104 LearnersLast updated on June 27th, 2025
Side Angle Side Calculator
A calculator is a tool designed to perform both basic arithmetic operations and advanced calculations, such as those involving trigonometry. It is especially helpful for completing mathematical school projects or exploring complex mathematical concepts. In this topic, we will discuss the Side Angle Side Calculator.
What is the Side Angle Side Calculator
The Side Angle Side Calculator is a tool designed for calculating the unknown side or angles of a triangle when two sides and the included angle are known. This is based on the Side-Angle-Side (SAS) theorem in trigonometry. SAS allows us to solve for the third side of a triangle or other angles if two sides and the angle between them are given.
How to Use the Side Angle Side Calculator
For calculating an unknown side or angle using the SAS calculator, follow the steps below:
Step 1: Input: Enter the two known sides and the included angle.
Step 2: Click: Calculate. By doing so, the given inputs will be processed.
Step 3: You will see the unknown side or angle in the output column.
Tips and Tricks for Using the Side Angle Side Calculator
Mentioned below are some tips to help you get the right answer using the Side Angle Side Calculator.
Know the formula:
Use the Law of Cosines to find the unknown side, which is c² = a² + b² - 2ab * cos(C), where ‘a’ and ‘b’ are the known sides, and ‘C’ is the included angle.
Use the Right Units:
Ensure that the sides and angles are in the right units, such as degrees or radians for angles.
Enter correct Numbers:
When entering the sides and angle, make sure the numbers are accurate. Small mistakes can lead to big differences, especially with larger numbers.
Common Mistakes and How to Avoid Them When Using the Side Angle Side Calculator
Calculators mostly help us with quick solutions. For calculating complex math questions, students must know the intricate features of a calculator. Given below are some common mistakes and solutions to tackle these mistakes.
Mistake 1
Rounding off too soon
Rounding the decimal number too soon can lead to wrong results.
For example, if the calculated side is 15.67, don’t round it to 16 right away. Finish the calculation first.
Mistake 2
Entering incorrect values for the sides or angle
Make sure to double-check the numbers you are going to enter as the sides and the angle. If you enter a side as ‘6’ instead of ‘7’, the result will be incorrect.
Mistake 3
Mixing up the angle units
Ensure the angle is in the correct unit (degrees or radians) as required by the calculator. Using the wrong units will give the wrong result.
Mistake 4
Relying too much on the calculator
The calculator gives an estimate. Real objects may not be perfect, so the answer might be slightly different. Keep in mind that it's an approximation.
Mistake 5
Mixing up the positive and negative signs
Always check that you’ve entered the correct positive (+) or negative (–) signs. A small mistake, like using the wrong sign for a side, can completely change the result. Make sure the signs are correct before finishing your calculation.
For example, if a side is 29 cm, entering -29 cm instead of +29 cm could give you an incorrect result.
Side Angle Side Calculator Examples
Problem 1
Help Rachel find the third side of a triangle if two sides are 8 cm and 11 cm, and the included angle is 60 degrees.
The third side of the triangle is approximately 8.94 cm.
Explanation
To find the third side, we use the Law of Cosines: c² = a² + b² - 2ab * cos(C)
Here, ‘a’ is 8 cm, ‘b’ is 11 cm, and ‘C’ is 60 degrees.
c² = 8² + 11² - 2 * 8 * 11 * cos(60)
c² = 64 + 121 - 176 * 0.5
c² = 185 - 88
c² = 97
c = √97 ≈ 8.94 cm
Problem 2
Two sides of a triangle are 5 cm and 7 cm, with an included angle of 45 degrees. What is the length of the third side?
The length of the third side is approximately 5.41 cm.
Explanation
To find the third side, we use the Law of Cosines: c² = a² + b² - 2ab * cos(C)
Since ‘a’ is 5 cm, ‘b’ is 7 cm, and ‘C’ is 45 degrees, we can find the third side as c² = 5² + 7² - 2 * 5 * 7 * cos(45)
c² = 25 + 49 - 70 * 0.7071
c² = 74 - 49.497
c² = 24.503
c = √24.503 ≈ 5.41 cm
Problem 3
A triangle has sides of 9 cm and 12 cm, and the included angle is 30 degrees. Calculate the third side.
The third side is approximately 6.93 cm.
Explanation
For the third side, we use the Law of Cosines: c² = a² + b² - 2ab * cos(C)
Here, ‘a’ is 9 cm, ‘b’ is 12 cm, and ‘C’ is 30 degrees.
c² = 9² + 12² - 2 * 9 * 12 * cos(30) c² = 81 + 144 - 216 * 0.866
c² = 225 - 186.696
c² = 38.304
c = √38.304 ≈ 6.93 cm
Problem 4
Find the third side of a triangle with sides 10 cm and 15 cm and an included angle of 120 degrees.
The third side is approximately 20.81 cm.
Explanation
Using the Law of Cosines: c² = a² + b² - 2ab * cos(C) ‘a’ is 10 cm, ‘b’ is 15 cm, and ‘C’ is 120 degrees.
c² = 10² + 15² - 2 * 10 * 15 * cos(120)
c² = 100 + 225 + 300 * 0.5
c² = 325 + 150 c² = 475
c = √475 ≈ 20.81 cm
Problem 5
John needs to determine the third side of his triangular garden plot with sides 13 m and 14 m and an included angle of 75 degrees.
The third side of the triangular garden plot is approximately 15.84 m.
Explanation
Using the Law of Cosines: c² = a² + b² - 2ab * cos(C)
Here, ‘a’ is 13 m, ‘b’ is 14 m, and ‘C’ is 75 degrees.
c² = 13² + 14² - 2 * 13 * 14 * cos(75)
c² = 169 + 196 - 364 * 0.2588
c² = 365 - 94.2112
c² = 270.7888
c = √270.7888 ≈ 15.84 m
FAQs on Using the Side Angle Side Calculator
1.What is the formula for finding the third side in SAS?
2.What should I do if I enter an angle as ‘0’?
3.What will be the third side if the sides are 6 cm and 8 cm, with an included angle of 90 degrees?
4.What units are used to represent the sides and angles?
5.Can we use this calculator to find angles as well?
Important Glossary for the Side Angle Side Calculator
- Side: A straight line segment that forms part of a triangle.
- Angle: The space between two intersecting lines or surfaces at or close to the point where they meet.
- Included Angle: The angle formed between two known sides of a triangle.
- Law of Cosines: A formula used to calculate the third side of a triangle when two sides and the included angle are known: c² = a² + b² - 2ab * cos(C).
- Radians: A unit for measuring angles, equivalent to the angle at the center of a circle subtended by an arc equal in length to the radius.
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Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
