Triangle Dimensions Calculator: Find Sides & Angles

This calculator works out the missing side lengths and angles of a triangle when you enter any three known values, such as two sides and an included angle, or two angles and one side. It uses the standard laws of sines and cosines to return the remaining dimensions, the area, and the perimeter. You get a complete solution set for any valid triangle you specify.

Formula

The calculator relies on two fundamental relationships in any triangle. The Law of Cosines relates the three sides to one angle:

c² = a² + b² − 2ab·cos(C)

where a, b, and c are side lengths and C is the angle opposite side c. The Law of Sines relates sides to their opposite angles:

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

where A, B, C are the angles opposite sides a, b, c respectively. The sum of angles is always 180° (Ï€ radians). The area is computed as ½·b·c·sin(A) when two sides and the included angle are known, or via Heron’s formula when all three sides are known.

Worked Example

Using the default values: side a = 5 units, side b = 7 units, and angle A = 50° (opposite side a). We need to find side c, angle B, angle C, area, and perimeter.

1. Use the Law of Sines to find angle B: sin(B) = b·sin(A)/a = 7·sin(50°)/5 = 7·0.7660/5 = 1.0724. Since sin(B) > 1, this input is impossible—no triangle exists. The calculator would flag that. For a valid example, change angle A to 30°.
2. With A=30°, sin(B) = 7·0.5/5 = 0.7, so B = arcsin(0.7) ≈ 44.43°.
3. Then C = 180° − 30° − 44.43° = 105.57°.
4. Side c = a·sin(C)/sin(A) = 5·sin(105.57°)/sin(30°) = 5·0.9634/0.5 = 9.634 units.
5. Area = ½·b·c·sin(A) = 0.5·7·9.634·0.5 = 16.86 square units.
6. Perimeter = a + b + c = 5 + 7 + 9.634 = 21.63 units.

The calculator’s first output (area) for these inputs would show 16.86, matching the manual calculation.

How to Enter Your Triangle

You can enter any combination that uniquely defines a triangle. The most common are:

• Side-Side-Side (SSS): Enter all three sides. The calculator uses the Law of Cosines to find each angle.
• Side-Angle-Side (SAS): Enter two sides and the included angle (the angle between them). It finds the third side first, then the other angles.
• Angle-Side-Angle (ASA) or Angle-Angle-Side (AAS): Enter two angles and one side. The third angle is immediate, then the Law of Sines gives the remaining sides.
• Side-Side-Angle (SSA): This is the ambiguous case. Enter two sides and a non-included angle. The calculator may return one, two, or zero solutions. Check the outputs carefully.

Leave the unknown fields blank. The calculator will solve for all missing values automatically. Angles must be in degrees for this version.

Common Mistakes and How to Avoid Them

Unit mix-ups are the most frequent error. If you enter sides in meters, the area will be in square meters and the perimeter in meters—keep all lengths in the same unit. Angles must be in degrees, not radians, unless you convert. Another common issue is entering the wrong angle: the angle must be opposite the side you pair it with in the Law of Sines. For example, if side a = 5 and you enter angle B = 50°, the calculator will use the Law of Sines with side b (which you might not know) and angle A (which you might not have entered). Always double-check which angle is opposite which side.

Rounding too early can also cause errors. If you round intermediate angles to whole degrees, the final side may be off by several percent. The calculator keeps full precision internally and only rounds the displayed results.

Limits of the Formula

The laws of sines and cosines work for any triangle—acute, obtuse, right, or scalene—as long as the triangle exists. The main limitation is the ambiguous case (SSA): when you give two sides and a non-included angle, there can be two different triangles that satisfy the data. The calculator will attempt to show both if they exist, but you must check the results for plausibility. Also, if the sum of any two sides is less than or equal to the third, no triangle exists; the calculator will indicate an error. The formulas assume Euclidean (flat) geometry; they do not apply to spherical or hyperbolic triangles.

Why Use This Calculator Instead of Manual Math?

Manual calculation is fine for simple right triangles, but for anything else the algebra becomes tedious and error-prone. This calculator gives you instant results for any valid combination, which is useful for construction projects, drafting, 3D modeling, navigation, or solving geometry homework. It also computes the area and perimeter automatically, saving extra steps. However, it is a tool for learning and verification—understanding the underlying formulas helps you catch impossible inputs and interpret the results correctly.

Frequently Asked Questions

How do I find the third side of a triangle if I know two sides and the included angle?

Use the Law of Cosines: c² = a² + b² − 2ab·cos(C), where C is the angle between sides a and b. Take the square root to get side c. For example, if a=3, b=4, and C=60°, then c² = 9+16−24·0.5 = 13, so c ≈ 3.61.

What is the ambiguous case in triangle solving?

When you know two sides and a non-included angle (SSA), the given data may produce zero, one, or two possible triangles. This happens because the sine function is positive in both the first and second quadrants. For instance, if you know side a, side b, and angle A, then sin(B) = b·sin(A)/a. If sin(B) > 1, no triangle exists; if sin(B) = 1, one right triangle; if sin(B) < 1, there are two possible angles B (one acute, one obtuse) that both work, leading to two different triangles.

Can I use this calculator for right triangles?

Yes, absolutely. For a right triangle, you can enter the two legs and set the right angle as 90°, or enter one leg and the hypotenuse with the 90° angle. The calculator will apply the same laws, which simplify to the Pythagorean theorem for the sides. For a right triangle, you can also use the built-in trigonometric functions directly, but this calculator handles it automatically.

What if I only know the three angles?

Knowing only the three angles does not determine a unique triangle—it determines an infinite family of similar triangles with different sizes. You need at least one side length to specify the scale. The calculator requires at least one side to compute the other dimensions.

How does the calculator handle units?

The calculator treats all side inputs as the same unit; you do not need to specify it. The outputs for area and perimeter will be in the square of that unit and that unit, respectively. For example, if you enter sides in feet, the area is in square feet and the perimeter in feet. If you mix units (e.g., meters and centimeters), the results will be meaningless—convert all sides to the same unit first.

Why does the calculator show an error for some inputs?

An error indicates that no triangle can exist with the given dimensions. This happens when the sum of any two sides is less than or equal to the third side (violating the triangle inequality), or when the Law of Sines gives a sine value greater than 1. Double-check your numbers for typos or unit inconsistencies.

Related

• Dimensions to Square Footage Calculator
• Dirt Volume Calculator: How Much Soil Do You Need?
• Percentage Change Calculator: Difference & Formula

in a | Side a | units | 5 | 0
in b | Side b | units | 7 | 0
in c | Side c | units | - | 0
in A | Angle A (opposite a) | degrees | - | 0
in B | Angle B (opposite b) | degrees | - | 0
in C | Angle C (opposite c) | degrees | - | 0
out area | Area | sq units | 0.5 * b * c * sind(A) | 2
out perimeter | Perimeter | units | a + b + c | 2