Geometry often requires accurate calculations to solve triangles and find missing sides or areas. Doing this by hand can be time-consuming, especially when applying trigonometric formulas. That’s why we created the Side Angle Side (SAS) Calculator – a free online tool that lets you quickly calculate the third side and the area of a triangle when you know two sides and the included angle.
Side Angle Side Calculator
What is the SAS Rule in Geometry?
In trigonometry, SAS (Side-Angle-Side) refers to a case where you know the length of two sides of a triangle and the measure of the angle between them.
With this information, you can determine:
- The length of the third side (using the Law of Cosines).
- The area of the triangle (using the formula with sine).
This method is widely used in mathematics, engineering, construction, navigation, and physics.
How Does the SAS Calculator Work?
The SAS calculator is based on two main formulas:
- Law of Cosines (to find the third side, C):
c=a2+b2−2ab⋅cos(C)c = \sqrt{a^2 + b^2 – 2ab \cdot \cos(C)}c=a2+b2−2ab⋅cos(C)
Here:
- aaa and bbb are the known sides
- CCC is the included angle (in degrees)
- ccc is the unknown third side
- Area of Triangle (using sine rule):
Area=12⋅a⋅b⋅sin(C)\text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin(C)Area=21⋅a⋅b⋅sin(C)
Features of the Online SAS Calculator
✔️ Easy-to-use input fields for two sides and an angle
✔️ Instant results for the third side and area
✔️ Reset option to start fresh
✔️ Works for all valid triangles (angle between 1° and 179°)
✔️ Accurate up to two decimal places
How to Use the SAS Calculator (Step-by-Step)
Follow these steps:
- Enter Side A → Input the first side length.
- Enter Side B → Input the second side length.
- Enter Angle → Provide the included angle (in degrees).
- Click “Calculate” → Instantly see the third side and area.
- Click “Reset” → Clear the fields and enter new values.
Example Calculation
Let’s say we have:
- Side A = 8 units
- Side B = 6 units
- Included Angle = 40°
Step 1: Find the third side using Law of Cosines
c=82+62−2⋅8⋅6⋅cos(40°)c = \sqrt{8^2 + 6^2 – 2 \cdot 8 \cdot 6 \cdot \cos(40°)}c=82+62−2⋅8⋅6⋅cos(40°) c=64+36−96⋅0.7660c = \sqrt{64 + 36 – 96 \cdot 0.7660}c=64+36−96⋅0.7660 c=100−73.54=26.46≈5.14c = \sqrt{100 – 73.54} = \sqrt{26.46} \approx 5.14c=100−73.54=26.46≈5.14
So, Side C ≈ 5.14 units
Step 2: Find the area using sine formula
Area=12⋅8⋅6⋅sin(40°)\text{Area} = \frac{1}{2} \cdot 8 \cdot 6 \cdot \sin(40°)Area=21⋅8⋅6⋅sin(40°) Area=24⋅0.6428≈15.43\text{Area} = 24 \cdot 0.6428 \approx 15.43Area=24⋅0.6428≈15.43
So, Area ≈ 15.43 square units
The calculator provides these results instantly without manual calculation.
Why Use an SAS Calculator?
- Saves Time: No need for manual trigonometry.
- Accuracy: Prevents calculation errors.
- Education Aid: Helps students learn triangle solving techniques.
- Practical Applications: Useful for architects, engineers, builders, and surveyors.
- Free & Online: No installation needed, works directly in browser.
Applications of SAS Rule
- Construction: Determining support beams in triangular structures.
- Navigation: Calculating distances when two paths and angle are known.
- Surveying: Measuring land plots using triangles.
- Engineering: Designing triangular parts with precision.
- Physics: Solving vector problems involving angles and forces.
20 Frequently Asked Questions (FAQs)
Q1. What does SAS stand for in geometry?
SAS means Side-Angle-Side, where two sides and the included angle of a triangle are known.
Q2. What formula does the SAS calculator use?
It uses the Law of Cosines to find the missing side and the sine formula to calculate the area.
Q3. Can I calculate area with only SAS?
Yes, if you know two sides and the included angle, you can directly find the area using 12absin(C)\frac{1}{2} ab \sin(C)21absin(C).
Q4. Is this calculator accurate?
Yes, results are precise up to two decimal places.
Q5. What is the maximum angle allowed?
The calculator accepts angles between 1° and 179°.
Q6. What happens if I enter 0 or negative values?
The calculator will show an error and ask for valid inputs.
Q7. Does the SAS calculator work for all triangles?
Yes, as long as valid sides and angle values are entered.
Q8. Can I use this for right triangles?
Yes, but a right triangle calculator may be simpler if one angle is 90°.
Q9. Is the SAS method different from SSS or ASA?
Yes. SAS requires two sides and the included angle. SSS requires all three sides, and ASA requires two angles and a side.
Q10. Can this calculator find perimeter?
It gives the third side, so you can easily find the perimeter by adding all three sides.
Q11. Is this tool free to use?
Yes, it is completely free and works online.
Q12. Do I need to install software?
No, it works directly in your web browser.
Q13. What units does it support?
You can use any unit (cm, m, inch, etc.) as long as you use the same unit for both sides.
Q14. Does it show step-by-step solutions?
It instantly shows results, but you can compare with manual formulas.
Q15. Can I use it for obtuse triangles?
Yes, the calculator works for both acute and obtuse triangles.
Q16. What if the angle is 180°?
That would form a straight line, not a triangle. The calculator will reject it.
Q17. Is the angle input in degrees or radians?
The calculator uses degrees.
Q18. Can I calculate more than one triangle at once?
You’ll need to reset and re-enter values for each triangle.
Q19. Is this calculator useful for students?
Yes, it’s an excellent learning tool for geometry and trigonometry.
Q20. How is the area result displayed?
It is displayed in square units, based on the unit you entered for sides.
Conclusion
The Side Angle Side (SAS) Calculator is a powerful yet simple online tool that helps you instantly find the third side and area of any triangle when two sides and the included angle are known. Whether you’re a student learning trigonometry, an engineer designing a structure, or just solving geometric problems, this calculator saves time and ensures accuracy.