Center
The fixed point in the circle is called the center.
Radius
Diameter
Circumference
Arc of a circle
Sector of a circle:
Semicircle
Properties related to Lines in a Circle
Chord
Properties of Chord
Tangent
Properties of Tangent
Important Properties of Circle – Related to Angles
Properties related to Angles in a circle
Inscribed Angle
Properties of Inscribed Angles
Central Angle
Property of Central Angles
The following are some mathematical formulae that will help you calculate the area and perimeter/circumference of a circle.
Perimeter:
Area:
Example of Application of the properties
Example 1: The lengths of two sides in a rightangle triangle other than hypotenuse are 6 cm and 8 cm. If this rightangle triangle is inscribed in a circle, then what is the area of the circle?
(a) 5 π
(b) 10 π
(c) 15 π
(d) 20 π
(e) 25 π
Correct answer is option (e)
Step 1: Given
 The lengths of two sides other than hypotenuse of a right triangle are 6 cm and 8 cm.
 This triangle is inscribed in a circle.
Step 2: To find
 Area of the circle.
Step 3: Approach and Working out
 Let us draw the diagrammatic representation.
By applying the property that the angle in a semicircle is 90º, we can say that AB is the diameter of the circle.
 And, once we find the length of the diameter, we can find the radius, and then we can find the area of the circle as well.
Applying Pythagoras theorem in △ ABC,
 AB² = AC² + BC²
 AB² = 6² + 8² = 36 +64 = 100
 AB = 10 cm
Since AB is the diameter, AB = 2R = 10
 Hence, R = 5 cm.
Area of the circle = π × R²= π × 5² = 25 π.
Hence, the correct answer is option E.
Example 2: In the diagram given below, O is the center of the circle. If OB = 5 cm and ∠ABC = 300 then what the length of the arc AC?
(a) 5π/6
(b) 5π/3
(c) 5π/2
(d) 5π
(e) 10π
Correct answer is option (b)
Step 1: Given
 OB = 5 cm
 ∠ABC = 30°
Step 2: To find
 Length of the arc
Step 3: Approach and Working out
 Length of the arc = (Central angle made by the arc/360°) × 2 × π × R.
To find the length of the arc, we need the value of two variable, the center angle made by the arc and the radius.
 We are already given radius as OB = 5cm
 We need to find the ∠AOC
On visualizing the diagram, the inscribed angle by the arc AC is ∠ABC, and the center angle by arc AC is ∠AOC.
 Hence, we can apply the property that the angle made at the center by an arc is twice the inscribed angle formed by the same arc.
 Thus, ∠AOC = 2 × ∠ABC = 2 × 30° = 60°
Now, we know the central angle formed by the arc as well.
 Hence, length of the arc AC =(Central angle made by the arc/360°) × 2 × π × R.
 =(60°/360°) × 2 × π × 5.
 =(1/6) × 2 × π × 5.
 =(5π/3) cm
Thus, the correct answer is option B.
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