krit.club logo

Understanding Quadrilaterals - Polygons and their Classification

Grade 8ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

A polygon is a simple closed curve made up of only line segments. Polygons are classified by the number of sides (or vertices) they have. For example, a polygon with 33 sides is a triangle, 44 sides is a quadrilateral, 55 sides is a pentagon, and so on.

Classification of polygons based on number of sides
•

A polygon is convex if no part of its diagonals lies in its exterior. In a concave polygon, at least one diagonal (or part of it) lies in the exterior of the polygon, and at least one interior angle is greater than 180∘180^\circ.

Comparison between convex and concave polygons
•

A regular polygon is both equiangular (all angles are equal) and equilateral (all sides are equal). An irregular polygon does not have all sides and angles equal.

Regular vs Irregular polygons
•

The diagonals of a polygon are line segments connecting two non-consecutive vertices. The number of diagonals increases with the number of sides.

Diagonals in a pentagon

📐Formulae

Sum of interior angles of a polygon = (n−2)×180∘(n - 2) \times 180^\circ

Each interior angle of a regular polygon = (n−2)×180∘n\frac{(n - 2) \times 180^\circ}{n}

Sum of exterior angles of any convex polygon = 360∘360^\circ

Each exterior angle of a regular polygon = 360∘n\frac{360^\circ}{n}

Number of diagonals in a polygon of nn sides = n(n−3)2\frac{n(n - 3)}{2}

Measure of an interior angle + Measure of its adjacent exterior angle = 180∘180^\circ

💡Examples

Problem 1:

Find the sum of the interior angles of a polygon with 1212 sides.

Solution:

Step 1: Identify the number of sides, n=12n = 12. Step 2: Use the formula for the sum of interior angles: S=(n−2)×180∘S = (n - 2) \times 180^\circ. Step 3: Substitute n=12n = 12 into the formula: S=(12−2)×180∘S = (12 - 2) \times 180^\circ. Step 4: Solve the expression: S=10×180∘=1800∘S = 10 \times 180^\circ = 1800^\circ.

Explanation:

To find the total sum of all angles inside any polygon, we subtract 22 from the total number of sides and multiply the result by 180180 degrees.

Problem 2:

Each interior angle of a regular polygon is 135∘135^\circ. Find the number of sides of the polygon.

Solution:

Step 1: Use the linear pair relationship to find the exterior angle. Exterior angle = 180∘−Interior angle180^\circ - \text{Interior angle}. Step 2: Calculate: Exterior angle = 180∘−135∘=45∘180^\circ - 135^\circ = 45^\circ. Step 3: Use the formula for the number of sides based on exterior angles: n=360∘Exterior anglen = \frac{360^\circ}{\text{Exterior angle}}. Step 4: Substitute the value: n=360∘45∘n = \frac{360^\circ}{45^\circ}. Step 5: Solve: n=8n = 8.

Explanation:

In a regular polygon, all interior angles are equal, which means all exterior angles are also equal. Since the sum of exterior angles is always 360∘360^\circ, dividing 360360 by the measure of one exterior angle gives the number of sides.

Problem 3:

Calculate the measure of each exterior angle of a regular hexagon.

Regular hexagon showing an exterior angle

Solution:

n=6n = 6 Sum of exterior angles=360∘\text{Sum of exterior angles} = 360^\circ Each exterior angle=360∘n\text{Each exterior angle} = \frac{360^\circ}{n} Each exterior angle=360∘6=60∘\text{Each exterior angle} = \frac{360^\circ}{6} = 60^\circ

Explanation:

Since a regular hexagon has 6 equal sides and 6 equal exterior angles, we divide the total sum of exterior angles (360∘360^\circ) by 6.

Problem 4:

Find the value of xx in the given quadrilateral where the interior angles are xx, 2x2x, 3x3x, and 4x4x.

Quadrilateral with angles expressed in terms of x

Solution:

Sum of interior angles of a quadrilateral=(4−2)×180∘=360∘\text{Sum of interior angles of a quadrilateral} = (4 - 2) \times 180^\circ = 360^\circ x+2x+3x+4x=360∘x + 2x + 3x + 4x = 360^\circ 10x=360∘10x = 360^\circ x=360∘10=36∘x = \frac{360^\circ}{10} = 36^\circ

Explanation:

The sum of interior angles of any quadrilateral is always 360∘360^\circ. We set up an equation with the given ratios and solve for xx.