Though it may look similar to other types of right triangles, the reason a 30-60-90 triangle is so special is that you only need three pieces of information in order to find every other measurement. And, finally, the side opposite the 90° angle will always be the largest side (the hypotenuse) because 90 degrees is the largest angle. The side opposite the 60° angle will be the middle length, because 60 degrees is the mid-sized degree angle in this triangle. The side opposite the 30° angle is always the smallest, because 30 degrees is the smallest angle. And the hypotenuse is 2 times the shortest leg, or $2√3$) (Why is the longer leg 3? In this triangle, the shortest leg ($x$) is $√3$, so for the longer leg, $x√3 = √3 * √3 = √9 = 3$. Because it is a special triangle, it also has side length values which are always in a consistent relationship with one another.įor example, a 30-60-90 degree triangle could have side lengths of: So let's get to it!Ī 30-60-90 triangle is a special right triangle (a right triangle being any triangle that contains a 90 degree angle) that always has degree angles of 30 degrees, 60 degrees, and 90 degrees. In this guide, we'll walk you through what a 30-60-90 triangle is, why it works, and when (and how) to use your knowledge of it. And a 30-60-90 triangle-pronounced 'thirty sixty ninety'-happens to be a very special type of triangle indeed. Each angle is a different size.Acute, obtuse, isosceles, equilateral…When it comes to triangles, there are many different varieties, but only a choice few that are 'special.' These special triangles have sides and angles which are consistent and predictable and can be used to shortcut your way through your geometry or trigonometry problems.

Two angles are the same size., and scalene triangles close scalene triangle Each side is a different length. All angles are 60°., isosceles triangles close isosceles triangle Two sides are equal in length. This gives the order of rotational symmetry.Ī unique set of properties relating to the comparative length of its sides and the comparative size of its angles help to identify equilateral triangles close equilateral triangle All sides are equal in length.

Count how many ways the triangle will fit into its outline in a full turn (360°).

This gives the number of lines of symmetry of the triangle.

Count how many ways the triangle can be cut into a pair of mirrored halves.

Different numbers of arcs indicate different angles.

The same number of arcs indicate equal angles.

Different numbers of hash marks indicate different lengths.

The same number of hashes indicate equal lengths.

To classify a triangle using comparative lengths or angles: in vertices close vertex The point at which two or more lines intersect (cross or overlap). The same number of marks indicate angles are equal in size.

Recognise that arcs close arcs (annotation) Curved marks inside the vertex of a shape.

The same number of marks indicate equal lengths.

Recognise that hash marks close hash marks Short lines marked on the side or edge of a shape.

Recognising line symmetry and rotational symmetry will also help. Understanding different types of angles and that angles in a triangle sum to 180° can be helpful when classifying a triangle. Other properties relate to the symmetry that the triangle has.are used to represent angles of equal measure. at vertices close vertex The point at which two or more lines intersect (cross or overlap). Arcs close arcs (annotation) Curved marks inside the vertex of a shape.are used to represent segments of equal length on diagrams.

Hash marks close hash marks Short lines marked on the side or edge of a shape.These properties can be annotated on a diagram: with three straight edges is a triangle close triangle A three-sided polygon.Ī triangle is classified by the comparative length of its edges close edge Side of a polygon or a 3D shape. Any polygon close polygon A closed 2D shape bounded by straight lines.

Obtuse isosceles triangle side lengths