10th Class Mathematics Areas Related to Circles

 Areas Related to Circles

• A circle is a closed curve in a plane drawn in such a way that every point on this curve is at a constant distance (r units) from a fixed point O inside it.

The fixed point O is called the centre of the circle and the constant distance r is called the length of radius of the circle.

• Circumference of a circle:

The perimeter of a circle is called its circumference. The length of the thread that winds tightly around the circle exactly once gives the circumference of the circle.

Circumference\[=2\pi r=\pi d\], where r = radius and d = diameter.

Here \[\pi \] (Pi) is a constant.

Note: The approximate value of n is taken as\[\frac{22}{7}\]or 3.14. However n is not a rational number. It is an irrational number and is defined as the ratio of circumference of a circle to its diameter.

• Area of a circle:

Area of a circle with radius r units is equal to\[\pi {{r}^{2}}sq\]units.

• Area of a ring:

The region enclosed between two concentric circles of different radii is called a ring.

Area of path formed\[=(\pi {{R}^{2}}-\pi {{r}^{2}})sq.\] units

      \[=\pi ({{R}^{2}}-{{r}^{2}})sq.units\]

      \[=\pi (R+r)(R-r)sq.units\]

• Length of an arc of a circle:

Let A and B be any two points on a circle. The length of the thread that will wrap along this arc from A to B is the length of AB written as\[\overset\frown{AB}\]?

In a circle of radius r, we have

\[\frac{l(\overset\frown{AB})}{Circumference}=\frac{{{x}^{o}}}{{{360}^{o}}}or\,l(\overset\frown{AB})=\frac{2\pi r{{x}^{o}}}{{{360}^{o}}}\]

• Area of a sector:

A sector of a circle is the region enclosed by an arc of a circle and two radii to its end points. Area of sector \[=\frac{{{x}^{o}}}{{{360}^{o}}}\times \pi {{r}^{2}}\]where x is sector angle and r is radius of circle.

• Segment of a circle:

A segment of a circle is the region enclosed by an arc of the circle and its chord.

Area of minor setment \[\text{AXB}=\]area of sector OAXB - area of \[\Delta \text{OAB}\]

Area of major segment AYB = area of circle - area of minor segment AXB