To find the length of side BC of a triangle, identify what you already know: the other two side lengths and the included angle, or two angles and another side. When two sides and the included angle are known, apply the Law of Cosines, where BC² equals AB² plus AC² minus two times AB times AC times the cosine of angle A. If you know two angles and one side, use the Law of Sines to set up a proportion and solve for BC. This evergreen explainer walks through these core methods with numbered steps, labeled diagrams in words, and simple examples you can reuse for homework, tests, and practical layout calculations.
Core Methods to Find Side BC
Three main toolsets help you find the length of side BC: the Law of Cosines for SAS, the Law of Sines for AAS or ASA, and the Pythagorean theorem when the triangle is right-angled. Choose the method based on which elements are given. If all three sides are involved and one angle is known or can be derived, Law of Cosines is direct. If two angles and any side are known, Law of Sines links angles to their opposite sides. For right triangles with a known hypotenuse and one leg, the Pythagorean theorem gives BC cleanly.
Using the Law of Cosines for SAS
Label your triangle with vertices A, B, and C, where side a lies opposite angle A (side BC), side b lies opposite angle B (side AC), and side c lies opposite angle C (side AB). When you know sides AB and AC and the included angle A, write BC² = AB² + AC² − 2 × AB × AC × cos(A). Compute the cosine, multiply, and subtract to isolate BC², then take the square root to find BC. This approach works for any triangle, not just right triangles, so it is widely applicable and reliable.
Using the Law of Sines for AAS or ASA
When you know two angles and one side, first find the third angle by subtracting the known angles from 180°. Then set up the proportion BC divided by sin(A) equals known side divided by sin(opposite angle). Solve for BC by multiplying sin(A) by the known side length and dividing by the sine of the angle opposite that known side. This method connects angles and side lengths directly without needing right angles.
Special Case: The Right Triangle Shortcut
If triangle ABC is right-angled at A and you know the hypotenuse and one leg, label carefully so the right angle is between the legs. The Pythagorean theorem states that AB² plus AC² equals BC² when angle A is 90° and BC is the hypotenuse. If BC is a leg instead, rearrange so the unknown leg equals the square root of the squared hypotenuse minus the squared known leg. Double-check which side is opposite the right angle before applying the formula.
Worked Examples to Follow
Example 1 (SAS): In triangle ABC, AB is 7 units, AC is 10 units, and angle A is 60°. Compute BC² as 49 + 100 − 2 × 7 × 10 × 0.5, which equals 79. Therefore BC is the square root of 79, roughly 8.89 units. Example 2 (AAS): In triangle ABC, angle A is 40°, angle B is 70°, and side AB is 6 units. Find angle C as 70°, then apply the Law of Sines: BC equals 6 × sin(40°) divided by sin(70°), yielding approximately 4.10 units. These examples show how the same labeled method works with different given information.
Avoiding Common Mistakes
- Ensure your calculator is in the correct angle mode (degrees or radians) before taking cos or sin values.
- Confirm that your known angle is the included angle when you use the Law of Cosines.
- Label sides opposite their correct vertices so proportions in the Law of Sines remain accurate.
- Check that side lengths satisfy the triangle inequality after computing BC.
Practical Applications and Units
In design, architecture, and surveying, knowing how to find the length of side BC of a triangle lets you compute distances that are otherwise hard to measure directly. Always state your answer in the requested units and include units consistently across all inputs. For area calculations, use this side length with the appropriate height or with Heron’s formula. With labeled diagrams and clear variable choices, these steps stay reliable across many real-world problems.