How to Calculate Degrees From 3 Sides

When you know all three sides of a triangle, you can calculate its angles using the Law of Cosines. This method is useful in geometry, engineering, and physics when you need to determine angles from known side lengths.

Introduction

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. This formula is particularly useful when you know all three sides of a triangle and need to find the angles. The formula is:

c² = a² + b² - 2ab cos(C)

Where:

c is the length of the side opposite angle C
a and b are the lengths of the other two sides
C is the angle opposite side c

This formula can be rearranged to solve for angle C:

cos(C) = (a² + b² - c²) / (2ab)

Then, angle C can be found by taking the arccosine of the result.

Once you have one angle, you can find the other two angles by applying the Law of Cosines again or by using the fact that the sum of angles in a triangle is 180 degrees.

The Formula

The Law of Cosines provides a direct way to calculate angles when all three sides are known. Here's how it works:

Identify the sides of the triangle: let's call them a, b, and c, where c is the side opposite the angle you want to find.
Plug the side lengths into the formula: cos(C) = (a² + b² - c²) / (2ab).
Calculate the value inside the parentheses first.
Divide by 2ab.
Take the arccosine of the result to get angle C in degrees.

Remember that the arccosine function will give you the angle in radians, so you may need to convert it to degrees using a calculator.

You can repeat this process for the other two angles by treating each side as the side opposite the angle you want to find.

Worked Example

Let's calculate the angles of a triangle with sides a = 5, b = 6, and c = 7.

Step 1: Calculate angle C

Using the formula cos(C) = (a² + b² - c²) / (2ab):

cos(C) = (5² + 6² - 7²) / (2 × 5 × 6) = (25 + 36 - 49) / 60 = (61 - 49) / 60 = 12 / 60 = 0.2

Now, take the arccosine of 0.2: C ≈ 78.46°

Step 2: Calculate angle A

Now treat side a as the side opposite angle A:

cos(A) = (b² + c² - a²) / (2bc) = (6² + 7² - 5²) / (2 × 6 × 7) = (36 + 49 - 25) / 84 = (85 - 25) / 84 = 60 / 84 ≈ 0.714

Now, take the arccosine of 0.714: A ≈ 44.41°

Step 3: Calculate angle B

Finally, treat side b as the side opposite angle B:

cos(B) = (a² + c² - b²) / (2ac) = (5² + 7² - 6²) / (2 × 5 × 7) = (25 + 49 - 36) / 70 = (74 - 36) / 70 = 38 / 70 ≈ 0.543

Now, take the arccosine of 0.543: B ≈ 56.13°

Verification

Let's check that the angles add up to 180°: 44.41° + 56.13° + 78.46° ≈ 179° (the slight difference is due to rounding).

In practice, you might get slight rounding differences when working with real-world measurements, but the angles should always sum to approximately 180°.

Common Errors

When calculating angles from three sides, there are several common mistakes to avoid:

1. Incorrect Side Assignment

Make sure you correctly identify which side is opposite which angle. The side opposite angle C is side c, opposite angle A is side a, and opposite angle B is side b.

2. Forgetting to Convert Radians to Degrees

The arccosine function returns a value in radians, but most people work in degrees. Remember to convert the result to degrees using your calculator.

3. Rounding Errors

When working with real-world measurements, rounding can accumulate. For precise work, keep more decimal places during intermediate calculations.

4. Angle Sum Verification

Always check that your calculated angles sum to approximately 180°. If they don't, there's likely an error in your calculations.

If your angles don't sum to 180°, double-check your side measurements and calculations. The Law of Cosines should always produce valid triangle angles.

FAQ

Can I use the Law of Cosines for any triangle?

Yes, the Law of Cosines works for any triangle, whether it's acute, right, or obtuse. It's particularly useful when you know all three sides but don't know any angles.

Do I need to know any angles to use the Law of Cosines?

No, the Law of Cosines is used when you know all three sides and want to find the angles. You don't need to know any angles beforehand.

What if my triangle has sides that don't form a valid triangle?

If the sum of any two sides is less than or equal to the third side, the sides don't form a valid triangle. In this case, the Law of Cosines won't produce meaningful results.

Can I use the Law of Cosines to find all three angles?

Yes, you can use the Law of Cosines to find one angle, then use the fact that the sum of angles in a triangle is 180° to find the other two angles.

Is there a simpler formula for calculating angles from sides?

The Law of Cosines is one of the most straightforward methods for this calculation. Other methods might involve more complex steps or additional information.