GRE Geometry | Circles

A Circle is a 2-dimensional figure. It is a closed figure in which boundary is equidistant from the center and distance from the center to the boundary is called radius and it remains the same throughout the figure. 

C is the center of circle. Perimeter of a circle is called circumference. 

Important properties of a circle: 

• Area of circle = π * r * r
• Perimeter of circle = 2 * π * r 
  or π * diameter
• diameter = 2 * radius

Semicircle:  

Important properties of a circle:  

• Area of semicircle = π * r * r / 2
• Perimeter of semicircle = * π * r 
  or π * diameter / 2

π is the ratio of circumference to the diameter of circle which is same for all circle or,  

= circumference / diameter 
= π or circumference / (2 * radius) 
= π 

π is a non repeating non terminating number which is equal to 22 / 7 or approximately 3.14. 

Chord: 
It is line segment which connects two points on the circle. Diameter is a special type of chord that passes through center of circle. 

AB and BD are chord of given circle. 

Arc: 
Consider any two point on a circumference of a circle, length between these two points including all points between them is called arc. Generally, an arc contains 3 points to avoid ambiguity. 

ADE is an arc of given circle. D is the point in between A and E to avoid ambiguity. 

How to find length of an arc? 
Length of the arc = (angle subtended by an arc at center / 360°)* circumference 

What is sector? 
Sector is the area covered by an arc. 
Area of sector = radius * angle subtends by sector. 
∠ ACED = 120° 
Area of sector = radius * angle subtends by sector. 

Tangent and point of tangency: 
A tangent is a line segment which lies on the same plane as a circle and touches the circle at only one point this point is called point of tangency. 

AP and BP are tangent of the circle, where A and B are the point of tangency. Radius is drawn to the point of tangency is perpendicular to the tangent. 

Tangent drawn from the same point are equal in length. 
or AP = BP. 

What is inscribed figure and circumscribed figure? 
Inscribed figure: When a circle covers all points of a polygon then the polygon is inscribed in a circle. 

Circumscribed figure: When a circle is inside a polygon, its perimeter touches all sides of polygon and every side of polygon is tangent to the circle then the figure is called circumscribed. 

What are concentric circles? 
Two or more circle with same of different radius are concentric circles. 

Example-1: 
What is the radius of circle having area 154 cm²? 

Solution:

Area of circle = π * radius * radius 
154 = 22 / 7 * radius * radius (take π = 22 / 7)
49 = radius * radius 
radius = 7 cm 

Example-2: 

If ∠ ADB = 65° then, what is the sum of ∠ ADB, ∠ ACB and ∠ AEB? 

Solution: 
Angle formed by two points in the same arc are equal. 

∠ ADB = ∠ ACB = ∠ AEB = 65°
∠ ADB + ∠ ACB + ∠ AEB = 195°

Example-3: 
What will be the perimeter of quadrilateral APBC if radius of circle is 5 cm and length of CP is equal to 13? 

Solution: 
Angle made by tangent at point of tangency along with radius is always 90° 
So, triangle APC is a right angle triangle. 
Apply Pythagoras theorem in triangle APC: 

AP2 + AC2 = PC2
AP2 = PC2 - AC2
AP2 = 132 - 52
AP2 = 169 - 25
AP2 = 144
AP = 12 

Since tangents from the same external point are equal in length.  

AP = BP = 12 cm

Perimeter of quadrilateral: 
= 12 + 12 + 5 + 5 
= 34 cm