Geo Triangle Calculator: Solve Any Triangle

A geo triangle calculator solves for the missing sides and angles of a triangle when you enter the known values—typically two sides and an included angle (SAS) or three sides (SSS). You get back the remaining sides, all three angles, perimeter, and area, with the formulas shown step by step.

Formula

The calculator uses two core formulas from trigonometry. When you know two sides and the included angle (SAS), the Law of Cosines finds the third side:

c² = a² + b² − 2ab·cos(C)

where a and b are the known sides, C is the angle between them (in degrees), and c is the side opposite angle C. After finding c, the Law of Sines or a second application of the Law of Cosines gives the remaining angles:

sin(A)/a = sin(B)/b = sin(C)/c

For the area, the formula is:

Area = ½·a·b·sin(C)

All angles are in degrees; all side lengths use the same unit you enter (meters, feet, etc.).

Worked Example

Using the defaults: a = 5, b = 7, C = 60°.

1. Find side c: c² = 5² + 7² − 2·5·7·cos(60°) = 25 + 49 − 70·0.5 = 74 − 35 = 39, so c = √39 ≈ 6.245.
2. Find angle A using the Law of Cosines: cos(A) = (b² + c² − a²)/(2bc) = (49 + 39 − 25)/(2·7·6.245) = 63/87.43 ≈ 0.7206, so A ≈ 43.9°.
3. Find angle B: since A + B + C = 180°, B = 180 − 43.9 − 60 = 76.1°.
4. Area = ½·5·7·sin(60°) = 17.5·0.8660 ≈ 15.16 square units.

The calculator rounds the same way: c = 6.24, A = 43.90°, B = 76.10°, area = 15.16.

When to Use a Geo Triangle Calculator

This tool is handy in surveying, construction, navigation, and physics problems where you need to find distances or angles indirectly. For example, if you know two sides of a triangular plot and the angle between them, you can find the third side without measuring it. It also helps students check homework or visualize how changing one parameter affects the whole triangle.

How to Choose the Right Inputs

You need at least three pieces of information (sides and/or angles) to define a triangle uniquely. The most common cases:

• SAS (side-angle-side): two sides and the included angle — exactly what this calculator uses.
• SSS (side-side-side): three sides; you can compute angles using the Law of Cosines.
• ASA/AAS: two angles and a side; use the Law of Sines.

Make sure the angle you enter is the one between the two sides, not an opposite angle. If you have a different combination, convert it first or use a general triangle solver.

Common Mistakes and Pitfalls

• Unit mix-ups: If you enter sides in meters, the area comes out in square meters. Don’t mix feet and meters without converting.
• Angle mode: This calculator uses degrees. If you’re used to radians, convert first (180° = Ï€ radians).
• Rounding too early: Rounding intermediate steps can throw off final angles by several degrees. Keep full precision until the end.
• Impossible triangles: The sum of any two sides must be greater than the third. If your inputs violate this, the calculator may produce an error or a nonsensical result.

Limitations of the Formula

The Law of Cosines works for any triangle, but it only gives one solution when you have SAS. However, if you use the Law of Sines with SSA (two sides and a non-included angle), you can get the “ambiguous case� with two possible triangles. This calculator avoids that by requiring the included angle. For SSS, the Law of Cosines always gives a unique set of angles (as long as the triangle inequality holds).

Alternatives and Related Tools

If you need to solve for a right triangle, a dedicated right-triangle calculator is simpler. For coordinate geometry, you might want a distance calculator. If you’re working with a triangle’s centroid, circumcenter, or other centers, you’ll need a specialized tool. This calculator focuses on the basic side-angle-area set.

Frequently Asked Questions

Can I use this calculator if I only know three sides?

Yes, but you’d need to enter two sides and the included angle. If you have three sides, you can first find one angle using the Law of Cosines: cos(A) = (b² + c² − a²)/(2bc), then use that angle as the included angle with the other two sides. Alternatively, use a dedicated SSS solver.

What is the ambiguous case?

It occurs when you know two sides and a non-included angle (SSA). Depending on the values, there may be zero, one, or two possible triangles. This calculator avoids that by requiring the included angle, so you always get a unique triangle.

How do I convert the area to different units?

If your sides are in feet, the area is in square feet. To get square meters, multiply by 0.0929. For square yards, divide by 9. Always convert side lengths first if you want the area in a specific unit.

Why does my calculator show an error for some inputs?

Most likely the triangle inequality is violated: one side is longer than the sum of the other two, or the angle is 0° or 180°. Check your numbers and make sure the angle is between 0 and 180 degrees.

Can I use this for non-Euclidean geometry?

No, the Law of Cosines applies to flat (Euclidean) triangles. On a sphere, you’d need spherical trigonometry, which has different formulas.

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in a | Side a | units | 5 | 0
in b | Side b | units | 7 | 0
in C | Angle C (between a and b) | degrees | 60 | 0
out c | Side c | units | sqrt(a^2 + b^2 - 2*a*b*cosd(C)) | 2
out A | Angle A | degrees | acosd((b^2 + c^2 - a^2) / (2*b*c)) | 2
out B | Angle B | degrees | acosd((a^2 + c^2 - b^2) / (2*a*c)) | 2
out area | Area | square units | 0.5*a*b*sind(C) | 2