Law of Cosines SSS Calculator

Enter three sides and solve every triangle angle. Review area, perimeter, validation, and clear steps. Export clean records for class, plans, or field notes.

Calculator

Side a is opposite angle A.
Side b is opposite angle B.
Side c is opposite angle C.
Use cm, m, ft, in, or any label.

Formula Used

The law of cosines finds an angle when all three sides are known.

Angle A: cos(A) = (b² + c² - a²) / (2bc)

Angle B: cos(B) = (a² + c² - b²) / (2ac)

Angle C: cos(C) = (a² + b² - c²) / (2ab)

Area: Area = √s(s - a)(s - b)(s - c), where s = (a + b + c) / 2.

How To Use This Calculator

  1. Enter side a, side b, and side c.
  2. Use the same length unit for all three sides.
  3. Choose degrees or radians for angle output.
  4. Select the decimal places needed in the result.
  5. Press Calculate to view results below the header.
  6. Use CSV or PDF buttons to save the same calculation.

Example Data Table

Side a Side b Side c Expected angle type Common use
7 cm 8 cm 9 cm Acute Classroom example
3 m 4 m 5 m Right Layout check
5 ft 5 ft 6 ft Acute isosceles Design sketch
4 in 7 in 10 in Obtuse Craft planning

A Practical SSS Angle Tool

This calculator helps when all three sides are known. It uses the law of cosines to find each missing angle. It also checks whether the sides can form a real triangle. That check is important. A small input mistake can create impossible geometry.

Why The Method Works

The law of cosines connects one angle with the three sides around it. For angle A, side a sits opposite the angle. Sides b and c meet at that angle. The formula rearranges the side relationship and isolates the cosine value. Then the inverse cosine gives the angle. The same method is repeated for angles B and C.

Useful Extra Results

A good SSS calculator should do more than show angles. This tool also returns perimeter, semiperimeter, area, height values, medians, inradius, and circumradius. These values help in construction sketches, classroom problems, surveying notes, craft planning, and layout checks. The classification note explains whether the triangle is acute, right, obtuse, equilateral, isosceles, or scalene.

Accuracy And Rounding

The calculator uses decimal precision chosen by the user. More decimals give finer results. Fewer decimals make reports easier to read. Very tiny differences may appear when sides are long or decimals are high. That is normal because angle calculations use inverse trigonometric functions.

How To Read The Output

Angles are shown in the selected unit. The sum should be close to 180 degrees, or pi radians. Area is found with Heron's formula. The perimeter is the total outside length. Heights show the perpendicular distance to each side. Medians show the line from each vertex to the midpoint of the opposite side.

Best Use Cases

Use this page when you know three side lengths. Do not use it when only two sides are known. Use consistent units for every side. If sides are entered in meters, area becomes square meters. If sides are entered in feet, area becomes square feet. The exported files are helpful for records, homework, estimates, and review notes.

Input Tips

Measure carefully before entering numbers. Avoid mixing inches with centimeters. Use the same scale for copied plans. When possible, keep original notes nearby. This makes checks easier and prevents typing errors during repeated calculations for every project.

FAQs

What is an SSS triangle calculation?

It is a triangle calculation using three known side lengths. The calculator finds all three angles and extra measurements from those sides.

Which formula finds the angles?

It uses the law of cosines. The formula compares the three side lengths and then uses inverse cosine to find each angle.

Can the sides be in any unit?

Yes. Use one unit for every side. Do not mix inches, feet, meters, or centimeters in the same calculation.

Why does the calculator reject some inputs?

The side lengths must satisfy the triangle inequality. Any two sides must add up to more than the third side.

How is the area calculated?

The area is calculated with Heron's formula after finding the semiperimeter. It works well when all three sides are known.

What does angle classification mean?

It describes the largest angle. A triangle can be acute, right, or obtuse based on the final angle values.

What does side classification mean?

It compares side lengths. Equal sides create equilateral or isosceles results. Three different sides create a scalene result.

Why use CSV or PDF export?

Exports make it easier to keep records. They help with homework, project notes, estimates, and repeated geometry checks.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.