you have, apply the formula.
I assume your polygon is a regular polygon, that is,
all sides are equal, and all interior angles are equal.
Let the number of sides be n and consider one of the isosceles
triangles consisting of a base of a , two equal sides and a central angle
of 360/n°
construct a perpendicular so you have a right-angled triangle
with base a/2 , a height of h, and the angle at the centre as 360/(2n)°
tan (360/(2n)) = a/2 / h
h = (a/2) / tan(180/n)°
area of one triangle = (1/2) base x height
= (1/2)(a/2)tan(180/n) = (1/4) a tan(180/n)
you have n of those, so total area =
(n/4)a tan(180/n) = 48
a n tan(180/n) = 192
a = 192/(ntan(180/n)
so once you know what n is in your polygon, you got it.
Oops. I forgot the factor of 2.
a = 1/2 √(A/n tan(π/n))
I think mathhelper forgot a factor of a. Area is in units^2, not just units.
yup, glad you caught that.
Would this work?
suppose we have a hexagon with side of 6, then each of the 6 triangles has an area of
(1/2)(6)(3√3) = 9√3
total area = 54√3 = appr 93.53
ratio of areas of similar shapes is proportional to the square of their sides.
93.53/48 = 36/a^2
a^2 = 48*36/93.53 = 18.47..
a = appr 4.3
To find the formula that shows the length of side a, we need some more information about the polygon. The given area alone does not provide sufficient information to determine the length of a specific side. In order to calculate the length of a side, we typically require either the number of sides, the length of one side, or additional information about the shape of the polygon.
However, if we assume that the polygon is a regular polygon, meaning that all sides and angles are equal, we can use the formula for the area of a regular polygon to derive a formula for the length of side a.
The area of a regular polygon can be calculated using the formula:
Area = (1/4) * n * a^2 * cot(π/n)
Where:
- n is the number of sides of the polygon
- a is the length of each side
Rearranging this equation, we can solve for the length of side a:
a^2 = (4 * Area) / (n * cot(π/n))
Taking the square root of both sides gives us the formula for side a:
a = √((4 * Area) / (n * cot(π/n)))
Keep in mind that this formula assumes the polygon is regular. If the polygon is not regular, we would need additional information or specific values to find the length of side a.