A circle is a closed curve where all points on the boundary (called the circumference) are at the same distance from a fixed point inside it.

• The fixed point inside the circle is called the center (O)

The radius is a straight line segment that connects the center of the circle to any point on its circumference.

Characteristics:

• Symbol: Usually represented as r

• All radii of a circle have the same length

• A circle has infinite radii (one to every point on the circumference)

• The radius is always half the diameter

• Radius = `"Diameter"/"2"`

 The diameter is a straight line segment that passes through the center of the circle and has both endpoints on the circumference.

Characteristics:

• The diameter passes through the center

• A circle has infinite diameters

• The diameter is the longest possible chord of a circle

• The diameter is twice the radius

• Diameter = 2 × Radius and

A chord is a straight line segment that connects any two points on the circumference of the circle.

Characteristics:

• A circle has infinite chords

• The diameter is the longest chord in any circle

• Chords closer to the centre are longer than chords farther from the center

Polygon: Polygon refers to a closed 2D shape which is made up of a finite number of line segments, but the perimeter is a one-dimensional measurement.

Perimeter of a rectangle = 2 × length + 2 × breadth

P = 2(1 + b) ⇒ (i) l = `P/2` − b, i.e., length = `"Perimeter"/2` − breadth

                         (ii) l = `P/2` − l, i.e., breadth = `"Perimeter"/2` − length

Perimeter of Square = Total boundary of the square

                                   = Side + Side + Side + Side

P = 4 × Side

Or: P = 4s (where 's' represents the side length)

side = ` "perimeter"/"4"`

Always include the correct linear unit (cm, m, mm, km, etc.)

Perimeter of a Triangle = 3 × length of a side.

The perimeter of a regular polygon = (length of one side) × number of sides.

The perimeter of an Irregular polygon = Sum of all sides of Irregular polygons.

We have to prove that

AQ = `1/2` (perimeter of ΔABC)

Perimeter of ΔABC = AB + BC + CA

= AB + BP + PC + CA

= AB + BQ + CR + CA

(∵ Length of tangents from an external point to a circle are equal ∴ BP = BQ and PC = CR)

= AQ + AR  ...(∵ AB + BQ = AQ and CR + CA = AR)

= AQ + AQ  ...(∵ Length of tangents from an external point are equal)

= 2AQ

⇒ AQ = `1/2` (Perimeter of ΔABC)

Hence proved.