Find the area of the regular pentagon

As n increases, the area of the regular polygon approaches the area of the circle.

Step-by-step explanation:

A regular polygon inscribed in a circle can be used to derive the formula for the area of a circle.

The polygon area can be expressed in terms of the area of a triangle.

Let s denote the side length of triangle.

r hypotenuse of right triangle.

h height of the triangle.

let n be the number of sides of the regular polygon.

The area of polygon is given by:

polygon area = n(12sh)

The correct statement is:

As n increases, the area of the regular polygon approaches the area of the circle.

" Since, as the number of sides of the polygon increases the polygon start approaching to the circumference of circle and as n tends to infinity the polygon will tend to take the shape of the circle "

As n increases, the area of the regular polygon approaches the area of the circle.

Step-by-step explanation:

A regular polygon inscribed in a circle can be used to derive the formula for the area of a circle.

The polygon area can be expressed in terms of the area of a triangle.

Let s denote the side length of triangle.

r hypotenuse of right triangle.

h height of the triangle.

let n be the number of sides of the regular polygon.

The area of polygon is given by:

polygon area = n(12sh)

The correct statement is:

As n increases, the area of the regular polygon approaches the area of the circle.

" Since, as the number of sides of the polygon increases the polygon start approaching to the circumference of circle and as n tends to infinity the polygon will tend to take the shape of the circle "

1). 238.95 (2nd one)

2). 151.5 (2nd one)

3).  106.5

4). 26 (3rd one)

5). 20

Explanation:

Not 100% sure, some of these I got online, others I didn't.

1). 238.95 (2nd one)

2). 151.5 (2nd one)

3).  106.5

4). 26 (3rd one)

5). 20

Explanation:

Not 100% sure, some of these I got online, others I didn't.

1. The problem statement tells you to find "the area of the hexagonal face".

2. If we assume the intent is to find the shaded area of the face only, it differs from the area of a regular hexagon in that there is a hole in the middle.

3. You must find the area of the regular hexagon, and subtract the area of the circular hole in the middle.

4. The formula for the area of a circle in terms of its radius is

... A = πr²

5. The formula for the area of a regular hexagon in terms of the radius of the circumcircle is

... A = (3√3)/2·r²

6. The radius of the circumcircle of the regular hexagon is given. No additional information is needed.

7. You can use the trig functions of the angles of an equilateral triangle to find the apothem, but there is no need for that when you use the formula of 5.

8. All this is unnecessary. The apothem is (8 mm)·(√3)/2 = 4√3 mm ≈ 6.9282 mm, the shorter leg is (8 mm)·(1/2) = 4 mm. The perimeter is 6·8 mm = 48 mm.

9. The area of the hexagon is

... A = 3√3/2·(8 mm)² = 96√3 mm² ≈ 166.277 mm²

10. The area of the circle is

... A = π·(4 mm)² = 16π mm² ≈ 50.265 mm²

11. The area of the hexagonal face is approximately ...

... 166.277 mm² - 50.265 mm² = 116.01 mm²

1. The problem statement tells you to find "the area of the hexagonal face".

2. If we assume the intent is to find the shaded area of the face only, it differs from the area of a regular hexagon in that there is a hole in the middle.

3. You must find the area of the regular hexagon, and subtract the area of the circular hole in the middle.

4. The formula for the area of a circle in terms of its radius is

... A = πr²

5. The formula for the area of a regular hexagon in terms of the radius of the circumcircle is

... A = (3√3)/2·r²

6. The radius of the circumcircle of the regular hexagon is given. No additional information is needed.

7. You can use the trig functions of the angles of an equilateral triangle to find the apothem, but there is no need for that when you use the formula of 5.

8. All this is unnecessary. The apothem is (8 mm)·(√3)/2 = 4√3 mm ≈ 6.9282 mm, the shorter leg is (8 mm)·(1/2) = 4 mm. The perimeter is 6·8 mm = 48 mm.

9. The area of the hexagon is

... A = 3√3/2·(8 mm)² = 96√3 mm² ≈ 166.277 mm²

10. The area of the circle is

... A = π·(4 mm)² = 16π mm² ≈ 50.265 mm²

11. The area of the hexagonal face is approximately ...

... 166.277 mm² - 50.265 mm² = 116.01 mm²

Follows are the python code to this question:

import math as m #import package  

n=int(input('Enter the number of sides:'))#defining n variable for input number of side

s= float(input('Enter the side: '))#defining s variable for input side value

area = (s*s *n)/(4 * (m.tan(m.pi/n)))#defining formula for calcaulting area

print (area)#print area

Output:

please find the attached file.

Step-by-step explanation:

In the above code, firstly, we import the math package as m, and after that, two-variable "n and s" is defined, which uses for input integer and float value from the user end.