# In the below figure, $ A B C D $ is a square of side $ 2 a $. Find the ratio between the circumferences."

Given:

\( A B C D \) is a square of side \( 2 a \).

To do:

We have to find the ratio between the circumferences.

Solution:

The square $ABCD$ is inscribed a circle.

Length of the side of the square $= 2a$

From the figure,

Diameter of the outer circle $AC =$ Diagonal of the square

$=\sqrt{2} \times 2 a$

$=2 \sqrt{2} a$

This implies,

Radius of the outer circle $R=\frac{\mathrm{AC}}{2}$

$=\frac{2 \sqrt{2} a}{2}$

$=\sqrt{2} a$

Diameter of the inner circle $=2a$

Radius of the inner circle $r=\frac{2a}{2}=a$

Therefore,

The ratio between the circumferences of the circles $=\frac{\text { circumference of outer circle }}{\text { circumference of inner circle }}$

$=\frac{2 \pi \mathrm{R}}{2 \pi r}$

$=\frac{\mathrm{R}}{r}$

$=\frac{\sqrt{2} a}{a}$

$=\frac{\sqrt{2}}{1}$

**The ratio between the circumferences is $\sqrt{2}:1$.**

- Related Articles
- In the below figure, \( A B C D \) is a square of side \( 2 a \). Find the ratio between the areas of the incircle and the circum-circle of the square."\n
- In the figure below, \( A B C D \) is a square with side \( 2 \sqrt{2} \mathrm{~cm} \) and inscribed in a circle. Find the area of the shaded region. (Use \( \pi=3.14) \)"\n
- In the figure given below .Is \( A B+B C+C D+D Aanswer."\n
- In figure below, \( P Q R S \) is a square of side \( 4 \mathrm{~cm} \). Find the area of the shaded square."\n
- In figure below, D is the mid-point of side BC and $AE \perp BC$. If \( B C=a, A C=b, A B=C, E D=x, A D=p \) and \( A E=h, \) prove that \( b^{2}+c^{2}=2 p^{2}+\frac{a^{2}}{2} \)."\n
- In figure below, D is the mid-point of side BC and $AE \perp BC$. If \( B C=a, A C=b, A B=C, E D=x, A D=p \) and \( A E=h, \) prove that \( b^{2}=p^{2}+a x+\frac{a^{2}}{4} \)."\n
- In figure below, D is the mid-point of side BC and $AE \perp BC$. If \( B C=a, A C=b, A B=C, E D=x, A D=p \) and \( A E=h, \) prove that \( c^{2}=p^{2}-a x+\frac{a^{2}}{4} \)."\n
- Find Angle A in the below figure."\n
- In the figure, $X$ is a point in the interior of square $A B C D$. $AXYZ$ is also a square. If $D Y=3\ cm$ and $AZ=2 \ cm$. Then $BY=?$."\n
- In the below figure, \( O A B C \) is a square of side \( 7 \mathrm{~cm} \). If \( O A P C \) is a quadrant of a circle with centre O, then find the area of the shaded region. (Use \( \pi=22 / 7 \) )"\n
- In the below figure, the square \( A B C D \) is divided into five equal parts, all having same area. The central part is circular and the lines \( A E, G C, B F \) and \( H D \) lie along the diagonals \( A C \) and \( B D \) of the square. If \( A B=22 \mathrm{~cm} \), find the circumference of the central part."\n
- In the below figure, a square \( O A B C \) is inscribed in a quadrant \( O P B Q \) of a circle. If \( O A=21 \mathrm{~cm} \), find the area of the shaded region."\n
- In the below figure, \( A B C D \) is a rectangle with \( A B=14 \mathrm{~cm} \) and \( B C=7 \mathrm{~cm} \). Taking \( D C, B C \) and \( A D \) as diameters, three semi-circles are drawn as shown in the figure. Find the area of the shaded region."\n
- In the below figure, \( O A C B \) is a quadrant of a circle with centre \( O \) and radius \( 3.5 \mathrm{~cm} \). If \( O D=2 \mathrm{~cm} \), find the area of the quadrant \( O A C B \)."\n
- In the below figure, \( A B C D \) is a trapezium of area \( 24.5 \mathrm{~cm}^{2} . \) In it, \( A D \| B C, \angle D A B=90^{\circ} \), \( A D=10 \mathrm{~cm} \) and \( B C=4 \mathrm{~cm} \). If \( A B E \) is a quadrant of a circle, find the area of the shaded region. (Take \( \pi=22 / 7) \)."\n

##### Kickstart Your Career

Get certified by completing the course

Get Started