Two types of special right triangles appear commonly in geometry, the "angle based" and the "side based" triangles. The two "angle based" triangles are the "45-45-90 triangle" and the "30-60-90 triangle." Four of the more common "side based" triangles are listed below. Knowing the ratios of the sides of these special right triangles allows one to quickly calculate various lengths in geometric problems. A triangle. ...

Angle-based

"Angle-based" special right triangles are specified by the angles of which the triangle is composed and the side lengths are generally deduced from the basis of the unit circle or other geometric methods. This form is most interesting in that it may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, & 60°. Illustration of a unit circle. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ...

45-45-90 triangle

The side lengths of a 45-45-90 triangle

This is a triangle whose three angles respectively measure 45°, 45°, and 90°. The sides are in the ratio Image File history File links 45-45-triangle. ... Image File history File links 45-45-triangle. ...

A simple proof. Say you have such a triangle with legs a and b and hypotenuse c. Suppose that a = 1. Since two angles measure 45°, this is an isosceles triangle and we have b = 1. The fact that follows immediately from the Pythagorean theorem. A right triangle and its hypotenuse, h, along with catheti, c1 and c2. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

30-60-90 triangle

The side lengths of a 30-60-90 triangle

This is a triangle whose three angles respectively measure 30°, 60°, and 90°. The sides are in the ratio Image File history File links 30-60-90. ... Image File history File links 30-60-90. ...

The proof of this fact is clear using trigonometry. Although the geometric proof is less apparent, it is equally trivial: Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ...

Draw an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D. Then ABD is a 30-60-90 triangle with hypotenuse of length 2, and base BD of length 1.

The fact that the remaining leg AD has length follows immediately from the Pythagorean theorem.

In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

Side-based

All of the special side based right triangles posses angles which are not necessarily rational numbers, but whose sides are always of integer length and form a Pythagorean triple. They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship. In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... The integers are commonly denoted by the above symbol. ... The Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ... Multiple is a comic book superhero in the Marvel Comics universe. ...

Common Pythagorean triples

There are several Pythagorean triples which are very well known, including:

Fibonacci triangles

Starting with 5, every other Fibonacci number {0,1,1,2,3,5,8,13,21,34,55,89,...} is the length of the hypotenuse of a right triangle with integral sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle. A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral. ...

The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely.

External links

• 3-4-5 triangle
• 30-60-90 triangle
• 45-45-90 triangle With interactive animations