Right Triangle Solver | Sides & Angles
Solve any right triangle given two sides or one side and one angle.
All calculations run live in your browser using standard trigonometric identities. Sides: a=opposite, b=adjacent, c=hypotenuse. Angle C = 90°.
What Is the Right Triangle Solver | Sides & Angles?
The Right Triangle Solver finds all missing sides, angles, area, and trig ratios from just two known values. Two input modes: angle + one side (uses SOHCAHTOA), or two sides (uses Pythagorean theorem for third side, arctan for angles). An SVG diagram labels all sides and angles.
- ›Sides: a = opposite to angle A, b = adjacent to angle A, c = hypotenuse
- ›Angle C = 90° always; angles A and B are the two acute angles (A + B = 90°)
- ›All six trig ratios shown: sin, cos, tan, csc, sec, cot for angle A
- ›Pythagorean verification confirms a² + b² = c²
Formula
SOHCAHTOA, Right Triangle Formulas
sin A = a/c
opp/hyp, SOH
cos A = b/c
adj/hyp, CAH
tan A = a/b
opp/adj, TOA
Pythagorean
a² + b² = c²
Angle B
B = 90° − A
Area
½ × a × b (legs)
How to Use
- 1Select input mode: "Angle + Side" or "Two Sides"
- 2Angle + Side mode: enter acute angle A (0°–90°) and select which side you know
- 3Two Sides mode: select the type of each side (a=opposite, b=adjacent, c=hypotenuse) and enter lengths
- 4Click "Solve Triangle", all sides, angles, area, and trig ratios appear
- 5The SVG diagram shows your solved triangle with labeled sides and angles
- 6Click Reset to start fresh
Example Calculation
Angle A = 30°, hypotenuse c = 10:
b (adjacent) = c × cos(30°) = 10 × 0.86603 = 8.66025
Angle B = 90° − 30° = 60°
Area = ½ × 5 × 8.66025 = 21.6506
Perimeter = 5 + 8.66025 + 10 = 23.66025
sin(30°)=0.5, cos(30°)=0.86603, tan(30°)=0.57735
Check: 5² + 8.66025² = 25 + 75 = 100 = 10² ✓
Special right triangles
30-60-90: sides in ratio 1 : √3 : 2. For hyp=10: opp=5, adj=8.660
45-45-90: sides in ratio 1 : 1 : √2. For leg=5: hyp=5√2≈7.071, area=12.5
Understanding Right Triangle | Sides & Angles
Common Trig Values Reference
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 = 0.500 | √3/2 = 0.866 | 1/√3 = 0.577 |
| 45° | √2/2 = 0.707 | √2/2 = 0.707 | 1 |
| 60° | √3/2 = 0.866 | 1/2 = 0.500 | √3 = 1.732 |
| 90° | 1 | 0 | undefined |
Frequently Asked Questions
What is SOHCAHTOA?
SOHCAHTOA is the foundation of right triangle trigonometry. "Opposite" and "adjacent" are defined relative to the angle you're working with, not the triangle in general.
- ›SOH: sin(A) = opposite / hypotenuse = a/c
- ›CAH: cos(A) = adjacent / hypotenuse = b/c
- ›TOA: tan(A) = opposite / adjacent = a/b
- ›"Adjacent" is the leg next to angle A; "opposite" is the leg across from angle A
What are the six trigonometric ratios?
All six ratios are shown by this calculator for angle A. They are related by reciprocal and quotient identities.
- ›csc A = 1/sin A = c/a (cosecant)
- ›sec A = 1/cos A = c/b (secant)
- ›cot A = 1/tan A = b/a = cos/sin (cotangent)
- ›Pythagorean identity: sin²A + cos²A = 1 always holds
What are the special right triangles?
Special triangles are worth memorizing because they appear constantly in math and give exact results.
- ›30-60-90: sin(30°)=½, cos(30°)=√3/2≈0.866, tan(30°)=1/√3≈0.577
- ›45-45-90: sin(45°)=cos(45°)=√2/2≈0.707, tan(45°)=1
- ›sin(60°)=√3/2≈0.866, cos(60°)=½, tan(60°)=√3≈1.732
- ›The 3-4-5 and 5-12-13 are special integer right triangles (Pythagorean triples)
How do I find an angle if I know two sides?
The inverse functions (arcsin, arccos, arctan) reverse the trig ratios to give the angle from a ratio.
- ›A = arctan(a/b), given legs a and b
- ›A = arcsin(a/c), given opposite a and hypotenuse c
- ›A = arccos(b/c), given adjacent b and hypotenuse c
- ›All three give the same angle; arctan is most convenient given two legs
What is the altitude to the hypotenuse?
The altitude to the hypotenuse is the perpendicular from vertex C to side c. It equals the geometric mean of the two hypotenuse segments it creates.
- ›h = ab/c (product of legs divided by hypotenuse)
- ›h is also the geometric mean: h² = p × q (where p,q are hypotenuse segments)
- ›a² = p × c and b² = q × c (leg² = adjacent segment × full hypotenuse)
- ›Area = ½ × a × b = ½ × c × h (two equivalent area formulas)
Can I use this for non-right triangles?
Every triangle type has an appropriate solver. Right triangles use SOHCAHTOA and the Pythagorean theorem. General triangles use the more general Laws of Sines and Cosines.
- ›Right triangle (one 90° angle): this solver, SOHCAHTOA
- ›Any triangle, SSS: use Law of Cosines, a²=b²+c²−2bc·cos(A)
- ›Any triangle, ASA/AAS/SAS: use Law of Sines, a/sin(A)=b/sin(B)
- ›Any triangle area: use Heron's formula, A=√(s(s−a)(s−b)(s−c))
What are real-world uses of right triangle trigonometry?
Right triangle trig is one of the most applied areas of mathematics, appearing in every quantitative field.
- ›Surveying: angle of elevation from known distance gives height
- ›Architecture: roof pitch = rise/run = tan(pitch angle)
- ›Navigation: compass bearing decomposed into N-S and E-W components
- ›Physics: force vectors decomposed into horizontal and vertical components
- ›Engineering: grade and slope calculations for roads and drainage