Triangle area with sine is one of 3 triangles formulas in the trigonometry section of this library, and it is used at high school level.
Why triangle area with sine works
Area is half the base times the height. Take side a as the base; the height from the far vertex down to it measures b sin C, because b is the hypotenuse of the little right triangle formed by that height. Substituting the height gives one half a b sin C, so this is the familiar area rule with the height rewritten.
What each symbol means
$a,b$ enclose angle $C$.
Triangle area with sine: when it holds
The angle must be the included angle between the chosen sides.
When it stops applying
The area alone cannot tell you the shape, because sin C and sin(180° − C) are equal. Sides 8 and 10 give an area of 20 whether the included angle is 30 degrees or 150 degrees, even though the second triangle is much wider.
Triangle area with sine: a worked example
Sides $8,10$ with included angle $30^\circ$ give $A=20$.
The mistake to avoid
What people do: Students grab any angle in the triangle and multiply it against the two side lengths they were handed.
Why it goes wrong: Only the angle wedged between those two sides controls how wide the triangle opens. Any other angle gives a height that does not belong to the base you chose.
Do this instead: Check that your angle touches both chosen sides. With sides 8 and 10 and the 30-degree angle between them, the area is one half times 8 times 10 times 0.5, which is 20 square units.
Triangle area with sine: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a,b$ enclose angle $C$.
- Check the conditions before substituting. The angle must be the included angle between the chosen sides.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Triangles
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where triangle area with sine comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Area of a Triangle — the lesson behind this formula: choose a base-height, trigonometric, or side-length formula.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about triangle area with sine
What do I do if I only know the three sides?
Use Heron's formula, or first find one angle with the law of cosines and then come back here. Both routes work; Heron's skips the trigonometry entirely if you already have all three lengths.
Does it matter which two sides I pick?
No, as long as the angle you use is the one between them. All three pairings give the same area, which makes for a quick way to double-check an answer.
Why is the answer largest at 90 degrees?
Because sin C peaks at 1 there. With two sides fixed, opening the hinge past a right angle starts folding the triangle flat again, and the area shrinks back toward zero at 180 degrees.
Can I use this for a parallelogram?
Yes, and you drop the one half. A parallelogram is two copies of the triangle glued along a diagonal, so its area is a b sin C with the same two sides and the angle between them.