This page examines the nature of two-dimensional or ‘plane’ polygons. A polygon is any shape comprised of straight lines that can be attracted on a flat surface, prefer a piece of paper. Together shapes include squares, rectangles, triangles and also pentagons but not one or any other shape that contains a curve.

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Understanding shapes is essential in mathematics. Girlfriend will definitely be compelled to learn about shapes in ~ school yet understanding the properties of shapes has countless practical applications in professional and also real-life cases too.

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

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

When working through polygons the main properties i beg your pardon are vital are:

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

Number the Sides

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

Three-Sided Polygons: Triangles

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

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

Triangles can additionally be explained in regards to their internal angles (see our page on Angles for an ext about specify name angles). The interior angles that a triangle always add up to 180°.

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

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


Four-Sided polygon - Quadrilaterals

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

The ax quadrangle is regularly used to define a rectangular enclosed the end space, for example ‘the freshers assembled in the college quadrangle’. The hatchet tetragon is regular with polygon, pentagon etc. You may come across it occasionally, but it is not commonly used in practice.

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

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


Square: 4 sides of equal length, four internal appropriate angles.

Rectangle: four internal best angles, opposite sides of equal length.

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

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

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

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

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

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

More than 4 Sides

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

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

Polygon Names

The surname of polygon are acquired from the prefixes of old Greek numbers. The Greek numerical prefix wake up in plenty of names of daily objects and concepts. These have the right to sometimes be beneficial in help you remember how many sides a polygon has. For example:

one octopus has actually eight foot – one octagon has actually eight sides. A te is ten year – a decagon has ten sides. The contemporary pentathlon has actually five events – a pentagon has five sides. An Olympic heptathlon has seven occasions – a heptagon has actually seven sides.

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

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

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

A regular polygon has actually equal length sides through equal angles between each side. Any other polygon is one irregular polygon, i beg your pardon by meaning has unequal size sides and also unequal angles between sides.

Circles and shapes that include curves are not polygons - a polygon, by definition, is consisted of of right lines. See our pages top top circles and also curved shapes because that more.


Angles between Sides

The angles in between the sides of forms are essential when defining and also working with polygons. Check out our web page on angle for an ext about just how to measure angles.

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

(number of political parties - 2) × 180°


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

5 - 2 = 3

3 × 180 = 540°.

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

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

540 ÷ 5 = 108°.

A regular pentagon as such has five angles each same to 108°.

The size of the Sides

As well as the number of sides and also the angles in between sides, the size of each side of shapes is additionally important.

The length of the political parties of a airplane shape permits you to calculate the shape’s perimeter (the distance around the exterior of the shape) and also area (the quantity of room inside the shape).


If your shape is a consistent polygon (such together a square in the instance above) then it is only necessary to measure one next as, by definition, the other sides that a regular polygon room the exact same length. That is usual to usage tick point out to display that every sides space an same length.

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

It is typical for some dimensions not to be shown for more facility shapes. In such cases missing dimensions can be calculated.


In the instance above, two lengths space missing.

The absent horizontal length deserve to be calculated. Take the much shorter horizontal well-known length native the longer horizontal well-known length.

9m - 5.5m = 3.5m.

The very same principle have the right to be provided to occupational out the absent vertical length. That is:

3m - 1m = 2m.

Bringing all the details Together: Calculating the Area of Polygons

The simplest and also most simple polygon for the objectives of calculating area is the quadrilateral. To achieve the area, you just multiple length by upright height.

For parallelograms, keep in mind that vertical elevation is NOT the size of the sloping side, yet the vertical distance between the 2 horizontal lines.

This is due to the fact that a parallelogram is basically a rectangle v a triangle reduced off one end and also pasted onto the other:

You can see that if you eliminate the left-hand blue triangle, and stick it onto the various 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-related out the area the a triangle, friend multiple length 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 fifty percent a rectangle.

To calculation the area of any type of regular polygon, the easiest way is to division it into triangles, and also use the formula because that the area the a triangle.

So, because that a hexagon, for example:

You have the right to see native the diagram the there are 6 triangles.

The area is:

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

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

See our page Calculating Area because that more, consisting of examples.

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

See more: How Big Is 1 Oz Of Cheese In Cups? How Do You Measure Cheese

Continue to: Calculating AreaCurved Shapes

See also:Three-Dimensional forms | Area, surface ar Area and also Volume reference Sheet straightforward Transformations of 2-Dimensional forms | optimistic and negative Numbers

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