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3. Everything / Maths, Science & Technology / Mathematics

Triangles

floor tiles constructed of tessellated triangles. Clever, eh?

This entry is about the triangle of Euclidean geometry1. It is defined as a planar (two-dimensional) figure bounded by three straight lines and therefore having three interior angles and three exterior angles. The sum of the interior angles of a triangle equals 180°2. It is a convex shape: if you draw a line from any point in the shape to any other, then the line is within the shape.

Triangles were among the first things studied in geometry. This is partly because they're so simple and partly because many shapes can be reduced to triangles in some way.

Other Definitions of the Word 'Triangle'

This article is about the type of triangle defined above. The following definitions also exist:

  • An instrument of percussion, usually made of a rod of steel, bent into the form of a triangle, open at one angle, and sounded by being struck with a small metallic rod.

  • A draughtsman's square, often called a setsquare, in the form of a right-angled triangle with the other angles being usually 30 and 60 degrees or 45 and 45 degrees.

  • A kind of frame formed of three poles stuck in the ground and united at the top, to which soldiers were bound when undergoing corporal punishment; this is no longer used.

  • A small constellation of stars called 'The Triangle' (Triangulum).

  • A group of three very bright stars visible from the Northern Hemisphere, the Summer Triangle.

  • A love triangle refers to a romantic relationship involving three people.

  • Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle.

  • Bermuda triangle: A triangular area of the Atlantic whose apices are Bermuda, Miami, and the Lesser Antilles. Reputed to be the site of numerous mysterious disappearances of planes and ships.

Classification Of Triangles

There are two methods of classifying triangles: by the relative lengths of their sides and according to the size of their largest internal angle.

When looking at the lengths of sides:

  • In an equilateral triangle all sides are of equal length. An equilateral triangle is also equiangular, ie, all its internal angles are equal - namely, 60°.

  • In an isosceles triangle two sides are of equal length. An isosceles triangle also has two equal internal angles.

  • In a scalene triangle all sides have different lengths. The internal angles in a scalene triangle are all different.

When looking at the size of the largest internal angle:

  • A right-angled triangle3 has one 90° internal angle (a right angle).

  • An obtuse triangle has one internal angle larger than 90° (an obtuse angle).

  • An acute triangle has internal angles that are all smaller than 90° (three acute angles).

Triangle Definitions

A vertex4 refers to one of the points of the triangle, where two sides meet.

Any one of the sides may be considered the base of the triangle. The perpendicular distance5 from a base to the opposite vertex is called an altitude6.

A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it.

The circumcenter is where three perpendicular bisectors meet in a single point; this point is the centre of the circumcircle, the circle passing through all three vertices.

The line segment joining the midpoint of a side to the opposite vertex is called a median.

The three medians intersect in a single point, the triangle's centroid. This is also the triangle's centre of gravity.

Two triangles are said to be similar if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are proportional. This occurs, for example, when two triangles share an angle and the sides opposite to that angle are parallel.

In a right-angled triangle the hypotenuse is the side opposite the right angle and is the longest side.

The Mathematics of Triangles

The mathematics of triangles is called trigonometry. It is a massive topic which will not be covered here. It is, however, covered in two other guide entries: one on the trigonometry of right-angled triangles and one on the trigonometric laws of sines and cosines.

The most famous mathematical theorem about triangles is Pythagoras's Theorem. This states that, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Using this theorem you can calculate the length of any side of a right-angled triangle, provided you already know the length of the other two sides.

Some Uses Of Triangles

If you put three points anywhere in space (not in a straight line) you have exactly one plane that goes through all three. This is unique to three points, any more points and there may not be a plane and any less and there are multiple planes. This is useful as it means that a three-legged stool will be stable no matter how uneven the ground7. It also helps simplify some calculations, for example, those in 3-D graphics.

In 3-D graphics, the surfaces of 3-D objects are broken down into triangles. Small numbers of triangles are used for flat surfaces, while large numbers are used to mould curved surfaces similar to the way a geodesic dome is constructed8. The triangle vertices are computed on an X-Y-Z scale and every one must be recomputed each time the object is moved.

Triangluation is using the trigonometry of sines cosines and Pythagoras's Theorem to find your location assuming you know two angles and a length of one side a triangle. Trilateration is how to work out your exact point in space assuming you know the distance to three (if you are working on a plane) or four (in 3D) points. Triangulation and trilateration are used for many purposes, including surveying, navigation, metrology (the science that deals with measurement), astrometry (the measurement of the positions and motions of celestial bodies), binocular vision and gun direction of weapons. They are the basis on which GPS (Global Positioning System) works.

Triangles have many uses in construction due to their strong and rigid shape. They are the basis of many buildings including bridges, monuments9 and domes. The geodesic dome is one of the most stable of geometric forms and is made of many triangles (as tetrahedrons) which distribute stress evenly to all elements of the dome, providing a high strength-to-weight ratio. The dome of the Eden Project is an example of such a geodesic dome, although this dome is of triangles arranged together as hexagons and pentagons.


1 Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions.
2 While this certainly appears to be true, it can't actually be proved, and non-Euclidean geometries in which the angles add up to less than or more than 180° have been found to be completely consistent, although not bearing much relevance to Planet Earth. They imply the existence of curved space.
3 Also sometimes known as a 'right triangle'.
4 Plural: Vertices.
5 Perpendicular means forming a right angle with.
6 This is also known in common language as the height.
7 As long as the ground is not so uneven that the stool actually falls over.
8 It takes an enormous number of triangles to simulate totally round objects.
9 For example: The Eiffel Tower, The Statue of Liberty and the Crystal Palace.

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Entry Data
Entry ID: A8412653 (Edited)

Written and Researched by:
The stone eater aka SigSig (SI) - webmaster

Edited by:
SchrEck Inc.


Date: 22   February   2006


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Referenced Guide Entries
Mathematics
Trigonometry
The Eiffel Tower, Paris, France
Global Positioning System (GPS)
The Eden Project, Cornwall, UK
Gravity
Binomial Theorem
Orchestral Percussion - the Kitchen Sink Department
The Trigonometric Laws of Sines and Cosines
The Statue of Liberty
Flight 19 - The 'Lost Patrol'
Curved Space and the Fate of the Universe
The Summer Triangle
Constellations: Triangulum 'the Triangle'
Constellation Overview


Related BBC Pages
Maths - BBC Learning


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