Find third side of triangle given 2 sides and angle calculator
The Correct Answer and Explanation is:
To find the third side of a triangle when you are given two sides and the angle between them, you can use the Law of Cosines.
Law of Cosines Formula:
If a triangle has sides a, b, and c, and angle C is the angle between sides a and b, then the third side c can be calculated as:c=a2+b2−2ab⋅cos(C)c = \sqrt{a^2 + b^2 – 2ab \cdot \cos(C)}c=a2+b2−2ab⋅cos(C)
Example:
Let’s say you are given:
- Side a = 7 cm
- Side b = 9 cm
- Included angle C = 60 degrees
Now apply the Law of Cosines:c=72+92−2⋅7⋅9⋅cos(60∘)c = \sqrt{7^2 + 9^2 – 2 \cdot 7 \cdot 9 \cdot \cos(60^\circ)}c=72+92−2⋅7⋅9⋅cos(60∘)c=49+81−126⋅0.5c = \sqrt{49 + 81 – 126 \cdot 0.5}c=49+81−126⋅0.5c=130−63c = \sqrt{130 – 63}c=130−63c=67c = \sqrt{67}c=67c≈8.19 cmc \approx 8.19 \text{ cm}c≈8.19 cm
Explanation
When you are given two sides of a triangle and the angle between them, you cannot apply the Pythagorean Theorem directly, since it only works for right triangles. In general triangles, especially when the angle is not 90 degrees, you must use the Law of Cosines to find the missing side. This rule adjusts the Pythagorean Theorem by subtracting the cosine of the included angle, scaled by twice the product of the two given sides.
This method works because the Law of Cosines generalizes the relationship between the lengths and angles in a triangle. The cosine function helps measure how much the shape of the triangle deviates from being a right triangle. For example, if the angle is exactly 90 degrees, the cosine of 90 degrees is zero, and the formula simplifies to the Pythagorean Theorem. But when the angle is more or less than 90 degrees, the cosine adjusts the final result accordingly.
This formula is essential in geometry and trigonometry because many real-life triangles are not right-angled. Whether designing a roof, calculating distances in navigation, or solving physics problems, the Law of Cosines provides the right tool for finding unknown sides or angles.
Once you plug in the given side lengths and the included angle, calculate the cosine of the angle, multiply everything accordingly, and take the square root of the result. This will give you the third side of the triangle.
