Right Triangle Calculator

Right Triangle Solver

Directly below is a set of javascript right triangle calculators that will determine the unknowns in right triangles. All you need to do is select the image that has your two unknown parameters circled in red, type the known quantities into the solver to the right of it, and click calculate. You can change your numbers, and move between the solvers at will without having to reload the page - just click calculate. Keep in mind that angles D and E will (have to) add up to 90 degrees.

Below the solvers is a little bit of information on Trigonometry, right triangles, and special triangles.

Angle D: Side d:

Known are Angle D and Side d.
Calculate for Angle E and Sides e & f

Angle D: Side e:

Known are Angle D and Side e.
Calculate for Angle E and Sides d & f.

Angle D: Side f:

Known are Angle D and Side f.
Calculate for Angle E and Sides d & e.

Side d: Side e:

Known are Side d and Side e.
Calculate for Angle D & E, and Side f.

Hypotenuse f: Side e:

Known are Side f and Side e.
Calculate for Angle D & E, and Side d.

Hypotenuse f: Side d:

Known are Side f and Side d.
Calculate for Angle D & E, and Side e.

Please note!

All of these calculations are as accurate as possible, but since they use Javascript functions, that accuracy may depend on your installation of Javascript! While They are certainly close enough, please do not use the results for anything that involves safety or trade - whatever that may be. If you have any problems, please do not hesitate to email me at info@righttrianglecalculator.com

A quick explanation of right triangles and special triangles.

While the subject and history of Trigonometry is broad and has uses and implications in many fields of mathematics, including a direct relationship with the study of circles, here I will try to give a very brief synopsis of Trigonometry as applied to the solving of the lengths of a right triangle's sides. This is one of the more common "everyday" uses of Trigonometry. This is certainly not a thorough academic overview of Trigonometry, and my apologies go out in advance to the Mathematicians.

A right triangle is simply, as it's name implies a triangle where one of it's sides is a 90 degree right angle. This triangle, and the functions that surround solving it, are one of the basis for Trigonometry, the study of triangles. Trigonometry is derived from the Greek, and literally means "Triangle Measuring"

The lengths of the sides of a right triangle have a defined ratio that depends on the other two angles in a triangle. Since the other two angles always also add up to 90 degrees, this can be simplified to mean the lengths of the sides of a triangle are directly related to either one of the other angles.

These ratios are contained within the Trigonometric functions of sine, cosine and tangent. The ratios themselves are defined within the Trigonometric "Tables", which were traditionally hand calculated tables of parameter and ratio combinations.

The functions of a right triangle constitute ratios of the sides. By ratio, we mean the lengths of the sides are dimensionless. IE: For any combination of angles, the sides will be a defined ration. Side a will be so many units different from side b, etc.

This concept may be most clear in the 3,4,5 right triangle that is so familiar to carpenters, surveyors and many others. In effect, whether the units are inches, meters, feet, furlongs or kilometers, if one side of a right triangle is 3 units, and the other side is four units, the hypotenuse (the side opposite the 90 degree angle) will be five units. And, of course the converse is also true. IF one side is 3 units, one side is four units, and the hypotenuse is 5 units - the angle opposite the hypotenuse must be 90 degrees. Truly a handy thing to remember. This example also defines the 30-60-90 triangle, called a "special triangle", as those are the angles included in the 3,4,5 triangle. Lets look at it pictorially.

Image of a 3-4-5 triangle

A 3-4-5 Right Triangle - This is one of many special triangles, known as Pythagorean Triples. The definition of a Pythagorean triple is a triangle where Side e squared plus side d squared equals side f squared. Lets check this (3x3=9, plus 4X4=16 equals 25, which is 5 squared). There are many of these. If you are interested, search for the Pythagorean Triple.

And, of course, this triangle can be scaled. IE: it can be 3-4-5, or 6-8-10 or 12-16-20.

Speaking of pictures, before we continue to learn how to solve for other angles, lets "look" at the definitions of a right triangles sides and angles. I am using the same designation letters the solver uses, although they are not particularly the "official" letters. I think it will help to keep things the same.

Image of a right triangle

This diagram defines the following sides:

  • The hypotenuse is the side that is opposite of the right angle, (side f above). The hypotenuse is always the longest side of a right-angled triangle.
  • The opposite side is the side that is opposite to the angle we are calculating in (angle A), (side a above).
  • The adjacent side is the side between both the angles of interest (angle A and right-angle C), (side b above).

As to the angles, remember from geometry that the included angles of a triangle always total 180 degrees. Since this is a right triangle we are interested in, the angles D and E total to (180-90) 90 degrees. Often the purpose of these calculations is to solve for an unknown angle, hence the origin of the terms side opposite and side adjacent.

If you do a bit of construction, here is an inexpensive calculator that will not only help solve these right triangle equations, but will do it in Inches/fractions/and meters. Here is what the Manufacturer says:

This unique foot/inch/fraction calculator will automatically solve rise, run diagonal, and slope of any triangle. These powerful triangle functions make this calculator a must for anyone dealing with roof rafters, foundation layout, and squaring up angles, walls, and floor plans. It converts easily between feet, inches, and fractions; inches and fractions; decimal feet; and meters. It has a handy auto-off feature and comes with a long-lasting lithium battery and instruction manual. One-year limited warranty.