Contents

# Side|Definition & Meaning

## Definition

In **two-dimensional** flat shapes, a **side** is one of the various one-dimensional **line** segments bounding the shape. The **sum** of the length of all sides of a flat shape defines its **perimeter**. In **3D** shapes, a side of the object is usually called a **face** instead, and each side of a 3D object is a flat **2D** shape instead of a **1D** line segment.

**Figure 1** shows the three **sides**, AB, BC, and CA, of an equilateral **triangle** ABC.

## Polygon

A polygon is a closed **two-dimensional** shape on a plane or flat surface made with **straight** lines. These straight lines are the **sides** of the polygon, also known as **edges**. A polygon has no **curved** side. The corner points where two edges or sides meet are known as the **vertices** of a polygon.

## Regular and Irregular Polygons

**Regular** polygons have all **sides** of equal length and **equal** internal angles. For example, an **equilateral** triangle is a regular polygon, as all its sides are equal, and all the internal **angles** measure** 60°**.

An **irregular** polygon either has sides of **unequal** length or unequal interior angles, or both. For example, a **scalene** triangle is an irregular polygon as it has three unequal sides.

## Perimeter

A polygon’s perimeter is the **total** length of all the **sides** of a polygon. The perimeter’s **unit** is the same as the unit of the side’s length of the **polygon**.

## Area

The area of a polygon can be found if the **perimeter** **P** and the **apothem** **A** of the polygon are known. The **apothem** is the length of the line segment from a **midpoint** of any side to the polygon’s **center**. Hence, the general formula for the **area** of a polygon is:

Area of a Polygon = (1/2)PA

## Polygons With Different Numbers of Sides

The different **types** of polygons, depending on their number of **sides** discussed as follows:

### Triangle

A triangle has **three sides** with three internal angles. The sum of the three internal **angles** of a triangle equals **180°**. The three types of triangles, depending on the **length** of their **sides**, are equilateral, isosceles, and scalene triangles. The **right-angle** triangle is categorized by an interior angle.

#### Equilateral Triangle

The equilateral triangle has all three sides of **equal** length. It also has all the internal angles equal to **60°**; hence it is a **regular** polygon.

If **g** is the **length** of an equilateral triangle’s **side**, the **area** of the triangle can be found by using the formula:

\[ \text{Area} = \frac{ \sqrt{3} }{4} {\text{g}}^2 \]

The **perimeter** of an equilateral triangle is:

Perimeter = 3g

#### Isosceles Triangle

A triangle with **two** sides **equal** and **one** side **unequal** is known as an isosceles triangle. If **c** is the length of one **side** of the two equal sides and **f** is the third side’s length, the **perimeter** of the isosceles triangle is given as:

Perimeter = (2c) + f

The unequal or third side is known as the base “**b**” of the isosceles triangle. The height “**h**” is the **vertical** line from the apex to the triangle’s base. An isosceles triangle’s **area** is given as:

Area = (1/2).b.h

#### Scalene Triangle

A scalene triangle is an **irregular** polygon having all **three** sides of **unequal** length. If **p**, **q**, and **r** are the length of three sides, the **perimeter** of a scalene triangle is given as:

Perimeter = p + q + r

A scalene triangle’s **area** is the same as that of the **isosceles** triangle.

#### Right-Angle Triangle

A **right-angle** triangle has a right(**90°**) angle. The three sides of the right-angle triangle are the hypotenuse, base, and perpendicular. The hypotenuse “**h**” is the **longest** side, the base “**b**” is the side **adjacent** to the hypotenuse, and the perpendicular “**p**” is the side **opposite** to the hypotenuse.

The **Pythagoras** theorem relates the length of three **sides** of the right-angle triangle as:

$h^2$ = $p^2$ + $b^2$

The **perimeter** of the right-angle triangle is:

Perimeter = h + p + b

If **θ** is an **acute** angle between the hypotenuse and the base, then the three **sides** are related to the **trigonometric** functions as:

sin θ = p/h

cos θ = b/h

tan θ = p/b

**Figure 2** shows the isosceles, scalene, and right-angle triangles.

### Quadrilaterals

Quadrilaterals are **four-sided** polygons. They can be **classified** into different types by varying the **side** lengths or the internal **angles** as given below:

#### Square

A square has four **equal** sides. Each internal angle measures **90°**, with the **sum** of all four internal angles equal to **360°**. If **e** is the length of a side of a square, its **perimeter** is given as:

Perimeter = 4e

The formula for a square’s** area** is:

Area = $e^2$

#### Rectangle

A rectangle is an **irregular** quadrilateral with two **opposite** sides of **equal** length and all internal angles equal to **90°**. If the length of the longer side is **L** and that of the shorter side is **W**, the **perimeter** of a rectangle is given by:

Perimeter = 2(L + W)

A rectangle’s **area** with length **L** and width **W** is given by:

Area = L.W

#### Rhombus

A rhombus has four **equal** sides, hence is also known as an **equilateral** quadrilateral. The two **opposite angles** of a rhombus are equal. If **a** is the length of a **side** of a rhombus, its perimeter is:

Perimeter = 4a

The **area** of a rhombus is **half** times the product of the lengths of its two **diagonals**.

#### Parallelogram

A parallelogram has **opposite** sides parallel to each other. The **parallel** sides are **equal** in length, and the opposite **angles** are also equal in measure. The square, rectangle, and rhombus can also be considered parallelograms.

A parallelogram’s **area** with base **B** and vertical height **H** is given as:

Area = B.H

The **perimeter** of the parallelogram is:

Perimeter = 2(r + s)

Where **r** is the **length** of the longer side and **s** is the shorter side’s length.

#### Trapezoid

A trapezoid has only **two** sides **parallel** to each other. These parallel sides are known as the **bases** of the trapezoid. If **u** and **v** are the length of the two bases and **h** is the vertical **height** between the two bases, the **area** of a trapezoid is:

Area = [(u + v).h]/2

A trapezoid’s **perimeter** is the sum of the length of its four sides.

#### Kite

A kite has two **adjacent** sides pairs of **equal** length. The two **angles** where the unequal sides meet are equal. A kite’s two **diagonals** intersect each other at a **90°** angle.

The **area** of a kite is half times the product of the length of its two **diagonals**. If **r** is the length of the shorter side and **s** is the length of the longer side, the **perimeter** of a kite is:

Perimeter = 2(r + s)

**Figure 3** shows different types of quadrilaterals.

### Pentagon

“**Penta**” means five, so a pentagon is a** five-sided** polygon with five angles. The sum of the internal angles of a pentagon is **540°**.

### Hexagon

A hexagon has **six sides** and six internal angles. Each angle in a regular hexagon measures **120°**, with a sum of **720°** of all the internal angles.

### Heptagon

A heptagon is a **seven-sided** polygon with the sum of the seven internal angles equal to **900°**. An internal angle measures **128.57°** in a regular heptagon.

### Octagon

An octagon has **eight** sides and eight internal angles. Each internal angle measures** 135°** in a regular octagon. The sum of the eight internal **angles** of an octagon is **1080°**.

### Nonagon

A nonagon is a **nine-sided** polygon with nine internal angles that sum to **1260°**. A regular nonagon has nine equal sides and angles, with each internal angle equal to **140°**.

### Decagon

“**Deca**” means ten, so a decagon has **ten sides** and ten vertices. A regular decagon has all the sides of equal length, with each interior angle equal to **144°**. The sum of the internal angles of a decagon equals **1440°**.

**Figure 4** shows all types of polygons(regular) with sides of equal length.

## Number of Diagonals of an m-sided Polygon

A **line** segment formed by joining two **non-adjacent** vertices in a polygon is known as a **diagonal**. Non-adjacent vertices are not on the same **side** of a polygon. The number of diagonals of an **m-sided** polygon can be found by using the formula:

Number of Diagonals = m(m – 3)/2

For example, the total number of **diagonals** that can be drawn in a **heptagon**(m=7) are:

Number of Diagonals in a Heptagon = 7(7-3)/2 = 14

**Figure 5** shows all the possible diagonals that can be drawn in a four, five, six, and seven-sided polygon.

## Polyhedrons

A polyhedron is a **three-dimensional** object with its **faces** as its sides. The edges of a polyhedron are the line segments joining two sides. For example, a **cube** is a regular polyhedron with a **square** as a polygonal face.

## An Example of Using Side Lengths to Find the Area and Perimeter

A **parallelogram** has a base length of **5 cm** and a height of **7 cm**. The length of the other side is **3 cm**. Calculate the **area** and **perimeter** of a parallelogram. Also, calculate the number of **diagonals** of a parallelogram.

### Solution

The **area** of a parallelogram is given by:

Area = B.H

Here,

B = 5 cm, H = 7 cm

Putting the **values** in the above equation gives:

Area = (5)(7) = **35 cm$^2$**

The **perimeter** of a parallelogram is given by:

Perimeter = 2(r + s)

As the **base** is a side of the parallelogram, so

r = 5 cm

And the length of the other **side** is given as follows:

s = 3 cm

Putting the **values** gives the perimeter of the parallelogram as:

Perimeter = 2(5 + 3) = 2(8) = **16 cm**

The number of **diagonals** of a parallelogram can be found by using the formula:

Number of Diagonals = m(m-3)/2

As a parallelogram is a quadrilateral(**four-sided**), so

m = 4

Putting the value of **m** gives:

Number of Diagonals = 4(4-3)/2 = 4/2 = **2**

*All the images are created using GeoGebra.*