वच्मि - Area and Perimeter of regular polygons

Vachmi

Rectangle

A rectangle is a quadrilateral with four right angles. The opposite sides of a rectangle are equal. A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular. A rectangle is cyclic i.e. all corners of a rectangle lie on a single circle.

Area of a rectangle = $length × breadth$

Area of a rectangle ABCD = $l × b$

Perimeter of a rectangle = $2 (length + breadth)$

Perimeter of a rectangle ABCD = $2 (l + b)$

Square

A square is a special case of a rectangle. A rectangle whose all the sides are equal is a square. A very interesting fact to be noted with respect to square is that a square has a larger area than any other quadrilateral with the same perimeter.

Area of a square = $side^2$

Area of a square ABCD = $a^2$

Perimeter of a square = $4 × side$

Perimeter of a square ABCD = $4 × a$

Circle

A circle is a plane figure whose boundary (circumference) consists of points equidistant from a fixed point (called its centre). The circle has the largest area of any two-dimensional object having the same perimeter.
A circle can circumscribe a rectangle, trapezium, triangle, square and kite.
A circle can be inscribed inside a square, triangle and kite.

Area of a circle = $π × radius^2$

Area of a circle with centre O and radius OA (r) = $π × r^2$

Perimeter (Circumference) of a circle = $2 × π × radius$

Perimeter (Circumference) of a circle with centre O and radius OA (r) = $2 × π × r$

Sector

A sector of a circle is an area enclosed by two radii and an arc. An arc is a part of the circumference of the circle. AB is an arc in the figure below and AOB is a sector.

Area of a Sector = $π × radius^2 × C/360$

Area of a Sector with centre O and radius OA (r) = $π × r^2 × C/360$
(when C is in degrees)

Perimeter of a sector = $(2 × radius) + $ (Length of arc AB)

Perimeter of a sector with radius r and arc AB = $(2 × r)$ + $2 π r ×$ Angle $\dfrac{(AOB)}{360}$
$= 2r + 2π r$ × $\dfrac{C}{180}$
(when C is in degrees)

Triangle

A triangle is a plane figure with three straight sides and three angles. Area of a triangle can be obtained if we know base and height of a triangle. The base and height are always at right angle to each other.
Alternatively, area can be obtained if we know all the three sides of a triangle.

Area of a triangle = $ 1/2 × height × base$

Area of a triangle ABC with height (AD) and base (BC) = $ 1/2 × AD × BC$
Area of a triangle ABC with height (h) and base (b) = $ 1/2 × b h $

Heron's formula - A greek mathematician, Heron, gave the formula of area of triangle as follows:

Let $a$, $b$ and $c$ be the sides of a △ ABC
Then, s = $\frac12 (a+b+c)$
Area of △ ABC = $\sqrt{s(s-a)(s-b)(s-c)}$

Perimeter of a triangle = sum of the lengths of all the three sides

Perimeter of a triangle ABC = $l(AB) + l(BC) + l(AC)$

Pentagon

A regular pentagon is a polygon with five equal sides. This also makes all the five angles in pentagon equal. If a pentagon is regular, then all the sides are equal in length, and five angles are of equal measures.
A pentagon can be considered as 5 equal triangles put together. Hence area of a pentagon is 5 times the area triangle.

Area of a pentagon = $ 5 × (1/2 × height × base)$

Area of a pentagon ABCDE = $ 5 × (1/2 × b × h)$

Area of a regular pentagon when only length of it's side is known = $ 5 × side^2 / (4 × tan 36 ° )$
= $ 5 × b^2 / (4 × tan 36 ° )$

Perimeter of a pentagon = sum of the lengths of all the five sides

Perimeter of a pentagon ABCDE = $l(AB) + l(BC) + l(CD) + l(DE) + l(AE)$