Answer
Verified
399.4k+ views
Hint:First assume the perimeter of the square and circle and then equate the perimeters and then find the value of the sides \[s\] of the square and then equate the area of the square and circle. The formula for the square and circle as:
Area of square: \[{{s}^{2}}\]
Area of circle: \[\pi {{r}^{2}}\] (\[r\]: radius)
Complete step by step solution:
As given in the question, the circumference of the circle is equal to the value of the perimeter of the square, both square and circumference are the measurement of the outer boundary of the figure. Let us see the perimeter of the square which can be defined by the formula as:
\[\Rightarrow 4s\]
Whereas the perimeter of the circle is given as:
\[\Rightarrow 2\pi r\]
Now equating the perimeter of the square and circle together, we get the value of the sides of the square as:
\[\Rightarrow 4s=2\pi r\]
\[\Rightarrow s=\dfrac{2\pi r}{4}\]
\[\Rightarrow s=\dfrac{\pi r}{2}\]
Now as we have gotten the value of the sides of the square in terms of radius, we now equate the area of the square with the area of the circle and write the equation as:
Area of square \[=\] Area of circle
Placing the values in the above equation, we get the area of the circle to the area of the square as:
\[\Rightarrow {{\left( \dfrac{\pi r}{2} \right)}^{2}}=\pi {{r}^{2}}\]
\[\Rightarrow \dfrac{\pi {{r}^{2}}}{4}=\pi {{r}^{2}}\]
\[\Rightarrow \pi {{r}^{2}}\times \dfrac{4}{\pi {{r}^{2}}}\]
\[\Rightarrow \dfrac{4}{1}\]
Therefore, the ratio of the area of the circle and square as \[4:1\].
Note:
The circumference is same as the perimeter but for circular objects whereas both the perimeter and circumference is same as both of them are the measurement of the outer length of the object.
Area of square: \[{{s}^{2}}\]
Area of circle: \[\pi {{r}^{2}}\] (\[r\]: radius)
Complete step by step solution:
As given in the question, the circumference of the circle is equal to the value of the perimeter of the square, both square and circumference are the measurement of the outer boundary of the figure. Let us see the perimeter of the square which can be defined by the formula as:
\[\Rightarrow 4s\]
Whereas the perimeter of the circle is given as:
\[\Rightarrow 2\pi r\]
Now equating the perimeter of the square and circle together, we get the value of the sides of the square as:
\[\Rightarrow 4s=2\pi r\]
\[\Rightarrow s=\dfrac{2\pi r}{4}\]
\[\Rightarrow s=\dfrac{\pi r}{2}\]
Now as we have gotten the value of the sides of the square in terms of radius, we now equate the area of the square with the area of the circle and write the equation as:
Area of square \[=\] Area of circle
Placing the values in the above equation, we get the area of the circle to the area of the square as:
\[\Rightarrow {{\left( \dfrac{\pi r}{2} \right)}^{2}}=\pi {{r}^{2}}\]
\[\Rightarrow \dfrac{\pi {{r}^{2}}}{4}=\pi {{r}^{2}}\]
\[\Rightarrow \pi {{r}^{2}}\times \dfrac{4}{\pi {{r}^{2}}}\]
\[\Rightarrow \dfrac{4}{1}\]
Therefore, the ratio of the area of the circle and square as \[4:1\].
Note:
The circumference is same as the perimeter but for circular objects whereas both the perimeter and circumference is same as both of them are the measurement of the outer length of the object.
Recently Updated Pages
Fill in the blanks with suitable prepositions Break class 10 english CBSE
Fill in the blanks with suitable articles Tribune is class 10 english CBSE
Rearrange the following words and phrases to form a class 10 english CBSE
Select the opposite of the given word Permit aGive class 10 english CBSE
Fill in the blank with the most appropriate option class 10 english CBSE
Some places have oneline notices Which option is a class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
When was Karauli Praja Mandal established 11934 21936 class 10 social science CBSE
Which are the Top 10 Largest Countries of the World?
What is the definite integral of zero a constant b class 12 maths CBSE
Why is steel more elastic than rubber class 11 physics CBSE
Distinguish between the following Ferrous and nonferrous class 9 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE