How to Solve a Triangle from Three Parts

How do you solve a triangle from three parts?

You need exactly three of the six parts, and they cannot all be angles. Sides are named a, b, and c. The angle opposite each side uses the same letter in capitals. Fill three boxes in the triangle calculator and leave the rest blank. The missing sides and angles come back as cards, and the figure is drawn with the same scale on both axes so a right angle looks square.

Three sides is SSS. Two sides and the angle between them is SAS. Two angles and the side between them is ASA. Two angles and a side that is not between them is AAS. Two sides and an angle that is not between them is SSA, and that last pattern can fit two triangles. The page names which pattern it used, so you can match the result to the line on the sheet.

A chat reply can quote the formulas. It will not keep both SSA figures on one scale while you change a side and watch the second triangle appear or vanish. That is the job of this page. The arithmetic stays in the browser. Nothing is uploaded.

When do you use the law of cosines?

Use the law of cosines when you know all three sides, or two sides and the included angle. For sides a, b, and c, the angle opposite a satisfies cos A = (b² + c² − a²) / (2bc). The same pattern, with the letters rotated, gives B and C. Once one angle is known from three sides, the other two follow, and they must add to 180 degrees with the first.

SAS is the same identity rearranged. If you know sides a and b and the included angle C, the missing side is c² = a² + b² − 2ab cos C. Try a = 3, b = 4, and C = 90 degrees. Cosine of 90 is 0, so the last term drops out and c is 5. That is the familiar 3-4-5 triangle, reached from two sides and the right angle instead of from three lengths.

The law of cosines does not produce a second candidate. Cosine is one-to-one between 0 and 180 degrees, so each valid side triple has one angle at each vertex. If the three lengths cannot meet, the formula is never reached. Each side must be shorter than the other two added together. Sides 2, 3, and 6 fail that test, and the page says so instead of printing a nonsense angle.

When do you use the law of sines?

Law of sines versus law of cosines comes down to the parts you know. Use the law of sines when you already know an angle and the side opposite it, and you need another pair. The rule is a / sin A = b / sin B = c / sin C. ASA and AAS both start by finding the third angle, because the three angles of a triangle add to 180 degrees. With three angles and one side, every other side is a scale factor times the sine of its angle.

Try angles A = 40 degrees and B = 60 degrees, with side c = 10 between them. That is ASA. Angle C is 80 degrees. Side a is then 10 times sin 40 / sin 80, and side b is 10 times sin 60 / sin 80. If the known side had been a instead of c, the same angles would be AAS: the known side sits opposite a known angle, not between the two known angles. The numbers change. The method does not.

Two angles that already add to 180 degrees or more leave no room for the third. The page stops there. It will not invent a negative angle or a side of length zero to force a picture onto the screen.

What is the ambiguous SSA case?

SSA means two sides and an angle that is not sandwiched between them. Sine of an acute angle equals sine of its supplement, so the law of sines can return two legal angles. One SSA triangle versus two is decided by a height test. Drop a perpendicular from the third vertex to the side of the given angle. Call that height h. If the opposite side is shorter than h, it never reaches, and there is no triangle. If it equals h, there is one right triangle. If it is longer than the other given side, there is one triangle. Between those, there are two.

The standard classroom numbers are angle A = 30 degrees, opposite side a = 7, and other side b = 10. The height is b times sin 30, which is 5. Side a is longer than 5 and shorter than 10, so both answers survive. One angle B is about 45.585 degrees. The other is about 134.42 degrees. The two angle C values add to 120 degrees, because each is what remains after 30 degrees and its own angle B. Both figures share side a and side b. Side c differs.

A solver that prints only the acute angle is the failure this page is built to show. The second triangle is named in a caution line, and it is drawn dashed on the same axes. When the inputs allow only one triangle, that line stays quiet. An obtuse given angle cannot be ambiguous: the opposite side has to be the longest side, or there is no triangle at all.

When does the Pythagorean theorem apply?

The Pythagorean theorem applies only when one angle is 90 degrees. Then the square of the hypotenuse equals the squares of the other two sides. Sides 3, 4, and 5 satisfy it, and so do 5, 12, and 13, and 8, 15, and 17. The right angle sits opposite the longest side. Enter 3, 4, and 5 with the angles blank and the card for the angle opposite 5 reads 90 degrees.

A near miss is not a right triangle. Sides 3, 4, and 6 form a triangle, because each side is shorter than the other two combined, but 9 + 16 is not 36. The angle opposite 6 is obtuse. The law of cosines still solves it. The Pythagorean theorem does not, and the page will not label that figure as right.

If you know the right angle and one side, you still need a second side or a second angle before the third part is determined. A lone hypotenuse is a family of right triangles, not one of them. Add the other leg, or add an acute angle, and the rest follows.

For example, check a 3-4-5 triangle against the Pythagorean theorem before you trust a square corner: 3² + 4² = 5². Input: sides 3, 4, and 5. Output: the angle opposite 5 is 90 degrees. Change the 5 to 5.1 and that angle reads about 92.41 degrees, so the corner is not square.

Common mistakes

What should you open next?

After the triangle is solved, an angle in radians is easier to trust on the unit circle. A quadratic that showed up in the same problem set belongs in the quadratic equation solver. A side ratio you want to keep exact, such as 3/5 instead of 0.6, belongs in the fraction calculator.

This page does not compute area, and it does not accept a triangle that is only a shape. Bring three parts that fix the size, read both answers when SSA is ambiguous, and leave the fourth box empty.