This page examines the nature of two-dimensional or ‘plane’ polygons. A polygon is any shape consisted of of straight lines that deserve to be attracted on a flat surface, prefer a item of paper. Such shapes incorporate squares, rectangles, triangles and also pentagons yet not one or any kind of other form that includes a curve.

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Understanding shapes is crucial in mathematics. You will absolutely be forced to learn about shapes at school but understanding the properties of shapes has countless practical applications in professional and also real-life instances too.

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

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

When working with polygons the key properties i m sorry are important are:

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

Number of Sides

Polygons space usually identified by the variety of sides that they have.

Three-Sided Polygons: Triangles

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

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

Triangles can also be explained in regards to their interior angles (see our page on Angles for much more about specify name angles). The internal angles of a triangle always include up come 180°.

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

Each the these will also be one of two people equilateral, isosceles or scalene.


Four-Sided polygons - Quadrilaterals

Four-sided polygons space usually referred to as quadrilaterals, quadrangles or sometimes tetragons. In geometry the hatchet quadrilateral is typically used.

The term quadrangle is frequently used to define a rectangular enclosed outdoor space, for example ‘the freshers assembled in the college quadrangle’. The ax tetragon is continuous with polygon, pentagon etc. You may come across it occasionally, however it is not generally used in practice.

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

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


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

Rectangle: four internal appropriate angles, opposite political parties of equal length.

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

Rhombus: A special form of parallel in i m sorry all 4 sides are the very same length, choose a square that has been squashed sideways.

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

Isosceles Trapezium (or trapezoid): two sides space parallel and also base angles space equal, meaning that non-parallel sides are additionally equal in length.

Kite: 2 pairs of adjacent sides space of same length; the shape has actually an axis the symmetry.

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

More than four Sides

A five-sided shape is dubbed a pentagon.

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

Polygon Names

The names of polygons are acquired from the prefixes of ancient Greek numbers. The Greek number prefix wake up in many names of daily objects and also concepts. These deserve to sometimes be helpful in helping you remember how plenty of sides a polygon has. Because that example:

an octopus has actually eight legs – one octagon has actually eight sides. A decade 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 actually seven sides.

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

There are names for many different species of polygons, and usually the variety of sides is more important than the surname of the shape.

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

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

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


Angles in between Sides

The angles in between the political parties of shapes are vital when defining and also working v polygons. Watch our web page on angle for more about how to measure angles.

There is a useful formula for finding the end the total (or sum) of internal angles for any kind of polygon, the is:

(number of sides - 2) × 180°


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

5 - 2 = 3

3 × 180 = 540°.

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

Furthermore, if the shape is a regular polygon (all angles and also length the sides space equal) then you can simply divide the amount of the internal angles by the number of sides to uncover each inner 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 number of sides and also the angles between sides, the length of each side of shapes is also important.

The size of the sides of a plane shape allows you to calculation the shape’s perimeter (the distance about the exterior of the shape) and also area (the amount of space inside the shape).


If your form is a constant polygon (such together a square in the example above) climate it is only crucial to measure one next as, by definition, the various other sides the a continuous polygon are the very same length. That is common to use tick clues to show that every sides room an equal length.

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

It is common for some dimensions not to be displayed for more complicated shapes. In such cases lacking dimensions have the right to be calculated.


In the instance above, 2 lengths space missing.

The absent horizontal length can be calculated. Take the shorter horizontal known length native the much longer horizontal known length.

9m - 5.5m = 3.5m.

The very same principle can be provided to work out the missing vertical length. The is:

3m - 1m = 2m.

Bringing all the details Together: Calculating the Area that Polygons

The simplest and most basic polygon because that the purposes of calculating area is the quadrilateral. To achieve the area, you just multiple size by vertical height.

For parallelograms, keep in mind that vertical elevation is NOT the size of the sloping side, however the upright distance between the two horizontal lines.

This is since a parallelogram is essentially a rectangle through a triangle reduced off one end and also pasted top top the other:

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

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

To work out the area the a triangle, friend multiple size by vertical elevation (that is, the vertical elevation from the bottom line to the peak point), and also halve it. This is essentially since a triangle is half a rectangle.

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

So, for a hexagon, because that example:

You can see indigenous the diagram that there are 6 triangles.

The area is:

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

You can additionally work the end the area of any type of regular polygon utilizing trigonometry, but that’s rather an ext complicated.

See our web page Calculating Area because that more, including examples.

You can likewise work out the area of any regular polygon making use of trigonometry, however that’s rather an ext complicated. See our Introduction come Trigonometry page for more information.

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Continue to: Calculating AreaCurved Shapes

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

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