Right Triangle Calculator - Sides and Angles (SOHCAHTOA)

A right triangle has one 90° corner. On this page that corner is always vertex C. The two sides that form the right angle are the legs a and b. The side across from C is the hypotenuse c, always the longest side. Angle A sits at vertex A and is opposite leg a. Angle B is opposite leg b. Because C is 90°, the two acute angles are complementary: A + B = 90°.

Similarity is not enough to print a unique set of lengths. If you only know A = 36.87° and B = 53.13°, every 3-4-5 triangle scaled by any positive factor matches those angles. That is why the solver requires at least one side. Two angles with no length are rejected. Enter exactly two of {a, b, c, A, B} and include a length among them.

The default pair a = 3, b = 4 is the classic integer right triangle: c = 5, A ≈ 36.87°, B ≈ 53.13°, area = 6. You can instead type a hypotenuse and one acute angle, a leg and the angle opposite it, or a leg and the hypotenuse. Oblique triangles (no right angle), SSA ambiguous cases, and the law of sines for a general triangle belong on the triangle calculator. If you only need c from two legs and do not care about angles, the Pythagorean theorem calculator is the narrower tool.

All trigonometry here is in degrees, not radians. A graphing calculator in radian mode will disagree with these angles unless you convert. Area is the Euclidean formula ½ab for legs meeting at the right angle - the same number as ½bc sin A after the missing parts are filled.

Fill exactly two boxes. Leave the other three empty so the tool knows which identity to apply. Results update as you type.

1. Pick a known pair. Two legs; a leg and the hypotenuse; a side and one acute angle. Do not enter A and B alone.
2. Match the labels. a is opposite A, b is opposite B, c is opposite the right angle. Swapping a and b swaps A and B.
3. Read the full solution in the right panel: both legs, hypotenuse, A, B in degrees, and area. Copy the block for a lab notebook or CAD note.
4. Clear to start a new pair. The last entries are stored in this browser for up to 30 days and are never uploaded.

A leg must be strictly shorter than the hypotenuse. Acute angles must lie strictly between 0° and 90°. Desktop layouts can drag the splitter; fullscreen hides the article so only the numbers remain.

SOHCAHTOA is only a memory hook. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. With the right angle at C, opposite A is a and adjacent to A is b, so tan A = a/b. Inverse tangent of 3/4 is arctan(0.75) ≈ 36.87°, which is why the default 3-4-5 triangle prints that angle.

When both legs are known, the hypotenuse is Math.hypot(a, b) (stable Pythagorean length) and A = atan2(a, b) in degrees. When a leg and the hypotenuse are known, the other leg is √(c² − a²) and the opposite angle is arcsin(a/c). When a hypotenuse and an acute angle are known, the opposite leg is c sin A and the adjacent leg is c cos A. When a leg and its opposite angle are known, c = a / sin A and b = a / tan A. If you supply B instead of A, the solver first sets A = 90° − B and then uses the same identities.

Inverse sine, cosine, and tangent each return one acute angle in (0°, 90°) for valid right-triangle ratios. That is enough here because both A and B are acute. A general triangle can have an obtuse angle; do not use this page for that geometry.

Worked default: a = 3, b = 4

• c = √(9 + 16) = 5
• tan A = 3/4 → A ≈ 36.87°
• B = 90° − 36.87° ≈ 53.13°
• area = ½ × 3 × 4 = 6

Worked pair: hypotenuse and angle, c = 10, A = 30

• a = 10 sin 30° = 5
• b = 10 cos 30° = 5√3 ≈ 8.660
• B = 60° - the 30-60-90 ratio 1 : √3 : 2

Two families show up on every geometry exam. A 45-45-90 (isosceles right) triangle has equal legs and hypotenuse leg × √2. Enter a = 1, A = 45 and you should see b = 1, c = √2 ≈ 1.414. A 30-60-90 triangle has sides 1 : √3 : 2. The side opposite 30° is half the hypotenuse. Enter a = 5 and A = 30 to recover b ≈ 8.660 and c = 10.

Integer triples such as 5-12-13, 8-15-17, and 7-24-25 are scaled Pythagorean triples. They satisfy a² + b² = c² with whole-number sides. This calculator does not search for triples; it just solves whatever two independent parts you type. If you enter 5 and 12 as legs, c will be 13 exactly (within floating-point rounding, which is formatted away when the value is an integer).

Slope is closely related: rise over run is tan of the inclination angle. A ramp with rise 3 and run 4 has slope 0.75 and inclination ≈ 36.87° - the same numbers as the default triangle. Convert a grade or a roof pitch with the slope calculator when you care about m = Δy/Δx rather than a full side list.

An angle of elevation is the acute angle between a horizontal line of sight and a line looking up. An angle of depression looks down; if the horizontal lines are parallel, the depression angle equals the elevation angle at the other end (alternate interior angles). Both sit in a right triangle: opposite side = vertical separation, adjacent side = horizontal distance, hypotenuse = line of sight.

Surveying a tree: you stand 40 m from the trunk, the clinometer reads 28° elevation to the top, and your eye is 1.6 m above ground. The height above eye level is 40 tan 28° ≈ 21.27 m, so the tree is about 22.9 m tall. Enter b = 40 and A = 28 (or a as the opposite height once you have it) to check the missing parts. The same pattern sizes a guy-wire, a ladder against a wall, and the guyed mast of an antenna.

Navigation and aviation use the same ratios for short distances where the Earth can be treated as flat. A glide slope of 3° over a 5 km ground run has altitude 5000 tan 3° ≈ 262 m. Long-range great-circle problems are not right triangles on a plane; do not force them into this solver.

Construction: a rafter is the hypotenuse; run and rise are the legs. A 6-in-12 roof pitch is a right triangle with b = 12 and a = 6, so A = arctan(0.5) ≈ 26.57°. Stair stringers follow the same pair: total rise and total run fix the hypotenuse length of the stringer before nosing details. Always keep units consistent - mixing feet and inches without conversion is the usual field error, not the trig itself.

If the given data are two sides and you only need the third side, Pythagoras is sufficient. You never have to mention an angle. If the given data include an acute angle, or you must recover an angle from two sides, you need sine, cosine, or tangent (or their inverses). This page always reports both the missing side and the missing angles so a homework “solve the triangle” prompt is finished in one pass.

A common exam trap is using tan with the wrong opposite and adjacent pair after rotating the sketch. Re-label from the angle you actually know: opposite is the side that does not touch that angle (except via the hypotenuse), adjacent is the other leg. Another trap is applying the law of sines to a right triangle unnecessarily. It still works - a/sin A = c/sin 90° = c - but it is just sin A = a/c in disguise.

Rounding: intermediate rounding of 36.9° instead of 36.87° will drift the last side by a few hundredths. This calculator keeps full IEEE double precision internally and displays angles to two decimal places, which matches typical SOHCAHTOA homework. If your teacher wants nearest degree, round A and B after you copy them, not before you compute c.

Use the Pythagorean theorem calculator when the question is only “find the missing side of a right triangle” and angles will not be graded. Use the triangle calculator for SSS, SAS, ASA, AAS, or SSA with a possible obtuse angle and the ambiguous case. Use the slope calculator when the input is two points or a rise/run pair and the output should be m, distance, and inclination rather than labeled sides a, b, c.

Area of a non-right triangle needs Heron or ½ab sin C; those identities live on the general triangle page. This right-angle tool will not accept an angle of 91° or a side that cannot close a Euclidean triangle. If a word problem is actually an isosceles triangle that is not right, splitting the base into two right triangles is the usual hand method - solve each half here after you draw the altitude.

Recovering an angle from two sides is inverse trigonometry: A = arctan(a/b) when both legs are known, A = arcsin(a/c) when the opposite leg and hypotenuse are known, A = arccos(b/c) when the adjacent leg and hypotenuse are known. All three agree on a consistent triangle; they disagree if you pair the wrong side with the wrong ratio. A calculator left in radian mode will print 0.64 instead of 36.87 for the 3-4-5 angle - that is not a different triangle, it is the same angle in radians (π/5 ≈ 0.628, close to arctan(0.75) ≈ 0.644). This page never switches to radians.

Units cancel in a ratio, so a and b may be metres, feet, or pixels as long as they match. Area then has those units squared. Mixing 3 m and 4 cm without converting yields a nonsense hypotenuse of “5” in mixed units. Word problems that give a bearing or a compass heading are still right triangles after you draw east-north axes: the northward leg and the eastward leg are a and b, the displacement is c, and the heading off east is A or B depending on which axis you called adjacent.

Ambiguous SSA does not occur for a right triangle in the same way it does for an acute/obtuse pair: once C is 90° and you have a legal pair including a side, there is one Euclidean solution. Illegal pairs are a ≥ c, an angle of 0° or 90° or more, or two angles with no side. Those are errors here, not a second triangle. If a story problem is not right-angled, stop and use the general triangle calculator with the law of sines or cosines.

Significant figures: reporting A = 36.8698976458° is honest internally and ridiculous on a quiz. Two decimal degrees is the display convention. If a course wants nearest tenth, round after you copy. Do not round a to 3.0 and b to 4.0 mid-way if the originals were 3.14 and 4.02; feed the given digits and round the outputs once.