Triangle Calculator: Sides, Angles and Area

Using the calculator

Choose whether the triangle is right-angled or any triangle, then fill in three values and leave everything else blank. The labelling is the standard one used in every exam paper: side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Getting that pairing right is most of the work.

In right-angled mode, angle C is already fixed at 90°, so side c is the hypotenuse and two more values are enough. In any-triangle mode you need three, and at least one has to be a side — three angles describe a shape but not a size.

The degrees and radians switch changes both what you type and what comes back. Everything below GCSE and IGCSE level is in degrees; radians are there for students who have moved on to A-level trigonometry and want the same working in the units their course uses.

• Three sides (SSS) — every angle comes from the cosine rule
• Two sides and the angle between them (SAS) — the cosine rule gives the third side
• Two sides and an angle that is not between them (SSA) — the ambiguous case, which may have two answers
• Two angles and any side (ASA or AAS) — the angle sum, then the sine rule
• Right-angled: any two of the three sides, or one side and one of the acute angles

Which rule it reaches for, and why

The choice of rule is not arbitrary, and the calculator shows the reasoning rather than the result alone. If you have a side together with the angle opposite it, the sine rule has the matched pair it needs. If the angle you know sits between the two sides you know, there is no such pair and the cosine rule is the only way in. Three sides with no angle is the cosine rule again, rearranged to give a cosine.

Right-angled triangles get the treatment they are taught with. Two sides give the third by Pythagoras; the first angle comes from whichever of sine, cosine or tangent connects the two lengths in play, and the tool says which ratio it picked and why that one links those two sides.

One subtlety is worth watching for in the working. After the cosine rule has produced a third side, the calculator applies the sine rule to the shorter of the two remaining sides. The angle facing the shorter side cannot be the obtuse one, so the inverse sine your calculator returns is certain to be the right answer — apply it to the longer side instead and you can quietly lose an obtuse angle.

• Sine rule — a side and its opposite angle are known
• Cosine rule — two sides and the angle between them, or all three sides
• Pythagoras — a right angle and two sides
• SOHCAHTOA — a right angle, one side and one angle, or two sides and a wanted angle
• Area = ½ab sin C — two sides and the angle between them, no perpendicular height needed

What it deliberately will not do

It will not print NaN. Three values that no triangle can satisfy come back as an impossibility with the reason attached: angles that already total 180° or more, a longest side that exceeds the other two combined, or a sine rule that would need a sine above 1. Those are the moments a student most needs to be told that the measurements, not the arithmetic, are at fault.

It will not choose for you in the ambiguous case. Two sides and an angle that does not lie between them can genuinely describe two different triangles, because sin x and sin (180° − x) are equal. Both are returned and labelled. Picking one is a decision about the question — a diagram, an obtuse angle mentioned in the text — not a decision arithmetic can make.

It will not attach units, and it will not draw the triangle to scale. The numbers you type could be centimetres, metres or miles, so every length comes back unit-free and carries whatever unit went in. Values are held at full precision throughout and rounded only for display, so the area is never computed from a rounded side.

Where the marks are usually lost

The most common lost mark in trigonometry has nothing to do with trigonometry: it is a calculator left in radians. An answer of 0.644 where 36.87 was expected almost always means the mode, not the method, and the fix takes one keypress. Setting this tool to radians shows you what that looks like, which is a quicker way to recognise it in an exam than reading about it.

Next comes mislabelling. Side a must face angle A. When a question names its vertices P, Q and R, or gives a diagram with the letters in an unfamiliar order, it is worth relabelling on paper before any rule is written down — the sine rule is unforgiving about pairs.

Rounding early is the third. Round an intermediate angle to one decimal place, feed it into the next line, and the final answer can drift beyond the accuracy the mark scheme allows. Keep full precision on the calculator and round once, at the end, as the working here does.

• Calculator in radians when the question is in degrees
• Side a not matched to angle A after the vertices have been renamed
• Using the sine rule on the longer side and missing an obtuse angle
• Rounding an intermediate value and carrying the error forward
• Assuming a diagram is right-angled because it looks it — only the marked square counts