area of a nonagon with a perimeter of 126in
The area A of any regular polygon of n sides with side length s, apothem a is
A=nas/2.
Since ns=perimeter, p, we can also write the area as
A=ap/2
An apothem is the distance from the centre of the polygon to the side.
We assume the given nonagon is regular, i.e. all sides and interior angles are equal.
A nonagon has 9 sides. Each side is of length 126/9=14". We need first to find the apothem as follows.
The central angle for each side is 360/9=40°.
Each side of the nonagon forms an isosceles triangle with the centre of the nonagon. Its half vertical angle equals 40/2=20°.
The height of this triangle is the apothem and can be calculated by the trigonometric formula
a=(s/2)/tan(20)=7cot(20)
Using the formula above:
Area = ap/2=126*7cot(20)=882cot(20°)
To find the area of a nonagon (a polygon with nine sides) with a given perimeter of 126 inches, we need to use the formula for the area of a general polygon.
Step 1: Find the length of each side.
Since a nonagon has nine sides, the perimeter is the sum of the lengths of all sides. Therefore, each side length is calculated by dividing the perimeter by nine: 126 in ÷ 9 = 14 in.
Step 2: Determine the apothem of the nonagon.
The apothem is the distance from the center of the nonagon to the midpoint of one of its sides. Unfortunately, we don't have enough information to directly calculate the apothem.
However, we can find the approximate value of the apothem by dividing the length of a side by the tangent of half of the interior angle of the nonagon. The interior angle of a nonagon is given by (9-2) * 180° / 9 = 140°.
Using the formula, the approximate apothem is calculated as follows:
apothem ≈ side length / tan(1/2 * interior angle)
≈ 14 in / tan(1/2 * 140°)
You can calculate this using a scientific calculator or online calculator to get an approximate value.
Step 3: Calculate the area.
The area of the nonagon can be calculated using the formula: area = (perimeter × apothem) / 2.
Using the perimeter of 126 inches and the approximate value of the apothem, you can plug the values into the formula to find the estimated area.
To find the area of a nonagon (a polygon with nine sides), we need to know the length of one of its sides or some additional information. In this case, since we only have the perimeter, we need to find the length of one side first.
To find the length of one side, we can divide the perimeter by the number of sides (nine in this case).
Given that the perimeter is 126 inches,
Length of one side = Perimeter / Number of sides
= 126 in / 9
= 14 in
Now that we know the length of one side, we can calculate the area of the nonagon. However, be aware that there are multiple methods to do so. Let's go through two common methods: using trigonometry and using the apothem.
Method 1: Using Trigonometry
One way to find the area of the nonagon is by splitting it into triangles and then using trigonometry.
Step 1: Divide the nonagon into triangles. You can draw lines from one vertex of the nonagon to all other non-adjacent vertices.
This will create nine triangles within the nonagon.
Step 2: Find the area of one triangle.
Since the nonagon is regular (all sides and angles are equal), each triangle within it is isosceles. To find the area of one triangle, we need the height, which can be calculated using the trigonometric relationship of an isosceles triangle.
To find the height:
Height = (Side length / 2) * tan(180° / 9)
= (14 in / 2) * tan(20°)
≈ 2.409 in (rounded to three decimal places)
Step 3: Calculate the area of one triangle.
Area of one triangle = (1/2) * base * height
= (1/2) * 14 in * 2.409 in
≈ 16.833 in² (rounded to three decimal places)
Step 4: Calculate the area of the entire nonagon.
Since there are nine identical triangles, the total area of the nonagon is obtained by multiplying the area of one triangle by the number of triangles.
Area of the nonagon = 9 * Area of one triangle
≈ 151.497 in² (rounded to three decimal places)
Method 2: Using Apothem
Another approach to finding the area of a nonagon is by using the apothem (the distance from the center of the nonagon to any of its sides).
Step 1: Find the apothem.
To calculate the apothem, we can use the following formula:
Apothem = (Side length / 2) / tan(180° / 9)
= (14 in / 2) / tan(20°)
≈ 6.620 in (rounded to three decimal places)
Step 2: Calculate the area of the nonagon.
The area of a nonagon can be found using the formula:
Area = (Perimeter * Apothem) / 2
= (126 in * 6.620 in) / 2
≈ 416.340 in² (rounded to three decimal places)
Therefore, the approximate area of the nonagon with a perimeter of 126 inches is 416.340 square inches.