This web page examines the properties of two-dimensional or ‘plane’ polygons. A polygon is any type of shape comprised of straight lines that can be drawn on a level surface, choose a piece of paper. Such shapes encompass squares, rectangles, triangles and also pentagons however not one or any other form that includes a curve.

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Understanding shapes is important in mathematics. Friend will certainly be compelled to learn about shapes at school yet understanding the nature of forms has many practical applications in professional and also real-life situations too.

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

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

When working with polygons the key properties which are important are:

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

Number the Sides

Polygons room usually characterized by the variety of sides that they have.

Three-Sided Polygons: Triangles

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

Equilateral – every the sides space equal lengths, and all the inner angles room 60°. Isosceles – has two same sides, with the third one a different length. 2 of the internal angles room equal. Scalene – all 3 sides, and all three inner angles, are different.

Triangles can likewise be described in regards to their internal angles (see our page on Angles for an ext about naming angles). The inner angles the a triangle always add up to 180°.

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

Each that 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 occasionally tetragons. In geometry the ax quadrilateral is frequently used.

The ax quadrangle is often used to define a rectangle-shaped enclosed outdoor space, for instance ‘the freshers assembled in the college quadrangle’. The ax tetragon is continual with polygon, pentagon etc. You may come throughout it occasionally, however it is not typically used in practice.

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

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


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

Rectangle: 4 internal best angles, opposite political parties of equal length.

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

Rhombus: A special form of parallelogram in i beg your pardon all 4 sides space the same length, prefer a square that has been squashed sideways.

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

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

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

Irregular Quadrilateral: a four-sided form where no sides are equal in length and no internal angles are the same. All inner angles still add up to 360°, as with all other regular quadrilaterals.

More than four 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 name of polygons are obtained from the prefixes of old Greek numbers. The Greek numerical prefix wake up in plenty of names of daily objects and also concepts. These can sometimes be beneficial in helping you mental how plenty of sides a polygon has. For example:

an octopus has actually eight foot – an octagon has eight sides. A decade is ten year – a decagon has actually ten sides. The contemporary pentathlon has five events – a pentagon has 5 sides. One Olympic heptathlon has actually seven events – a heptagon has seven sides.

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

There space names for plenty of different types of polygons, and usually the number of sides is more important 보다 the surname of the shape.

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

A regular polygon has equal length sides with equal angles between each side. Any other polygon is one irregular polygon, i m sorry by an interpretation has unequal length sides and unequal angles between sides.

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


Angles between Sides

The angles in between the political parties of forms are necessary when defining and working through polygons. Watch our web page on angles for an ext about exactly how to measure angles.

There is a helpful 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 certainly be:

5 - 2 = 3

3 × 180 = 540°.

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

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

540 ÷ 5 = 108°.

A regular pentagon because of this has 5 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 shapes is also important.

The size of the political parties of a aircraft shape enables you to calculate the shape’s perimeter (the distance roughly the outside of the shape) and also area (the amount of an are inside the shape).


If your shape is a consistent polygon (such together a square in the instance above) then it is only important to measure one side as, by definition, the various other sides of a continuous polygon space the same length. The is common to use tick point out to present that every sides are an equal length.

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

It is usual for part dimensions not to be presented for more complicated shapes. In together cases absent dimensions can be calculated.


In the example above, 2 lengths room missing.

The lacking horizontal length can be calculated. Take the much shorter horizontal well-known length from the much longer horizontal recognized length.

9m - 5.5m = 3.5m.

The exact same principle deserve to be used to work-related out the missing vertical length. The is:

3m - 1m = 2m.

Bringing every the info Together: Calculating the Area that Polygons

The simplest and most an easy polygon because that the purposes of calculating area is the quadrilateral. To attain the area, you simply multiple size by vertical height.

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

This is due to the fact that a parallel is basically a rectangle with a triangle reduced off one end and also pasted ~ above the other:

You deserve to see that if you eliminate the left-hand blue triangle, and also stick that onto the other end, the rectangle i do not care a parallelogram.

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

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

To calculation the area of any kind of regular polygon, the easiest means 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 indigenous the diagram that there are 6 triangles.

The area is:

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

You can likewise work the end the area of any type of regular polygon using trigonometry, however that’s rather an ext complicated.

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

You can also work out the area of any type of regular polygon using trigonometry, yet that’s rather an ext complicated. View our Introduction to Trigonometry web page for much more information.

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Continue to: Calculating AreaCurved Shapes

See also:Three-Dimensional shapes | Area, surface Area and Volume referral Sheet straightforward Transformations the 2-Dimensional shapes | positive and negative Numbers

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