Key takeaways
- A = ½ab·sin(C), using two sides and the angle between them.
- C must be the included angle — the one directly between sides a and b.
- When C = 90°, sin(C) = 1 and the formula becomes the familiar A = ½ × base × height.
- Don't forget the ½, or the area comes out exactly double.
In this guide
What is the SAS area formula?
SAS stands for Side-Angle-Side. Normally, a triangle's area needs a base and a height (Area = ½ × base × height), but the height isn't always easy to measure directly. This formula swaps the height for a second side and the angle between the two sides, which is often much easier to find.
Why does this area formula matter?
Measuring a triangle's height requires it to be perpendicular to the base — not always practical on a slanted plot of land or an awkward shape. Knowing two sides and the angle between them is often much easier to measure on-site, and this formula turns that into an exact area.
Land surveyors rarely get to drop a perfectly vertical measuring stick onto an irregular plot. Two fence lengths and the angle between them are usually far easier to measure than a true height.
The formula
| Symbol | What it means | Example |
|---|---|---|
| A | The area of the triangle | 17.36 m² |
| a, b | Two sides of the triangle | 6 m, 9 m |
| C | The angle between sides a and b, in degrees | 40° |
How to find the area step by step
- 1Multiply side a by side b.
- 2Multiply that result by ½.
- 3Find sin(C) for the included angle.
- 4Multiply everything together:
A = ½ab·sin(C).
Enter your own two sides and included angle below to see the area calculated step by step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting the ½. Without it, you'd get exactly twice the actual area.
- Mixing up degrees and radians when finding sin(C) — check which mode your calculator is using.
- Assuming you need a right angle. This formula works for any triangle, not only right triangles.
Tips and tricks
- If you also want the third side of the same triangle, use the Law of Cosines calculator.
- For a triangle where all three sides are known instead of an angle, see Heron's formula calculator.
- For a triangle where only a base and a true height are known, the plain triangle area calculator is simpler.
Where you'll use it in real life
- Land surveying: a plot with two known fence lengths and the angle between them.
- Sail and roof design: triangular panels where two edge lengths and their angle are known.
- Construction: bracing panels cut at a specific angle rather than a right angle.
- Navigation: the area swept out between two travel legs at a known angle.
Quick summary
The SAS area formula finds a triangle's area from two sides and the angle between them, without ever measuring the height. Multiply the two sides, multiply by sin of the included angle, then halve it. Try the calculator above with your own triangle.
Worked example
A triangular garden plot has two fence sides measuring 6 meters and 9 meters, meeting at a 40° angle. What is the area of the garden?
- 01A = ½ab·sin(C)
- 02A = ½(6)(9)·sin(40°)
- 03A = 27 × 0.6428
- 04A = 17.36 m²
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I find a triangle's area with two sides and an angle?
Use A = ½ab·sin(C), where a and b are the two known sides and C is the angle directly between them.
Does this work for right triangles too?
Yes. If C = 90°, sin(90°) = 1, so the formula becomes A = ½ab — the familiar ½ × base × height formula, since the two sides are then perpendicular.
Why does this formula need the included angle specifically?
The included angle (the one between sides a and b) determines how much the two sides spread apart, which is what sin(C) captures — using a different angle would give the wrong area.
What if I know all three sides but no angle?
Use Heron's formula instead, which finds the area directly from the three side lengths.
