Find the area of a regular 36-sided polygon inscribed in a circle of radius 20
360/36 = 10 degrees center angle for each of the 36 isosceles triangles with equal sides of length 10
10 cos 5 degrees = h
sin 5 = (b/2)/ 10 so b/2 = 10 sin 5
A of each of the 36 = h b/2 = 10 cos 5 * 10 sin 5 = 100 sin 5 cos 5
A = 36 * 100 sin 5 cos 5 = 314
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check, should be a touch less than circle area
pi r^2 = 100 * 3.14159
LOL close enough
the apothem is 20 cos5° = 19.923
the side length is 2*20sin5° = 3.486
the area is 1/2 * perimeter * apothem = 1250.1
area of a circle of radius 20 = 400π = 1256.6
To find the area of a regular polygon inscribed in a circle, we need to know the length of one of its sides.
Let's calculate the length of one side of the 36-sided polygon by dividing the circumference of the circle by 36:
Circumference = 2 * π * radius = 2 * 3.14159 * 20 ≈ 125.6637
Length of one side = Circumference / 36 ≈ 125.6637 / 36 ≈ 3.4907
Now that we know the length of one side, we can calculate the area of the regular 36-sided polygon.
The area of a regular polygon with n sides, where s is the length of one side, is given by the formula:
Area = (n * s^2) / (4 * tan(π/n))
Substituting the values:
Area = (36 * (3.4907)^2) / (4 * tan(π/36))
Using a calculator, we can evaluate this expression:
Area ≈ 36 * 12.1601 / (4 * 0.0524) ≈ 10.3923 / 0.2095 ≈ 49.5862
Therefore, the area of the regular 36-sided polygon inscribed in a circle of radius 20 is approximately 49.5862 square units.
To find the area of a regular polygon inscribed in a circle, we need to know the radius of the circle.
In this case, the given radius is 20.
The area of a regular polygon can be calculated using the formula:
Area = (1/2) x Apothem x Perimeter
To find the apothem (distance from the center of the polygon to the midpoint of any side), we need to use some trigonometry.
The formula to find the apothem of a regular polygon is:
Apothem = Radius x Cosine (180° / Number of Sides)
In this case, the polygon has 36 sides, so the apothem can be calculated as follows:
Apothem = 20 x Cosine (180° / 36)
Using a calculator, we can compute:
Apothem ≈ 18.867
Now, we need to find the perimeter of the polygon. In a regular polygon, all sides are equal, so we can use the formula:
Perimeter = Number of Sides x Length of Each Side
To find the length of each side, we need to use the formula:
Length of Each Side = 2 x Radius x Sine (180° / Number of Sides)
In this case, the length of each side can be calculated as follows:
Length of Each Side = 2 x 20 x Sine (180° / 36)
Using a calculator, we can compute:
Length of Each Side ≈ 11.547
Finally, we can calculate the area of the polygon using the formula mentioned earlier:
Area = (1/2) x Apothem x Perimeter
Plugging in the values we calculated:
Area ≈ (1/2) x 18.867 x (36 x 11.547)
Using a calculator, we can compute the area:
Area ≈ 378.371 square units
Therefore, the area of the regular 36-sided polygon inscribed in a circle of radius 20 is approximately 378.371 square units.