# 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 cm^{2}?

**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:

AP^{2}+ AC^{2}= PC^{2}AP^{2}= PC^{2}- AC^{2}AP^{2}= 13^{2}- 5^{2}AP^{2}= 169 - 25 AP^{2}= 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