This page examines the properties of two-dimensional or ‘plane’ polygons. A polygon is any kind of shape consisted of of straight lines that can be attracted on a flat surface, like a piece of paper. Together shapes incorporate squares, rectangles, triangles and pentagons but not one or any kind of other shape that includes a curve.

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Understanding shapes is important in mathematics. You will certainly be compelled to learn around shapes in ~ school but understanding the properties of shapes has numerous practical applications in professional and also real-life instances too.

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

You might well need to recognize shapes once doing house improvements and DIY, when gardening and even once planning a party.

When working with polygons the main properties i beg your pardon are necessary are:

The number that sides of the shape. The angles in between the sides of the shape. The length that the political parties of the shape.

Number that Sides

Polygons are usually defined by the number of sides that they have.

Three-Sided Polygons: Triangles

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

Equilateral – every the sides space equal lengths, and all the interior angles are 60°. Isosceles – has two same sides, through the 3rd one a various length. Two of the inner angles are equal. Scalene – all 3 sides, and also all three interior angles, room different.

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

A triangle v onlyacuteinternal angles is referred to as an acute (or acute-angled) triangle. One with oneobtuseangle and two acute angles is referred to as obtuse (obtuse-angled), and one through aright angleis recognized as right-angled.

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


Four-Sided polygons - Quadrilaterals

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

The term quadrangle is frequently used to explain a rectangle-shaped enclosed outdoor space, for instance ‘the freshers assembled in the university quadrangle’. The hatchet tetragon is continual with polygon, pentagon etc. You may come across it occasionally, however it is not frequently used in practice.

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

The inner angles of every quadrilaterals add up come 360°.


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

Rectangle: 4 internal ideal angles, opposite political parties of same length.

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

Rhombus: A special kind of parallel in which all 4 sides are the exact same length, choose a square that has been squashed sideways.

Trapezium (or trapezoid): two sides space parallel, yet the other two sides are not. Next lengths and also angles are not equal.

Isosceles Trapezium (or trapezoid): 2 sides space parallel and base angles are equal, an interpretation that non-parallel political parties are also equal in length.

Kite: two pairs of nearby sides space of equal length; the shape has actually an axis of symmetry.

Irregular Quadrilateral: a four-sided form where no sides space equal in length and no interior angles space the same. All interior angles still include up to 360°, as with all other constant quadrilaterals.

More than 4 Sides

A five-sided shape is dubbed a pentagon.

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

Polygon Names

The surname of polygon are derived from the prefixes of old Greek numbers. The Greek number prefix wake up in numerous names of day-to-day objects and also concepts. These can sometimes be beneficial in helping you mental how numerous sides a polygon has. For example:

an octopus has eight foot – an octagon has eight sides. A te is ten years – a decagon has actually ten sides. The modern-day pentathlon has five occasions – a pentagon has 5 sides. An Olympic heptathlon has seven events – a heptagon has seven sides.

The ‘poly-‘ prefix simply way ‘multiple’, so a polygon is a form with multiple sides, in the same method that ‘polygamy’ method multiple spouses.

There room names for plenty of different varieties of polygons, and usually the variety of sides is much more important 보다 the surname of the shape.

There are two main species of polygon - regular and irregular.

A regular polygon has equal length sides v equal angles between each side. Any kind of other polygon is an irregular polygon, which by definition has unequal size sides and unequal angles between sides.

Circles and shapes that incorporate curves room not polygons - a polygon, by definition, is consisted of of straight lines. Check out our pages ~ above circles and curved shapes for more.


Angles between Sides

The angles between the sides of forms are important when defining and also working with polygons. Watch our page on angle for an ext about exactly how to measure angles.

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

(number of sides - 2) × 180°


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

5 - 2 = 3

3 × 180 = 540°.

The amount of interior angles for any (not complex) pentagon is 540°.

Furthermore, if the shape is a regular polygon (all angles and also length the sides space equal) climate you have the right to simply division the sum of the interior angles by the number of sides to discover each internal angle.

540 ÷ 5 = 108°.

A regular pentagon thus has 5 angles each same to 108°.

The length of the Sides

As well as the variety of sides and also the angles in between sides, the length of every side of forms is additionally important.

The length of the political parties of a airplane shape permits you to calculation the shape’s perimeter (the distance around the exterior of the shape) and also area (the amount of an are inside the shape).


If your form is a consistent polygon (such together a square in the instance above) then it is only vital to measure one side as, by definition, the other sides the a regular polygon space the same length. The is common to use tick marks to display that every sides are an equal length.

In the example of the rectangle we essential to measure 2 sides - the 2 unmeasured sides room equal to the two measured sides.

It is usual for some dimensions not to be shown for more facility shapes. In together cases missing dimensions have the right to be calculated.


In the instance above, two lengths room missing.

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

9m - 5.5m = 3.5m.

The exact same principle can be supplied to job-related out the missing vertical length. That is:

3m - 1m = 2m.

Bringing every the details Together: Calculating the Area that Polygons

The simplest and also most basic polygon for the functions of calculating area is the quadrilateral. To attain the area, you merely multiple size by upright height.

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

This is because a parallel is essentially a rectangle with a triangle cut off one end and pasted onto the other:

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

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

To occupational out the area that a triangle, friend multiple size by vertical elevation (that is, the vertical height from the bottom heat to the peak point), and also halve it. This is essentially due to the fact that a triangle is half a rectangle.

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

So, for a hexagon, because that example:

You deserve to see from the diagram the there are 6 triangles.

The area is:

Height (red line) × size of side (blue line) × 0.5 × 6 (because there are 6 triangles).

You can additionally work out the area of any regular polygon utilizing trigonometry, however that’s rather much more complicated.

See our web page Calculating Area for more, including examples.

You can additionally work the end the area of any regular polygon making use of trigonometry, however that’s rather more complicated. Check out our Introduction to Trigonometry page for much more information.

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Continue to: Calculating AreaCurved Shapes

See also:Three-Dimensional shapes | Area, surface ar Area and also Volume referral Sheet basic Transformations that 2-Dimensional forms | hopeful and an adverse Numbers

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