This web page examines the nature of two-dimensional or ‘plane’ polygons. A polygon is any shape consisted of of right lines that can be drawn on a flat surface, favor a item of paper. Together shapes encompass squares, rectangles, triangles and also pentagons however not one or any kind of other shape that consists of a curve.

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Understanding shapes is vital in mathematics. You will certainly be compelled to learn about shapes in ~ school however understanding the nature of forms has numerous practical applications in professional and real-life situations too.

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

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

When working through polygons the key properties i beg your pardon are important are:

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

Number the Sides

Polygons room usually characterized by the number of sides the they have.

Three-Sided Polygons: Triangles

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

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

Triangles can additionally be described in regards to their interior angles (see our page on Angles for more about specify name angles). The inner angles the a triangle always include up to 180°.

A triangle v onlyacuteinternal angles is dubbed an acute (or acute-angled) triangle. One through oneobtuseangle and two acute angle is called obtuse (obtuse-angled), and also one v 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 described as quadrilaterals, quadrangles or periodically tetragons. In geometry the hatchet quadrilateral is generally used.

The term quadrangle is regularly used to explain a rectangle-shaped enclosed out 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 frequently used in practice.

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

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


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

Rectangle: 4 internal right angles, opposite sides of same length.

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

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

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

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

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

Irregular Quadrilateral: a four-sided shape where no sides room equal in length and no inner angles are the same. All internal angles still add up to 360°, just like all other constant quadrilaterals.

More than 4 Sides

A five-sided form is dubbed a pentagon.

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

Polygon Names

The name of polygons are obtained from the prefixes of ancient Greek numbers. The Greek numerical prefix occurs in numerous names of everyday objects and concepts. These deserve to sometimes be helpful in helping you mental how plenty of sides a polygon has. Because that example:

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

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

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

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

A regular polygon has actually equal length sides with equal angles between each side. Any type of other polygon is an irregular polygon, i beg your pardon by an interpretation has unequal length sides and also unequal angles in between sides.

Circles and shapes that include curves room not polygons - a polygon, by definition, is made up of directly lines. View our pages ~ above circles and curved shapes because that more.


Angles in between Sides

The angles between the political parties of forms are important when defining and working with polygons. Check out our web page on angles for an ext about how to measure angles.

There is a useful formula for finding out the complete (or sum) of interior angles for any type 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 sum of interior angles for any type of (not complex) pentagon is 540°.

Furthermore, if the form is a regular polygon (all angles and length that sides space equal) climate you deserve to simply division the amount of the inner angles by the variety of sides to find each inner angle.

540 ÷ 5 = 108°.

A regular pentagon because of this has 5 angles each equal to 108°.

The size of the Sides

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

The length of the sides of a plane shape enables you to calculation the shape’s perimeter (the distance around the exterior of the shape) and also area (the lot of an are inside the shape).


If your shape is a consistent polygon (such together a square in the example above) climate it is only necessary to measure up one next as, by definition, the various other sides of a continuous polygon space the very same length. That is usual to use tick clues to show that every sides room an equal length.

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

It is usual for part dimensions not to be shown for more complex shapes. In such cases lacking dimensions deserve to be calculated.


In the instance above, two lengths room missing.

The absent horizontal length deserve to be calculated. Take the shorter horizontal recognized length native the longer horizontal recognized length.

9m - 5.5m = 3.5m.

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

3m - 1m = 2m.

Bringing all the information Together: Calculating the Area that Polygons

The simplest and also most basic polygon because that the objectives of calculating area is the quadrilateral. To obtain the area, you simply multiple size by upright height.

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

This is since a parallel is essentially a rectangle with a triangle cut off one end and also pasted top top the other:

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

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

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

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

So, because that a hexagon, because that example:

You deserve to see native 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 kind of regular polygon using trigonometry, however that’s rather an ext complicated.

See our page Calculating Area for more, including examples.

You can additionally work the end the area of any kind of regular polygon using trigonometry, but that’s rather an ext complicated. See our Introduction come Trigonometry web page for much more information.

See more: A Device That Converts Electrical Energy Into Mechanical Energy

Continue to: Calculating AreaCurved Shapes

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

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