Answer
Verified
387.6k+ views
Hint: To find the ratio of the areas of the circle and polygon, first find the relation between the radius of the circle and the length of the side of the polygon. Then put them in the formula for their respective areas, and then get the ratio.
Complete step-by-step answer:
Let the radius of the circle be r and the side length of the polygon be a.
We know that perimeter of circle = $2\pi r$
Perimeter of an n sided polygon = $n.a$
As the two perimeters are given to be equal-
$
2\pi r = n.a \\
\dfrac{{2\pi r}}{n} = a \\
$ …..relation 1
This way we found the side length a in terms of r.
Now, in order to calculate the ratio of the areas of the two surfaces, put the value of r and a in their respective formulas.
Area of circle = $\pi {r^2}$
Area of an n-sided polygon = $\dfrac{1}{4}n{a^2}\cot \left( {\dfrac{\pi }{n}} \right)$
Substituting the value of ‘a’ in the above formula from relation 1-
$
\dfrac{1}{4}n{a^2}\cot \left( {\dfrac{\pi }{n}} \right) \\
\dfrac{1}{4}n{\left( {\dfrac{{2\pi r}}{n}} \right)^2}\cot \left( {\dfrac{\pi }{n}} \right) \\
\dfrac{1}{4}n\left( {\dfrac{{4{\pi ^2}{r^2}}}{{{n^2}}}} \right)\cot \left( {\dfrac{\pi }{n}} \right) \\
\left( {\dfrac{{{\pi ^2}{r^2}}}{n}} \right)\cot \left( {\dfrac{\pi }{n}} \right) \\
$
Now the ratio of the circle and the polygon becomes-
$
\;\pi {r^2}:\left( {\dfrac{{{\pi ^2}{r^2}}}{n}} \right)\cot \left( {\dfrac{\pi }{n}} \right)\; \\
1:\left( {\dfrac{\pi }{n}} \right)\cot \left( {\dfrac{\pi }{n}} \right) \\
\tan \left( {\dfrac{\pi }{n}} \right):\left( {\dfrac{\pi }{n}} \right) \\
$
So the correct option is A.
Note: The formula for the area of a polygon can be determined by inscribing a circle inside the polygon. For direct calculation purposes, we use the final formula directly. Rest the question is based on simple calculations.
Complete step-by-step answer:
Let the radius of the circle be r and the side length of the polygon be a.
We know that perimeter of circle = $2\pi r$
Perimeter of an n sided polygon = $n.a$
As the two perimeters are given to be equal-
$
2\pi r = n.a \\
\dfrac{{2\pi r}}{n} = a \\
$ …..relation 1
This way we found the side length a in terms of r.
Now, in order to calculate the ratio of the areas of the two surfaces, put the value of r and a in their respective formulas.
Area of circle = $\pi {r^2}$
Area of an n-sided polygon = $\dfrac{1}{4}n{a^2}\cot \left( {\dfrac{\pi }{n}} \right)$
Substituting the value of ‘a’ in the above formula from relation 1-
$
\dfrac{1}{4}n{a^2}\cot \left( {\dfrac{\pi }{n}} \right) \\
\dfrac{1}{4}n{\left( {\dfrac{{2\pi r}}{n}} \right)^2}\cot \left( {\dfrac{\pi }{n}} \right) \\
\dfrac{1}{4}n\left( {\dfrac{{4{\pi ^2}{r^2}}}{{{n^2}}}} \right)\cot \left( {\dfrac{\pi }{n}} \right) \\
\left( {\dfrac{{{\pi ^2}{r^2}}}{n}} \right)\cot \left( {\dfrac{\pi }{n}} \right) \\
$
Now the ratio of the circle and the polygon becomes-
$
\;\pi {r^2}:\left( {\dfrac{{{\pi ^2}{r^2}}}{n}} \right)\cot \left( {\dfrac{\pi }{n}} \right)\; \\
1:\left( {\dfrac{\pi }{n}} \right)\cot \left( {\dfrac{\pi }{n}} \right) \\
\tan \left( {\dfrac{\pi }{n}} \right):\left( {\dfrac{\pi }{n}} \right) \\
$
So the correct option is A.
Note: The formula for the area of a polygon can be determined by inscribing a circle inside the polygon. For direct calculation purposes, we use the final formula directly. Rest the question is based on simple calculations.
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Which type of bond is stronger ionic or covalent class 12 chemistry CBSE
What organs are located on the left side of your body class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
When people say No pun intended what does that mea class 8 english CBSE