Senast uppdaterad: 19 augusti 2026
SAS-triangelkalkylator
Skapare
Dharmendra SinghGranskare
Skapare
Dharmendra SinghGranskare
Quick Answer
The SAS triangle calculator solves a triangle from two side lengths and the included angle. It uses the law of cosines to find the third side, then the law of sines to compute the remaining angles, and it also reports area and perimeter. Because SAS determines a unique triangle, it is a dependable geometry workflow for surveying, design, and classroom triangle solving.
For an SAS triangle, first use the law of cosines to find the third side, then use the law of sines and the 180-degree angle sum to finish the triangle.
Viktiga Punkter
- SAS means two sides and the included angle are known.
- The law of cosines is the correct first step for SAS.
- The law of sines helps recover the remaining angles once the third side is known.
- The area formula 1/2 ab sin C comes from the same inputs.
- SAS data defines a unique triangle when the measurements are valid.
Skapare
Dharmendra SinghGranskare
Skapare
Dharmendra SinghGranskare
Formel
c = √(a² + b² - 2ab cos C), sin A / a = sin C / c, B = 180° - A - C
Där:
- a=Known side a(units)
- b=Known side b(units)
- C=Included angle(degrees)
- c=Third side opposite angle C(units)
- A=Angle opposite side a(degrees)
- B=Angle opposite side b(degrees)
Lösta exempel
Sides 8 and 11 with included angle 35°
A modest acute SAS example showing how the third side is found before the remaining angles.
- 1Use the law of cosines: c = √(8² + 11² - 2·8·11·cos 35°) ≈ 6.389776.
- 2Use the law of sines: sin A / 8 = sin 35° / 6.389776, so A ≈ 45.899293°.
- 3Compute B = 180° - 35° - 45.899293° ≈ 99.100707°.
- 4Area = 1/2 · 8 · 11 · sin 35° ≈ 25.237363 square units.
Sides 7 and 10 with included angle 120°
An obtuse SAS triangle where the law of cosines makes the long opposite side obvious.
- 1Compute c = √(7² + 10² - 2·7·10·cos 120°) ≈ 14.798649.
- 2Use the law of sines to find A ≈ 24.182474°.
- 3Then B = 180° - 120° - 24.182474° ≈ 35.817526°.
- 4Area = 1/2 · 7 · 10 · sin 120° ≈ 30.310889 square units.
Sides 12 and 9 with included angle 50°
Another acute example showing the same cosines-then-sines workflow.
- 1Compute c = √(12² + 9² - 2·12·9·cos 50°) ≈ 9.282127.
- 2Use sin A / 12 = sin 50° / 9.282127 to get A ≈ 82.032871°.
- 3Then B = 180° - 50° - 82.032871° ≈ 47.967129°.
- 4Area = 1/2 · 12 · 9 · sin 50° ≈ 41.3664 square units.
Introduktion
The SAS Triangle Calculator solves a triangle when you know two side lengths and the included angle between them. This is one of the cleanest triangle-solving cases because it determines a unique triangle: first you use the law of cosines to find the missing third side, and then you use the law of sines to recover the remaining angles. That combination makes SAS problems common in geometry classes, surveying sketches, structural layouts, navigation diagrams, and any setting where two connected lengths and the angle between them are already known.
What SAS means
SAS stands for Side-Angle-Side. The angle must be the angle included between the two known sides, not an angle opposite one of them. That “included angle” detail is what makes the problem a unique triangle-solving case instead of an ambiguous one.
Known: side a, side b, and included angle C.
Unknown: third side c and the two remaining angles.
The included angle is the one touching both known sides.
SAS determines exactly one triangle when the inputs are valid.
Step 1: find the third side with the law of cosines
Because the included angle is known, the law of cosines is the most direct starting formula. It combines the two known sides and the cosine of the included angle to produce the side opposite that angle. Once c is known, the rest of the triangle is much easier to finish.
Step 2: find the remaining angles with the law of sines
After computing the third side, use the law of sines to match one known side-angle pair with an unknown angle. Then use the triangle angle sum to get the final angle. This sequence is numerically stable and easy to verify by hand.
Area formula for an SAS triangle
The same inputs that solve the triangle also give the area directly: Area = 1/2 · a · b · sin C. That is helpful because you can compute the area before or after solving the remaining angles and compare the result as a consistency check.
Worked example
With a = 8, b = 11, and C = 35°, the law of cosines gives c ≈ 6.389776. The law of sines then gives A ≈ 45.899293°, and the angle sum gives B ≈ 99.100707°. The area comes out to about 25.237363 square units.
Common SAS mistakes
The biggest SAS error is using an angle that is not included between the two known sides. Another is switching the side opposite the known angle with one of the adjacent sides when applying the law of sines.
Make sure angle C lies between side a and side b.
Keep degrees and radians separate; this calculator expects degrees.
Do not round the third side too early before using it in the law of sines.
Check that the three final angles add to 180 degrees.
When SAS is the right triangle method
Use SAS when the information naturally comes from two connected lengths and the angle between them. If you know three sides instead, Heron’s formula or a three-side solver is more direct. If you know two angles and one side, an ASA or AAS calculator is the better fit.
Real-world applications
SAS geometry appears in land surveying, roof framing, mechanical linkages, triangulated supports, and navigation sketches. In each case, two measured lengths and a turning angle can define a triangle that still needs a third length or internal-angle breakdown.
Manual solving versus calculator use
Solving one or two SAS problems by hand is the best way to learn the sequence of formulas. After that, a calculator is useful for repeated checking, decimal-heavy measurements, and quick comparison of multiple design options.
Write the law of cosines before substituting numbers.
Carry a few extra decimal places until the end.
Use the area formula as a quick cross-check.
Verify the perimeter after all three sides are known.
Snabbreferenskort
SAS triangle quick reference
Snabbreferens • SAS-triangelkalkylator
First c = √(a² + b² - 2ab cos C), then use sin A / a = sin C / c and B = 180° - A - C.Giltigt intervall: Positive side lengths with an included angle strictly between 0° and 180°.
Vanliga Värden
⚠ Se upp
- •The angle must be the included angle between the two known sides.
- •Angles of 0° or 180° do not form a valid triangle.
- •Rounding the third side too early can distort the later angles.
- •Keep angle units in degrees throughout the calculation.
Proffstips
- →Use the area formula as a quick validation check.
- →Solve the third side before attempting the missing angles.
- →Label the triangle clearly so you do not mix up the opposite side-angle pairs.
- →Compare your result with an estimate: a small included angle should usually give a smaller opposite side.
Vanliga Frågor
What does SAS stand for?
SAS stands for Side-Angle-Side, meaning you know two side lengths and the included angle between them.
Why do you use the law of cosines first?
The included angle makes the law of cosines the direct way to find the missing third side before you solve the remaining angles.
Why use the law of sines after that?
Once one full side-angle pair is known, the law of sines is a convenient way to compute another angle.
Can the included angle be obtuse?
Yes. SAS still works when the included angle is obtuse, and the law of cosines naturally produces the longer opposite side.
Does the angle need to be in degrees?
This calculator expects the input angle in degrees, which matches the labels and worked examples on the page.
How can I check the result?
Add the three angles to confirm they total 180 degrees, and compare the area from one-half ab sin C with the solved side lengths.
When should I use a different triangle calculator?
Use a different calculator when your given information is not SAS, such as SSS, ASA, AAS, or right-triangle data.