Areas and Properties of circle
Let us first discuss some formulas on areas in a circle.
1. Area of a circle:
In the above circle with centre O, the line segment OA is radius.
If the length of the radius is r, then the formula for finding area of the circle is Πr2
Note:
If‘d’ stands for length of the diameter of a circle, then area of the circle in terms of d is Πd2/4 {d = 2r}
2. Circumference of a circle
The circumference of a circle is the distance required to traverse one full round on the circle.
Circumference of a circle is also called perimeter of the circle.
The formula for finding circumference of a circle is 2Πr
Note:
1. The circumference of a circle is the length of a string shaped out into the circle
2. The Perimeter of a circle is same as its circumference. Therefore,
Perimeter of a circle = 2Πr
3. Area, Circumference and Perimeter of a semi circle.
A semi circle covers half the amount of the area in a plane as covered by a complete circle.
In the figure above, a semi circle is shown with centre O and of radius r.
The amount of region covered by a semi circle in a plane is half as much as the region covered by a full circle of a same radius r.
Therefore,
1. Area of a semi circle is ½Πr2
Circumference of a semi circle of radius ‘r’ is also half the circumference as that of a complete circle with the same radius ‘r’.
2. Therefore, circumference of a semi circle is Πr
Note:
Perimeter of a semi circle is the sum of the lengths of its circumference and diameter. Therefore,
Perimeter of a semi circle is Πr + 2r
3. Length of an arc of a circle
Consider the circle above with centre O. Let the curved path AB, arc AB, be of length ‘l’. Also, let the angle between the two radii OA and OB be angle θ.
We say, arc AB subtends angle θ at centre O of the circle.
Let angle θ be measured in degrees.
Then, the formula for finding length of the arc AB is
4. Area of sector OAB
In the circle above, OAB is a sector of the circle. The angle arc AB subtends at centre O is θ.
Now, the area of the sector OAB is
Inscribe a triangle in a circle
If the three vertices of a triangle drawn inside a circle lie on the the circle, then the triangle is said to be inscribed in the circle.
Consider triangle ABC inscribed in the circle with centre O as shown in the figure above.
One of the sides of triangle ABC, BC lies on the diameter of the circle, i.e. BC is both a side of the triangle and also diameter of the circle.
Then angle at vertex A is a right angle.
Now, sector BAC is a semi circle in which triangle ABC is inscribed with one of the sides of the triangle lying on the diameter of the circle.
So, triangle ABC is right triangle, with the right angle formed at vertex A. Therefore, the two sides AB and AC are the legs of the right triangle, while the side BC, the diameter is hypotenuse of the right triangle.
6. Concentric circles
Two or more circles that have a common centre are called concentric circles.
In the figure above, two circles are shown.
The outer circle with radius R is bigger than the inner circle of radius r.
The two circles have a common centre O.
Two or more circles having a common centre are termed concentric circles.
7. Tangents drawn from a same point to a same circle are equal in length
In the following figure, two straight lines are drawn from an exterior point P to touch the circle O at points Q and R respectively.
The two straight lines PQ and PR are tangents to the circle O.
Each of the two tangents touches the circle exactly at one point.
Q and R are the points of contact of the two tangents with the circle O.
The two tangents PQ and PR are equal in length.
Also, as noted earlier, the angles at the points of contact Q and R of the two tangents PQ and PR with the radii OQ and OR respectively are 900 each.
