Trig Triangle Calculator - Solve Any Right Triangle

A trig triangle calculator is a single tool that solves any right triangle from two values you already know, when the pair includes a side. Type two legs, a leg with the hypotenuse, a side with an acute angle, or the area with one leg, and the panel returns the rest at once using sin, cos, tan, and the Pythagorean theorem.

• • Find a missing side: Type the two legs you have and read the hypotenuse, or type the hypotenuse plus a leg to read the other leg.
• • Find an acute angle: Type two sides to get both acute angles in degrees, or one side with an acute angle for the complementary angle and remaining sides.
• • Recover a triangle from area and one side: If you know the area plus one leg, the tool recovers the missing leg, the hypotenuse, and both angles using area = a*b/2 plus the Pythagorean theorem.
• • Check a textbook answer: Compare your homework against a 3-4-5, 5-12-13, 30-60-90, or 45-45-90 triangle.

A right triangle has two legs (a and b) at the right angle, a hypotenuse (c) opposite it, and two acute angles (alpha and beta). The right angle fixes one corner and the triangle has five remaining quantities, so a side-containing pair is enough for the Pythagorean theorem and SOH-CAH-TOA to pin down the rest.

When the problem already gives the full right triangle and the user wants one missing value, Right Triangle Calculator handles the same leg-hypotenuse-angle workflow with a tighter form.

The calculator reads all six inputs, treats any zero value as unknown, and then picks the formula branch matching whichever side-containing pair you supplied. After the branch runs, it derives the remaining angle from beta = 90 - alpha and the area from a*b/2.

• sideA: Length of leg a, opposite angle alpha. Enter 0 if unknown.
• sideB: Length of leg b, opposite angle beta. Enter 0 if unknown.
• sideC: Length of hypotenuse c, the longest side. Enter 0 if unknown.
• angleAlpha: Acute angle alpha in degrees, between leg b and the hypotenuse.
• angleBeta: Acute angle beta in degrees, between leg a and the hypotenuse.
• area: Triangle area in square units. Enter 0 unless you know it with exactly one leg.

Branch detection runs in source order, so a side-and-angle pair never triggers the two-side branch. Pairs with no side, such as two angles or area with an angle, return a clear error because two angles alone never set the size of a real triangle.

After any branch, the panel re-derives the missing angle using beta = 90 - alpha to prevent round-trip errors. The side-and-side branches rely on the Pythagorean theorem: in any right triangle a^2 + b^2 = c^2, so c = sqrt(a^2 + b^2) and a = sqrt(c^2 - b^2) when the other leg is known (Wikipedia: Pythagorean theorem).

If the next step is to evaluate the six trig functions of one of the resulting angles, Trigonometry Calculator returns sin, cos, tan, csc, sec, and cot together with the unit-circle quadrant and reference angle.

Four ideas make the result panel read correctly for any pair of known values.

Reciprocal trig functions are not needed here because every quantity in the panel comes from the three primary ratios plus the Pythagorean theorem, and the area only needs a*b/2.

When the workflow needs the three primary ratios sin, cos, and tan of one of the angles in the triangle, Sin Cosine Tangent Calculator returns them side by side in one panel.

Six short steps give a complete right triangle from the two values you already have.

1. 1 Pick the two known quantities: Look at the sides, the two acute angles, and the area you know. You need a side-containing pair (two sides, a side with an acute angle, or area with one leg) to solve the triangle.
2. 2 Type the known values: Enter the two known numbers in their matching fields. Leave every other field at 0 so the tool knows those are unknown.
3. 3 Match the unit of the inputs: Use the same length unit for all three sides. Angles must be in degrees; area in matching square units.
4. 4 Watch for a clear error if the input is impossible: A hypotenuse shorter than a leg, or an angle reaching 90 degrees, returns an explicit error explaining what to change.
5. 5 Read all five unknowns at once: The result panel shows leg a, leg b, hypotenuse c, alpha, beta, and area, each rounded to fixed decimal places.
6. 6 Switch one input to a different branch: Change the known pair from two sides to a side and an angle and the tool picks a different branch automatically.

Type sideA = 5 and sideB = 12, leave the rest at 0. The panel shows sideC = 13, alpha = 22.62 deg, beta = 67.38 deg, area = 30, the classic 5-12-13 right triangle.

If the task is the inverse and the user already has all three sides and just wants the area, Triangle Area Calculator returns the same a*b/2 result for a right triangle and the Heron form for any triangle.

Combining all four input families into one form removes the need to switch tools when the known pair changes. The trig triangle calculator fits textbook, exam, and quick use.

• • Side-containing pairs solve the triangle: Type two sides, a side with an acute angle, or the area with one leg, and the panel returns the rest at once.
• • Pythagorean theorem built in: Side-and-side and area-plus-leg branches apply the Pythagorean theorem internally.
• • SOH-CAH-TOA in one place: Side-plus-angle branches use sin, cos, and tan directly.
• • Internal-consistency check: After any branch the panel re-derives beta = 90 - alpha and area = a*b/2.
• • Friendly error messages: Impossible inputs return a specific message that tells the user which input to fix.

The result panel keeps all five unknowns visible at once, so a quick scan tells the user whether the answer matches a textbook ratio such as 1 : sqrt(3) : 2 or 1 : 1 : sqrt(2).

When the answer is supposed to match a 30-60-90 or 45-45-90 ratio, Special Right Triangles Calculator returns the exact side and angle values of those two reference triangles without recomputing them from scratch.

Four factors determine the result and two mark the edge of validity. Knowing them keeps this trig triangle calculator accurate on every branch.

• • The calculator handles right triangles only. Oblique triangles need the law of sines and law of cosines instead of SOH-CAH-TOA.
• • All arithmetic uses double-precision floating point, so inputs above about 1e6 lose precision in the final decimal places.

The Pythagorean theorem and SOH-CAH-TOA are stable pure-math identities, so the result does not depend on a current rate or external standard. The side-and-angle branches use the right-triangle definition of sin, cos, and tan as opposite over hypotenuse, adjacent over hypotenuse, and opposite over adjacent (Wikipedia: Trigonometric functions).

If the triangle turns out to be a 30-60-90 reference case, 30-60-90 Triangle Calculator confirms the side ratio 1 : sqrt(3) : 2 and the angle values 30, 60, and 90 directly.