Triangle Area from Two Sides and the Angle
Base times height over two needs a height, which is often the hard thing to measure. With two sides and the angle between them, a sine gives the area directly. The third side comes from the cosine rule.
Your numbers
Area
41.366cm²
Side c
9.282cm
Perimeter
30.282cm
Formula
Area = Side a × Side b × sin Included angle ÷ 2
The angle has to be the one between the two sides. Any other angle breaks the formula.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| Side a (cm)The length of the first side. | 9 | 0 and up |
| Side b (cm)The length of the second side. | 12 | 0 and up |
| Included angle (°)The angle where the two sides meet; another angle breaks the formula. | 50 | 0 ~ 180 |
Step by step
Quick reference table
Results when only Side a (cm) changes and everything else stays put.
| Side a (cm) | Area (cm²) | Side c (cm) | Perimeter (cm) |
|---|---|---|---|
| 4.5 | 20.683 | 9.738 | 26.238 |
| 6.75 | 31.025 | 9.243 | 27.993 |
| 9 | 41.366 | 9.282 | 30.282 |
| 13.5 | 62.05 | 10.862 | 36.362 |
| 18 | 82.733 | 13.795 | 43.795 |
What each result means
| Result | At default values |
|---|---|
| Area (cm²)The space a shape covers — length squared. | 41.366 |
| Side c (cm)The third side; it must be shorter than the other two combined. | 9.282 |
| Perimeter (cm)The total length around the outside of a shape. | 30.282 |
Common mistakes
The angle has to be the one between the two sides. Any other angle breaks the formula.
Glossary
- Side a
- The length of the first side.
- Side b
- The length of the second side.
- Included angle
- The angle where the two sides meet; another angle breaks the formula.
- Area
- The space a shape covers — length squared.
- Side c
- The third side; it must be shorter than the other two combined.
- Perimeter
- The total length around the outside of a shape.
Frequently asked questions
Q. How is Triangle Area from Two Sides and the Angle calculated?
Area = Side a × Side b × sin Included angle ÷ 2 — Base times height over two needs a height, which is often the hard thing to measure. With two sides and the angle between them, a sine gives the area directly. The third side comes from the cosine rule.
Q. Can you walk through an example?
With Side a 9cm, Side b 12cm, Included angle 50°, the answer is Area 41.366cm².
Q. What do I need to enter?
Enter Side a, Side b, Included angle. The result recalculates as you type, and an empty box counts as zero.
Q. How much does the answer move if I change a number?
Changing only Side a (cm) moves the answer to Side a (cm) 4.5 → Area (cm²) 20.683 and Side a (cm) 18 → Area (cm²) 82.733. The table below lays out five steps.
Q. How are the numbers rounded?
Money is shown to the nearest whole unit, percentages to one decimal place and everything else to two. What you see is rounded; the calculation itself carries the unrounded value forward.
Q. Anything to watch out for?
The angle has to be the one between the two sides. Any other angle breaks the formula.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.