Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector

A chord $AB$ of a circle of radius $10 \mathrm{cm}$ makes a right angle at the centre of the circle. Find the difference between the areas of the major and minor segments of the circle. (use $\mathrm{\pi } = 3.14$)

A hand fan is made up of cloth fixed in between the metallic wires. It is in the shape of a sector of a circle of radius $21 \mathrm{cm}$ and an angle of $120°$ as shown in the figure. Calculate the area of the cloth used and also find the total length of metallic wire required to make such a fan.

A circle is divided into $5$ sectors with area equal to $1:2:3:4:5$. Find the angle subtended at the center of the circle by the largest sector.

If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the segment, the four points are con-cyclic (i.e., or the same circle)

The diagram shows a triangle $ABC$ inside a semicircle.

$A, B$, and $C$ are the points on the semicircle. $AB$ is the diameter of the semicircle. Angle $ACB=90°$, Angle $BAC=50°$, $\mathrm{AC}=18 \mathrm{cm}$.

Work out the perimeter of the semicircle. Give your answer correct to $2$ significant figures.

A hand fan is made up of cloth fixed in between the metallic wires. It is in shape of a sector of a circle of radius$21 cm$and of angle $120°$ as shown in figure. Calculate the area of the cloth used and also find the total length of metallic wire required to make such a fan.

The angle of a sector with angle $45°$ and radius$=14 \mathrm{cm}$ is _____ ${\mathrm{cm}}^2$.

Find the area of the blue region?

Find the area of the shaded region in the given figure.

What is the arc length in circle.

In the figure, $OAPB$ is a quadrant of a circle and $OAP$ is a sector. If $\angle AOP= 30°$ and the area of shaded region is $462 {\mathrm{cm}}^2$, then find the length of the arc $PB$

Find the area of the shaded portion.

In the given figure, $\mathrm{O}$ is the centre of the circle. If $\angle ADC=118°$, then the value of $x$ is :

In the given figure, $\mathrm{O}$ is the centre of the circle and $BA=AC$. If $\angle ABC=50°$, find $\angle BOC$ and $\angle BDC$.

When the circumference of a toy balloon, is increased from $20 \mathrm{cm}$ to $25 \mathrm{cm}$, its radius ( in $\mathrm{cm}$) is increased by:

Find the measure of the central angle in degrees. (Enter the value to the nearest integer).

In the figure, if $O$ is the centre of the circle, $\angle AOB$ is:

In figure, $\mathrm{PQ}$ is the diameter of the circle with centre $\mathrm{O}$. If $\angle \mathrm{PQR}=65°, \angle \mathrm{RPS}=40°$ & $\angle \mathrm{PQM}=50°$, find $\angle \mathrm{QPR}, \angle \mathrm{PRS} \& \angle \mathrm{QPM}$.

Sneha had a rectangular tablecloth with one side measuring 30 cm which she wanted to keep on her circular table of radius 25 cm. After keeping it on the table, she realised that the corners of the tablecloth just touched the edge of the circular table as shown in the figure.

Find the area of the table not covered by the tablecloth. 

(Note: Use $\mathrm{\pi }$ = 3.14.)

Shown below are two paths from Madrid to New York. The dotted line is the shortest distance, approximately $600 \mathrm{km}$, on a $2\mathrm{D}$ map. The other solid circular arc is the shortest distance on a $3\mathrm{D}$ globe and the actual path taken by a flight. The arc subtends an angle of $60°$ at the centre of the circular arc.

What is the actual distance covered by the flight? (Take $\mathrm{\pi }=3.14$, if needed.)