Special Right Triangles
Solve 30-60-90 and 45-45-90 special right triangles from any known side length. Find all sides, hypotenuse, and area with this free online geometry calculator.
About This Calculator
This free online special right triangles calculator solves 30-60-90 and 45-45-90 triangles from any known side length. Simply select the triangle type (30-60-90 or 45-45-90), choose which side you know (short leg, long leg, or hypotenuse for 30-60-90; leg or hypotenuse for 45-45-90), enter its value, and the calculator instantly determines all remaining sides and the area using the fixed side ratios of these special triangles. This tool is ideal for geometry students, trigonometry learners, test preparation for the SAT, ACT, and GRE, and professionals in construction and engineering who need quick triangle solutions.
Special right triangles are right triangles with consistent angle-based side ratios that make calculations predictable without needing the Pythagorean theorem. For 30-60-90 triangles, the sides follow the ratio 1 : sqrt3 : 2, where the hypotenuse is twice the short leg and the long leg equals the short leg times sqrt3. These triangles arise from cutting an equilateral triangle along its altitude. For 45-45-90 triangles (isosceles right triangles), both legs are equal and the hypotenuse equals the leg times sqrt2, following the ratio 1 : 1 : sqrt2. These triangles come from cutting a square along its diagonal. Both types of triangles appear frequently in geometry problems, trigonometry, standardized tests, and real-world applications such as roof rafter calculations, stair design, bridge trusses, structural engineering supports, navigation, and computer graphics rendering.
To use this calculator, select the triangle type, choose which side length you already know from the dropdown menu, enter its numerical value, and click Calculate. The calculator computes the lengths of all sides and the area using the correct ratio formulas. For example, in a 30-60-90 triangle with a short leg of 5 units, the long leg equals 5sqrt3 ≈ 8.66 units, the hypotenuse equals 10 units, and the area equals 1/2 x 5 x 8.66 ≈ 21.65 square units. In a 45-45-90 triangle with a leg of 5 units, the hypotenuse equals 5sqrt2 ≈ 7.07 units and the area equals 1/2 x 5^2 = 12.5 square units. The known side dropdown dynamically adjusts to show only relevant options for the selected triangle type, making the interface clear and easy to navigate.
Frequently Asked Questions
What are special right triangles?
Special right triangles are right triangles with fixed angle ratios that produce predictable side lengths without requiring the Pythagorean theorem. The two most common are the 30-60-90 triangle and the 45-45-90 (isosceles) triangle.
What are the side ratios of a 30-60-90 triangle?
In a 30-60-90 triangle, if the short leg is x, then the long leg is xsqrt3, and the hypotenuse is 2x. The sides are in the ratio 1 : sqrt3 : 2.
What are the side ratios of a 45-45-90 triangle?
In a 45-45-90 triangle, both legs are equal. If each leg is x, then the hypotenuse is xsqrt2. The sides are in the ratio 1 : 1 : sqrt2.
Can I enter any known side to solve the triangle?
Yes, select which side you know (short leg, long leg, or hypotenuse for 30-60-90; leg or hypotenuse for 45-45-90) and enter its length. The calculator derives the remaining sides and area.
Is this tool free?
Yes, all calculators on Calculy are completely free to use with no registration required.
How do you find the area of a special right triangle?
For a 30-60-90 triangle, area = (1/2) x short leg x long leg = (1/2) x x x xsqrt3 = x^2sqrt3/2. For a 45-45-90 triangle, area = (1/2) x leg x leg = leg^2/2.
Where are special right triangles used in real life?
Special right triangles are used in construction (roof rafters, stairs), engineering (bridge trusses, structural supports), navigation, computer graphics, and standardized tests like the SAT, ACT, and GRE.
What is the difference between 30-60-90 and 45-45-90 triangles?
A 30-60-90 triangle has angles 30 deg, 60 deg, and 90 deg with side ratio 1:sqrt3:2, where the hypotenuse is twice the shortest leg. A 45-45-90 triangle has angles 45 deg, 45 deg, and 90 deg with side ratio 1:1:sqrt2, where both legs are equal and the hypotenuse equals leg times sqrt2.