1 Marks Question

Question 11 Mark

If the numerical value of the area of a circle is equal to the numerical value of its circumference, find its radius.

Answer

$\because$ Numerical value of area of circle = Numerical value of circumference
$\therefore$ $\pi\text{r}^2=2\pi\text{r}$
or $r = 2$ units

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Question 21 Mark

Write the formula for the area of a sector of angle $\theta$ (in degrees) of a circle of radius $r$.

Answer

Area of a sector of a circle whose radius $= r$
and angle at centre $=\theta,$ Will be $\pi\text{r}^2\times\frac{\theta}{360^\circ}$

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Question 31 Mark

If a square is inscribed in a circle, what is the ratio of the areas of the circle and the square?

Answer

We have the following situation

Let $BD$ be the diameter and diagonal of the circle and the square respectively.
We know that area of the circle $=\pi\text{r}^2$
Area of the square $=$ side$^2$
As we know that diagonal of the square is the diameter of the square.
Diagonal $= 2r$
Side of the square $=\frac{\text{diagonal}}{\sqrt{2}}\ \dots(1)$
Substituting Diagonal $= 2r$ in equation $(1)$ we get,
Side of the square $=\frac{2\text{r}}{\sqrt{2}}$
Now we will find the ratio of the areas of circle and square.
$\frac{\text{Area of circle}}{\text{Area of square}}=\frac{\pi\text{r}^2}{\Big(\frac{2\text{r}}{\sqrt{2}}\Big)^2}$
Now we will simplify the above equation as below,
$\frac{\text{Area of circle}}{\text{Area of square}}=\frac{\pi\text{r}^2}{\frac{4\text{r}^2}{2}}$
$\frac{\text{Area of circle}}{\text{Area of square}}=\pi\text{r}^2\times{\frac{2}{4\text{r}^2}}$
Hence, $\frac{\text{Area of circle}}{\text{Area of square}}=\frac{\pi}{2}$
Therefore, ratio of areas of circle and square is $\pi:2.$

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