Grouped Calculator
Special Right Triangle Calculator
Solve special right triangles quickly.
Calculator Mode
Back To Right Triangle Calculator45-45-90: c Calculator
This calculator finds Hypotenuse c using c = a × √2.
Enter inputs to calculate Hypotenuse c.
Hypotenuse c
Result-
Solution Steps
Formula: c = a × √2
45-45-90: a Calculator
This calculator finds Leg a using a = c / √2.
Enter inputs to calculate Leg a.
Leg a
Result-
Solution Steps
Formula: a = c / √2
30-60-90: Long Leg Calculator
This calculator finds Long Leg using long leg = a × √3.
Enter inputs to calculate Long Leg.
Long Leg
Result-
Solution Steps
Formula: long leg = a × √3
30-60-90: c Calculator
This calculator finds Hypotenuse c using c = 2a.
Enter inputs to calculate Hypotenuse c.
Hypotenuse c
Result-
Solution Steps
Formula: c = 2a
30-60-90 and 45-45-90 Triangle Solver
This page is optimized for 45-45-90 and 30-60-90 triangles, where side ratios are fixed. It gives instant results without full general-triangle solving.
Quick Side Relationships
Each mode applies special-triangle identities like c = a * sqrt(2), a = c / sqrt(2), long leg = short leg * sqrt(3), and c = 2a, depending on the triangle type.
Which Mode To Pick: 45-45-90 triangles have equal legs with hypotenuse equal to leg times root 2. 30-60-90 triangles have sides in the ratio 1 : root 3 : 2. Recognizing these patterns lets you solve without a calculator and verify calculator results instantly.
Where This Calculator Helps
- Fast exam solving when triangle type is known from the angle pattern.
- Quick sanity checks before using full trigonometric workflows.
- Drafting and fabrication tasks that commonly use standard angle triangles.
Input Tips for Better Results
- Confirm whether your triangle is 45-45-90 or 30-60-90 before selecting a mode.
- In 30-60-90 problems, identify the short leg first to avoid ratio mistakes.
- Keep radicals exact during intermediate steps, then round at the end if needed.
Pro Tip: Use special-triangle output as a benchmark to check measurements from field data.
How To Use This Calculator
- Choose the tab that matches your known values before entering numbers.
- Enter values in consistent units and verify that your triangle inputs are valid.
- Review the calculated result, then cross-check with a related calculator when accuracy matters.
- Use related pages such as Right Triangle Side Calculator and Trigonometric Ratios Calculator (Sin, Cos, Tan, Sec, Csc, Cot) for advanced checks.
Calculator Modes Available
- 45-45-90: c: Hypotenuse of a 45-45-90 triangle.
- 45-45-90: a: Leg of a 45-45-90 triangle.
- 30-60-90: Long Leg: Longer leg in a 30-60-90 triangle.
- 30-60-90: c: Hypotenuse in a 30-60-90 triangle is twice the short leg.
Common Mistakes and Quick Fixes
- Mixing units in a single calculation. Keep all values in one unit system before solving.
- Choosing a mode that does not match known inputs. Start with the closest mode to your available values.
- Rounding too early. Keep full precision until the final result output.
- Skipping verification. Recheck using one related calculator before using results in high-stakes work.
Frequently Asked Questions
Answers to the most common right-triangle solving questions.
01 Which special triangles are covered on this page?
This page focuses on 45-45-90 and 30-60-90 triangles, where side ratios are fixed and solving is faster than general-triangle methods.
02 What is the key ratio for 45-45-90 triangles?
If a is a leg, then hypotenuse c = a * √2. Conversely, a = c / √2. Both are available as dedicated quick modes.
03 What is the key ratio for 30-60-90 triangles?
If the short leg is a, then long leg = a * √3 and hypotenuse c = 2a.
04 When should I use special-triangle solving instead of trig?
Use special-triangle modes whenever the angle pattern is known. It is faster, cleaner, and often more exact than full trig workflows.
05 How can I verify special-triangle outputs?
After solving with ratio rules, you can validate with the Side Calculator or Trig Ratios Calculator as a second check.
Related Calculators
Right Triangle Side Calculator
Find any missing side (a, b, or c).
Open Tool
Trigonometric Ratios Calculator (Sin, Cos, Tan, Sec, Csc, Cot)
Calculate sine, cosine, tangent, secant, cosecant and cotangent for any right triangle angle with step-by-step solutions.
Open Tool
Right Triangle Angle Calculator
Find acute angles A and B of a right triangle using inverse sine, cosine and tangent with step-by-step solutions.
Open Tool
Further Reading
Related Special Right Triangle Calculators
Use these focused calculators if you want to solve a specific special right triangle type.