# Concept: Quandrilaterals & Polygons

CONTENTS

Sum of all the angles of a quadrilateral is 360°

Sum of all the exterior angles of a quadrilateral is 360°

Area = ½ × (one diagonal) × (Sum of heights on this diagonal)

Area = ½ × d

_{1}d_{2}× 𝑆𝑖𝑛𝜃Sum of the 4 sides > Sum of the 2 diagonals

**General Properties of Parallelograms**

Opposite sides are parallel and equal

Opposite angles are equal

Area = base × height

Diagonals bisect each other

Adjacent angles are supplementary

Each diagonal divides the ||

^{gm}in two congruent triangles.AC

^{2}+ BD^{2}= AB^{2}+ BC^{2}+ CD^{2}+ DA^{2}Parallelogram with same base and between same parallel lines will have same area

All properties of a ||

^{gm}.All angles are equal to 90°.

Both diagonals are equal and bisect each other.

- AC = BD = $\sqrt{{l}^{2}+{b}^{2}}$
- OA = OB = OC = OD
If P is a point inside the rectangle, then

Area =

*l*×*b*Figure formed by joining the midpoints is a rhombus.

PA^{2} + PC^{2} = PB^{2} + PD^{2}

All properties of a rectangle.

Both diagonals are equal (a√2) and bisect each other at 90°.

Diagonals bisect the vertex angles.

Area = a

^{2}Figure formed by joining the midpoints is a square.

All properties of a ||

^{gm}.Diagonals are not necessarily equal but bisect each other at 90°

Diagonals bisect the vertex angles.

d

_{1}^{2}+ d_{2}^{2}= 4a^{2}Area = ½ × d

_{1}d_{2}Figure formed by joining the midpoints is a rectangle.

Kite is not a parallelogram.

2 pair of Adjacent sides are equal.

Here, AD = AB and CD = CB
Bigger diagonal bisects the smaller diagonal at 90°.

Area = ½ × d

_{1}d_{2}Opposite angles along the smaller diagonal are equal.

Here, ∠D = ∠B

One pair of opposite sides (bases) is parallel.

Area = ½ × ℎ𝑒𝑖𝑔ℎ𝑡 × (𝑠𝑢𝑚 𝑜𝑓 𝑏𝑎𝑠𝑒𝑠)

The length of a line dividing the oblique sides in the ratio a : b is given by

Diagonals divide the median in the ratio of length of parallel sides

$\frac{\mathrm{a}}{\mathrm{a\; +\; b}}$ × CD + $\frac{\mathrm{b}}{\mathrm{a\; +\; b}}$ × AB

**Isosceles Trapezium**

If oblique sides are equal, it is called an isosceles trapezium.

If a trapezium is inscribed inside a circle, it will always be an isosceles trapezium, i.e. its non−parallel sides will be equal

AD = BC and ∠D = ∠C

A quadrilateral whose all 4 vertices lie on the same circle is called a cyclic quadrilateral. The sum of any two opposite angles of a cyclic quadrilateral is 180°.

Area of a cyclic quadrilateral = $\sqrt{\mathrm{(s\; -\; a)(s\; -\; b)(s\; -\; c)(s\; -\; d)}}$

where s = semiperimeter = $\frac{\mathrm{a\; +\; b\; +\; c\; +\; d}}{2}$

A polygon is a closed figure bounded by three or more straight lines, known as sides. It has as many vertices as the number of sides, with no three of them collinear.

A line joining two non−adjacent vertices is known as the diagonal.

No. of diagonals in a ‘n’ sided polygon: $\frac{\mathrm{n(n\; -\; 3)}}{2}$

Sum of internal angles of a ′n′ sided polygon: (n - 2)π

Sum of exterior angles of a ′n′ sided polygon: 2π

A convex polygon has all angles < 180°.

A concave polygon has at least one angle > 180°.

Measure of each interior angle: $\frac{\mathrm{n(n\; -\; 2)\pi}}{2}$

Measure of each exterior angle: $\frac{\mathrm{2\pi}}{\mathrm{n}}$

Area of a regular ′n′ sided polygon: n × (area of smaller triangle)

Inradius of a n sided regular polygon = $\frac{a}{2}$cot$\left(\frac{\mathrm{\pi}}{n}\right)$

Circumradius of a n sided regular polygon: $\frac{a}{2}$cosec$\left(\frac{\mathrm{\pi}}{n}\right)$

i.e., n × $\frac{{a}^{2}}{4}$Cot$\left(\frac{\mathrm{\pi}}{n}\right)$

Height of a regular hexagon: a√3

Distance between opposite vertices: 2a

Area of a regular hexagon: 6 × $\frac{\sqrt{3}{a}^{2}}{4}$

- For a fixed perimeter, the area of a polygon with higher number of sides will always be more than the area of a polygon with lesser number of sides.
- For any fixed area, the perimeter of a regular polygon with lesser number of sides will always be more than that of a regular polygon with a greater number of sides.
- If the circumference of a circle is the same as the perimeter of a regular polygon, then the area of the circle will always be more than the area of the polygon.