Overview: Polygons

Table of Contents
1. What is a polygon?
2. Basic terms
3. Types of polygons
4. Classification of polygons
5. Polygon formulas: perimeter and area
6. Sum of angles in a polygon
7. Solved examples of polygons
8. Applications and quick problem tips
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The polygon is a flat shape formed by a finite number of straight line segments joined end-to-end to enclose a region. Each segment is called a side and the meeting point of two sides is a vertex. Examples of polygons include the triangle (3 sides) and the quadrilateral (4 sides). Polygons lie in two dimensions and are classified by the number of sides, by side- and angle-properties, and by whether they are regular or irregular, convex or concave. This document explains the definition, types, classifications, basic formulas (area, perimeter), angle-sum results, and worked examples suitable for a Class 7 / quantitative aptitude level.

What is a polygon?

A polygon is a two-dimensional closed figure made up of straight line segments. Each segment is a side or edge. The endpoints of sides are the vertices (corners) of the polygon. Polygons are named by the number of their sides: a 3-sided polygon is a triangle, a 4-sided polygon is a quadrilateral, a 5-sided polygon is a pentagon, and so on. Curved figures such as circles and ellipses are not polygons because they have no straight sides or vertices.

Basic terms

• Side (edge): a straight line segment forming part of the boundary.
• Vertex (corner): the point where two sides meet.
• Interior angle: the angle formed inside the polygon at a vertex between two adjacent sides.
• Exterior angle: the angle formed by extending one side at a vertex and measuring the angle between the extension and the adjacent side.
• Diagonal: a line segment joining two non-adjacent vertices.

Types of polygons

Polygons are described by their number of sides and by special properties of sides and angles. Below we describe commonly used types and give definitions and key properties.

Triangles (3-sided polygons)

The sum of the three interior angles of any triangle is 180°. Triangles are classified by sides and by angles:

Equilateral Triangle

A triangle with all three sides equal is an equilateral triangle. In an equilateral triangle each interior angle equals 60

Isosceles triangle

An isosceles triangle has two equal sides. The angles opposite the equal sides are equal. For example, in triangle ABC, if AB = AC then ∠B = ∠C.

Scalene Triangle

A scalene triangle has all three sides of different lengths and all three interior angles different.

Right-angled triangle

A triangle with one interior angle equal to 90° is a right-angled triangle. The side opposite the right angle is the hypotenuse.

• Triangles by angles can also be acute (all angles < 90°) or obtuse (one angle > 90°).
• Further triangle properties and area formulas are treated in separate triangle lessons; here we note the basic classifications used in polygon study.

Quadrilaterals (4-sided polygons)

Any four non-collinear points joined in order form a quadrilateral. Special quadrilaterals include parallelograms, rectangles, squares, rhombuses and trapeziums. Key properties are listed below.

Parallelogram

A parallelogram has two pairs of opposite sides that are parallel and equal in length. Diagonals bisect each other. Adjacent interior angles are supplementary (sum to 180°).

Rectangle

A rectangle is a parallelogram with all interior angles equal to 90°. Opposite sides are equal and diagonals are equal and bisect each other.

Square

A square has four equal sides and four right angles. It is both a rectangle and a rhombus. Diagonals equal each other, bisect each other at right angles, and bisect the interior angles.

Rhombus

A rhombus has four equal sides. Opposite sides are parallel and opposite angles are equal. Diagonals bisect each other at right angles and they bisect the interior angles.

Trapezium (trapezoid)

A trapezium has at least one pair of parallel sides (in some regions the name trapezoid is used). Other sides, angles and diagonals have no fixed equalities unless further conditions are given.

Other polygons by number of sides

• Pentagon: 5 sides; each interior angle in a regular pentagon is 108°.
• Hexagon: 6 sides; each interior angle in a regular hexagon is 120°.
• Heptagon: 7 sides.
• Octagon: 8 sides.
• Nonagon: 9 sides.
• Decagon: 10 sides.
• Hendecagon: 11 sides.
• Dodecagon: 12 sides.

Classification of polygons

Regular and irregular

• Regular polygon: all sides and all interior angles are equal; examples include the equilateral triangle and the square.
• Irregular polygon: sides or angles (or both) are not all equal; examples include scalene triangles and rectangles (equal angles but unequal adjacent sides).

Convex and concave

• Convex polygon: every interior angle is less than 180° and no line segment between two points of the polygon goes outside it.
• Concave polygon: at least one interior angle is greater than 180° and some diagonals lie outside the polygon.

Simple and complex (self-intersecting)

• Simple polygon: boundary does not cross itself; it has a single non-self-intersecting closed boundary.
• Complex polygon: sides intersect each other (also called self-intersecting polygons).

Polygon formulas: perimeter and area

Perimeter is the total length around the polygon. Area is the measure of the region enclosed by the polygon. Common formulas are:

• Perimeter of a triangle: \( \text{Perimeter} = a + b + c \) where a, b, c are the side lengths.
• Perimeter of a square: \( 4a \) where a is the side length.
• Perimeter of a rectangle: \( 2(L + B) \) where L is length and B is breadth.
• Perimeter of a parallelogram: \( 2(\text{adjacent side}_1 + \text{adjacent side}_2) \).
• Perimeter of a rhombus: \( 4s \) where s is the side length.

Area formulas for commonly used polygons:

• Area of a triangle: \( \dfrac{1}{2}\times \text{base}\times \text{height} \).
• Area of a rectangle: \( \text{Length}\times \text{Breadth} \).
• Area of a square: \( a^{2} \) where a is the side length.
• Area of a parallelogram: \( \text{base}\times \text{height} \).
• Area of a rhombus: \( \dfrac{1}{2}\times d_{1}\times d_{2} \), where \(d_{1}\) and \(d_{2}\) are the diagonals.
• Area of a regular polygon: \( \dfrac{1}{2}\times \text{perimeter}\times \text{apothem} \). The apothem is the perpendicular distance from the centre to a side.

Sum of angles in a polygon

Interior and exterior angles are fundamental to polygon geometry. The following results hold for any simple n-sided polygon (n ≥ 3).

Sum of interior angles

The sum of the interior angles of an n-sided polygon is:
\( (n-2)\times 180^\circ \)

For a regular n-sided polygon each interior angle equals the total divided by n:
\( \dfrac{(n-2)\times 180^\circ}{n} \)

Sum of exterior angles

• The sum of one exterior angle at each vertex (taken consistently, all clockwise or all anticlockwise) for any simple polygon equals \(360^\circ\).
• For a regular n-sided polygon each exterior angle equals \( \dfrac{360^\circ}{n} \).
• For any polygon at a vertex, interior angle + corresponding exterior angle = \(180^\circ\) (for the exterior angle formed by extending one side).

Solved examples of polygons

Example 1: A quadrilateral with four sides the sum of all the interior angles is?

Example 2: The exterior and interior angle ratio of a regular polygon is 2:3. Determine the polygon.
Solution: Let the exterior angle be \(2x\) and the interior angle be \(3x\).
Exterior + interior = \(180^\circ\).
\(2x + 3x = 180^\circ\)
\(5x = 180^\circ\)
\(x = 36^\circ\)
Exterior angle \(= 2x = 72^\circ\).
Number of sides \(= \dfrac{360^\circ}{\text{exterior angle}} = \dfrac{360^\circ}{72^\circ} = 5\).
The polygon is a pentagon.

Example 3: Obtain the number of sides of a polygon whose sum of interior angles is given by 540 degrees.
Solution: \( \text{Sum of interior angles} = 180^\circ(n-2) \)
Given \(180^\circ(n-2)=540^\circ\).
\(n-2 = \dfrac{540^\circ}{180^\circ} = 3\)
\(n = 3 + 2 = 5\)
The polygon has 5 sides (a pentagon).

Example 4: Each exterior angle of a polygon measured to 60 degrees determines the polygon?
Solution: Each exterior angle \(= \dfrac{360^\circ}{n}\).
Given \(60^\circ = \dfrac{360^\circ}{n}\).
\(n = \dfrac{360^\circ}{60^\circ} = 6\).
The polygon is a hexagon.

Applications and quick problem tips

• Use the interior angle sum formula \( (n-2)\times 180^\circ \) whenever interior angles are involved in a polygon problem.
• To find the number of sides from an exterior angle use \( n = \dfrac{360^\circ}{\text{exterior angle}} \).
• Perimeter problems require summing side lengths or using \(4a\) for squares and \(2(L+B)\) for rectangles.
• For regular polygons, the relation between interior and exterior angles simplifies many problems; each exterior angle is \(360^\circ/n\) and each interior angle is \(180^\circ - 360^\circ/n\).
• When area is required for regular polygons and you know the apothem, use \( \text{Area} = \dfrac{1}{2}\times \text{perimeter}\times \text{apothem} \).

Summary: A polygon is a closed planar figure made of straight sides. Learn the standard angle-sum formulas, area formulas for basic shapes (triangle, rectangle, square, rhombus), and the perimeter formulas. Classifying polygons as regular/irregular, convex/concave, and simple/complex helps select the right formulas when solving problems.