Triangle Tools
Triangle Side Calculator
Enter two angles and the side opposite one of them (AAS) to calculate the remaining sides and angle of the triangle using the Law of Sines.
What is a Triangle Side Calculator?
A Triangle Side Calculator is a tool used to determine the unknown lengths of the sides of a triangle when certain other properties, such as other sides and/or angles, are known. Specifically, this calculator focuses on the scenario where you know two angles and one side (typically Angle-Angle-Side or AAS), and it employs the Law of Sines to find the missing side lengths and the third angle. You might also use a Triangle Side Calculator for Angle-Side-Angle (ASA) or Side-Angle-Side (SAS) scenarios with different formulas (like the Law of Cosines for SAS).
This calculator is particularly useful for students learning trigonometry, engineers, architects, and anyone who needs to solve for triangle dimensions in geometry or real-world problems. It simplifies the process of applying the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.
Common misconceptions include thinking any three known values can solve a triangle (e.g., three angles – AAA only determines similarity, not size) or that there's always one unique solution (the SSA case can be ambiguous). Our Triangle Side Calculator here focuses on the unambiguous AAS case.
Triangle Side Calculator Formula and Mathematical Explanation (Law of Sines)
When you know two angles (let's say A and B) and one side (say, side 'a' opposite angle A – the AAS case), you can find the remaining angle and sides using the following steps:
- Find the third angle (C): The sum of angles in any triangle is 180 degrees. So, Angle C = 180° – Angle A – Angle B.
- Use the Law of Sines: The Law of Sines states:
a / sin(A) = b / sin(B) = c / sin(C)
Where 'a', 'b', and 'c' are the lengths of the sides opposite angles A, B, and C, respectively. - Calculate Side b: From the Law of Sines, b / sin(B) = a / sin(A), so b = (a * sin(B)) / sin(A).
- Calculate Side c: Similarly, c / sin(C) = a / sin(A), so c = (a * sin(C)) / sin(A).
- Calculate Perimeter: Perimeter P = a + b + c.
- Calculate Area: Area can be calculated using Area = 0.5 * a * b * sin(C), or other combinations.
The Triangle Side Calculator performs these calculations automatically.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| A, B, C | Angles of the triangle | Degrees | 0° – 180° (sum = 180°) |
| a, b, c | Lengths of sides opposite angles A, B, C | Length units (e.g., m, cm, inches) | Positive values |
| P | Perimeter | Length units | Positive values |
| Area | Area of the triangle | Square length units | Positive values |
Practical Examples (Real-World Use Cases)
Let's see how our Triangle Side Calculator works with some examples.
Example 1: Surveying Land
A surveyor stands at a point and measures the angle to two distant points, A and B, and also knows the distance to point A from one of the other points after measuring a baseline and some other angles. Let's simplify and assume they know Angle A = 40°, Angle B = 70°, and the side opposite Angle A (side a) is 100 meters.
- Angle A = 40°
- Angle B = 70°
- Side a = 100 m
Using the Triangle Side Calculator:
- Angle C = 180° – 40° – 70° = 70°
- Side b = (100 * sin(70°)) / sin(40°) ≈ 146.19 m
- Side c = (100 * sin(70°)) / sin(40°) ≈ 146.19 m (Since Angle B = Angle C, Side b = Side c – it's an isosceles triangle)
- Perimeter ≈ 100 + 146.19 + 146.19 = 392.38 m
- Area ≈ 0.5 * 100 * 146.19 * sin(70°) ≈ 6868.5 sq m
Example 2: Navigation
A boat observes a lighthouse at Angle A = 35° relative to its course. It travels 5 km (side c), and then observes the lighthouse at Angle B = 50° relative to its new position and course direction (this is more complex, let's use a straight AAS example). Imagine two observation points and a target, forming a triangle. From point 1, angle to target is 35°. From point 2, angle to target is 50°. Distance between point 1 and 2 is 5km, and let's assume this forms side 'a' opposite an angle A=35°, and angle B=50°. Wait, if the distance is between the points, that's not 'a' or 'b'. Let's rephrase: A boat is at point P. It sees a lighthouse L at an angle. It travels, and from point Q, it sees L at another angle. We know PQ and two angles. Let's say Angle PLQ is unknown, Angle LPQ=35, Angle LQP=50, PQ=5km. Angle PLQ = 180-35-50=95. So a=LQ, b=LP, c=PQ=5. a/sin(35)=5/sin(95), b/sin(50)=5/sin(95). No, this is ASA or AAS depending on which side is 5km. If PQ=5km, and we know angles at P and Q relative to L, that's ASA. Let's stick to AAS: Angle A=35°, Angle B=50°, Side a=5km.
- Angle A = 35°
- Angle B = 50°
- Side a = 5 km
Using the Triangle Side Calculator:
- Angle C = 180° – 35° – 50° = 95°
- Side b = (5 * sin(50°)) / sin(35°) ≈ 6.68 km
- Side c = (5 * sin(95°)) / sin(35°) ≈ 8.68 km
- Perimeter ≈ 5 + 6.68 + 8.68 = 20.36 km
- Area ≈ 0.5 * 5 * 6.68 * sin(95°) ≈ 16.63 sq km
How to Use This Triangle Side Calculator
Using the Triangle Side Calculator is straightforward:
- Enter Angle A: Input the value of the first known angle in degrees into the "Angle A" field. This is the angle opposite the known side 'a'.
- Enter Angle B: Input the value of the second known angle in degrees into the "Angle B" field.
- Enter Side a: Input the length of the side opposite Angle A into the "Side a" field.
- Calculate: The calculator will automatically update the results as you type if the inputs are valid. You can also click the "Calculate" button.
- Read the Results: The calculator will display:
- The calculated Angle C.
- The lengths of Side b and Side c.
- The Perimeter of the triangle.
- The Area of the triangle.
- A table summarizing inputs and outputs.
- A bar chart visualizing the side lengths.
- Reset: Click "Reset" to clear the fields to their default values.
- Copy: Click "Copy Results" to copy the main results and inputs to your clipboard.
Ensure that the sum of Angle A and Angle B is less than 180 degrees, and Side a is a positive number, otherwise, you'll see error messages.
Key Factors That Affect Triangle Side Calculator Results
The results from the Triangle Side Calculator are directly influenced by the input values:
- Value of Angle A: This angle, along with Side a, sets the scale for the triangle via the a/sin(A) ratio in the Law of Sines. A larger Angle A (for a fixed Side a) means sin(A) is larger, making the ratio smaller, leading to potentially smaller other sides.
- Value of Angle B: This angle determines sin(B), directly affecting the length of Side b.
- Sum of Angle A and B: This sum determines Angle C (180 – A – B), which in turn affects sin(C) and thus the length of Side c. If A+B is close to 180, C is small, and Side c will be small relative to others if a/sin(A) is moderate.
- Length of Side a: This is the known length and directly scales the other sides. Doubling Side a, while keeping angles the same, will double Side b and Side c, and quadruple the area.
- Accuracy of Input Measurements: Small errors in measuring the initial angles or side can lead to inaccuracies in the calculated values, especially when angles are very small or close to 90 or 180 degrees.
- Units Used: Ensure the unit for Side a is consistent. The output units for Side b, Side c, and Perimeter will be the same, and the Area will be in square units of that measure. The Triangle Side Calculator doesn't convert units, it just uses the numbers provided.
Frequently Asked Questions (FAQ)
- What is the Law of Sines?
- The Law of Sines is a formula relating the lengths of the sides of any triangle to the sines of its angles. It states a/sin(A) = b/sin(B) = c/sin(C).
- When can I use this Triangle Side Calculator?
- This calculator is designed for the AAS (Angle-Angle-Side) case: you know two angles and the side opposite one of them. You can also derive ASA from this (if you know A, c, B, find C=180-A-B, then you have AAS for side c and angles A, B, C).
- What if I know two sides and one angle?
- If you know two sides and the included angle (SAS), you use the Law of Cosines first. If you know two sides and a non-included angle (SSA), it's the ambiguous case, and there might be 0, 1, or 2 solutions; this specific Triangle Side Calculator isn't set up for SSA. You'd need a triangle solver that handles SSA.
- What if I know three sides (SSS)?
- You would use the Law of Cosines to find the angles. See our triangle angle calculator.
- What if I know three angles (AAA)?
- Three angles only define the shape (similarity), not the size of the triangle. You cannot find the side lengths without at least one side.
- Why does the sum of Angle A and Angle B have to be less than 180?
- The sum of all three angles in a Euclidean triangle is always 180 degrees. If two angles already sum to 180 or more, the third angle would be zero or negative, which is impossible for a triangle.
- Can I enter angles in radians?
- No, this Triangle Side Calculator expects angles in degrees. You would need to convert radians to degrees (degrees = radians * 180 / π) before inputting.
- What are the limitations of this calculator?
- It's primarily for the AAS case and assumes a valid triangle can be formed with the given inputs. It doesn't handle the ambiguous SSA case directly or SSS/SAS inputs. For SSS/SAS, check out our general geometry calculator section.
Related Tools and Internal Resources
- Triangle Area Calculator: Calculate the area of a triangle using various formulas (SSS, SAS, base-height).
- Pythagorean Theorem Calculator: Find the missing side of a right-angled triangle.
- Right Triangle Calculator: Solve right triangles given different inputs.
- Triangle Angle Calculator: Find angles given three sides (SSS) using the Law of Cosines.
- Geometry Formulas: A collection of common geometry formulas.
- Math Calculators: Our main hub for various math-related calculators.