This web page examines the nature of two-dimensional or ‘plane’ polygons. A polygon is any shape comprised of directly lines that can be drawn on a level surface, choose a item of paper. Together shapes incorporate squares, rectangles, triangles and also pentagons yet not one or any type of other form that contains a curve.

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Understanding shapes is essential in mathematics. You will certainly be forced to learn about shapes in ~ school but understanding the properties of forms has countless practical applications in professional and also real-life cases too.

Many specialists need to know the nature of shapes, consisting of engineers, architects, artists, real-estate agents, farmers and also construction workers.

You may well need to know shapes when doing residence improvements and DIY, as soon as gardening and also even once planning a party.

When working v polygons the main properties which are vital are:

The number of sides of the shape. The angles in between the sides of the shape. The length the the sides of the shape.

Number of Sides

Polygons are usually defined by the variety of sides the they have.

Three-Sided Polygons: Triangles

A three-sided polygon is a triangle. over there are several different species of triangle (see diagram), including:

Equilateral – all the sides space equal lengths, and all the internal angles are 60°. Isosceles – has actually two equal sides, v the 3rd one a various length. 2 of the internal angles are equal. Scalene – all three sides, and also all three interior angles, room different.

Triangles can additionally be described in terms of their interior angles (see our web page on Angles for more about specify name angles). The interior angles that a triangle always add up to 180°.

A triangle through onlyacuteinternal angle is dubbed an acute (or acute-angled) triangle. One through oneobtuseangle and two acute angles is dubbed obtuse (obtuse-angled), and also one with aright angleis well-known as right-angled.

Each the these will also be either equilateral, isosceles or scalene.


Four-Sided polygon - Quadrilaterals

Four-sided polygons room usually described as quadrilaterals, quadrangles or sometimes tetragons. In geometry the term quadrilateral is commonly used.

The hatchet quadrangle is regularly used to describe a rectangle-shaped enclosed outdoor space, for example ‘the freshers assembled in the university quadrangle’. The hatchet tetragon is continual with polygon, pentagon etc. You might come across it occasionally, yet it is not commonly used in practice.

The family of quadrilaterals consists of the square, rectangle, rhombus and also other parallelograms, trapezium/trapezoid and also kite.

The interior angles of all quadrilaterals include up to 360°.


Square: four sides of equal length, four internal best angles.

Rectangle: four internal right angles, opposite sides of same length.

Parallelogram: opposite sides room parallel, the contrary sides are equal in length, the opposite angles are equal.

Rhombus: A special form of parallelogram in which all four sides space the same length, favor a square that has actually been squashed sideways.

Trapezium (or trapezoid): two sides are parallel, but the various other two sides space not. Side lengths and angles are not equal.

Isosceles Trapezium (or trapezoid): two sides room parallel and base angles are equal, definition that non-parallel political parties are additionally equal in length.

Kite: 2 pairs of surrounding sides are of equal length; the shape has an axis that symmetry.

Irregular Quadrilateral: a four-sided shape where no sides room equal in length and also no interior angles are the same. All inner angles still include up come 360°, just like all other continual quadrilaterals.

More than 4 Sides

A five-sided form is referred to as a pentagon.

A six-sided form is a hexagon, a seven-sided shape a heptagon, while an octagon has actually eight sides…

Polygon Names

The names of polygons are acquired from the prefixes of ancient Greek numbers. The Greek number prefix wake up in numerous names of day-to-day objects and concepts. These can sometimes be beneficial in help you psychic how numerous sides a polygon has. For example:

one octopus has actually eight legs – an octagon has actually eight sides. A decade is ten year – a decagon has actually ten sides. The modern-day pentathlon has five occasions – a pentagon has five sides. One Olympic heptathlon has seven occasions – a heptagon has seven sides.

The ‘poly-‘ prefix simply method ‘multiple’, for this reason a polygon is a shape with lot of sides, in the same way that ‘polygamy’ way multiple spouses.

There are names for countless different types of polygons, and also usually the variety of sides is more important 보다 the name of the shape.

There space two main varieties of polygon - regular and also irregular.

A regular polygon has equal length sides v equal angles between each side. Any kind of other polygon is an irregular polygon, i m sorry by meaning has unequal size sides and also unequal angles between sides.

Circles and also shapes that incorporate curves are not polygons - a polygon, by definition, is made up of straight lines. Check out our pages on circles and also curved shapes for more.


Angles in between Sides

The angles in between the political parties of shapes are crucial when defining and also working with polygons. View our page on angles for more about just how to measure angles.

There is a beneficial formula because that finding the end the total (or sum) of interior angles for any type of polygon, that is:

(number of sides - 2) × 180°


For a pentagon (a five-sided shape) the calculation would be:

5 - 2 = 3

3 × 180 = 540°.

The sum of inner angles for any type of (not complex) pentagon is 540°.

Furthermore, if the shape is a regular polygon (all angles and length of sides space equal) climate you deserve to simply divide the sum of the inner angles by the variety of sides to uncover each internal angle.

540 ÷ 5 = 108°.

A regular pentagon because of this has five angles each same to 108°.

The size of the Sides

As well as the variety of sides and also the angles between sides, the size of every side of forms is likewise important.

The length of the political parties of a plane shape enables you to calculation the shape’s perimeter (the distance about the external of the shape) and area (the amount of an are inside the shape).


If your shape is a continuous polygon (such as a square in the instance above) climate it is only important to measure one next as, by definition, the other sides of a consistent polygon are the very same length. That is usual to use tick marks to show that every sides are an equal length.

In the instance of the rectangle we needed to measure 2 sides - the two unmeasured sides are equal come the two measured sides.

It is typical for some dimensions no to be shown for more complex shapes. In such cases missing dimensions can be calculated.


In the instance above, 2 lengths are missing.

The lacking horizontal length have the right to be calculated. Take it the much shorter horizontal well-known length native the much longer horizontal recognized length.

9m - 5.5m = 3.5m.

The very same principle deserve to be used to work out the missing vertical length. That is:

3m - 1m = 2m.

Bringing all the information Together: Calculating the Area of Polygons

The simplest and also most straightforward polygon for the functions of calculating area is the quadrilateral. To achieve the area, you simply multiple size by upright height.

For parallelograms, keep in mind that vertical height is NOT the length of the sloping side, however the vertical distance between the 2 horizontal lines.

This is because a parallelogram is essentially a rectangle with a triangle reduced off one end and also pasted onto the other:

You deserve to see the if you remove the left-hand blue triangle, and stick the onto the other end, the rectangle becomes a parallelogram.

The area is size (the height horizontal line) multiplied by height, the vertical distance in between the 2 horizontal lines.

To work out the area the a triangle, friend multiple size by vertical height (that is, the vertical elevation from the bottom heat to the top point), and halve it. This is essentially because a triangle is half a rectangle.

To calculation the area of any regular polygon, the easiest method is to division it into triangles, and use the formula for the area the a triangle.

So, because that a hexagon, because that example:

You have the right to see indigenous the diagram the there are six triangles.

The area is:

Height (red line) × length of next (blue line) × 0.5 × 6 (because there are six triangles).

You can likewise work out the area of any regular polygon making use of trigonometry, but that’s rather more complicated.

See our page Calculating Area for more, consisting of examples.

You can also work out the area of any kind of regular polygon utilizing trigonometry, but that’s rather more complicated. View our Introduction come Trigonometry page for much more information.

See more: How Many Different Values Of L Are Possible For An Electron With Principal Quantum Number N = 3?

Continue to: Calculating AreaCurved Shapes

See also:Three-Dimensional forms | Area, surface Area and Volume recommendation Sheet simple Transformations of 2-Dimensional shapes | hopeful and an adverse Numbers

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