Solve Triangle Calculator

Calculate all sides, angles, area, and perimeter of any triangle

SSS – Three Sides (a, b, c) SAS – Two Sides and Included Angle ASA – Two Angles and Included Side AAS – Two Angles and Non-Included Side SSA – Two Sides and Non-Included Angle

Understanding Triangle Solving

A triangle solver calculator is a powerful mathematical tool that helps you find all unknown properties of a triangle when given specific known values. Whether you know three sides, two sides and an angle, or two angles and a side, this calculator can determine all remaining measurements including sides, angles, area, and perimeter.

Why Use a Triangle Solver?

Triangle solving is fundamental in mathematics, engineering, architecture, navigation, and physics. This calculator eliminates the need for manual calculations using trigonometric formulas and provides instant, accurate results for:

• Engineering and Construction: Calculating structural dimensions and load distributions
• Navigation: Determining distances and bearings in triangulation
• Architecture: Designing roof trusses, staircases, and angular structures
• Surveying: Measuring land areas and property boundaries
• Physics: Analyzing force vectors and motion trajectories
• Education: Learning and verifying trigonometry homework

Triangle Solving Methods

1. SSS (Side-Side-Side)

When all three sides are known, you can calculate all angles using the Law of Cosines and then find the area and perimeter.

Law of Cosines:
cos(A) = (b² + c² – a²) / (2bc)
cos(B) = (a² + c² – b²) / (2ac)
cos(C) = (a² + b² – c²) / (2ab)

Example: Given a triangle with sides a = 7 units, b = 8 units, and c = 9 units:
• Angle A ≈ 50.7°
• Angle B ≈ 57.9°
• Angle C ≈ 71.4°
• Area ≈ 26.83 square units
• Perimeter = 24 units

2. SAS (Side-Angle-Side)

When two sides and the included angle are known, you can find the third side using the Law of Cosines, then calculate remaining angles.

Formula:
c² = a² + b² – 2ab·cos(C)
Then use Law of Sines for remaining angles

Example: Given a = 5 units, b = 6 units, and angle C = 60°:
• Side c ≈ 5.57 units
• Angle A ≈ 53.13°
• Angle B ≈ 66.87°
• Area ≈ 12.99 square units

3. ASA (Angle-Side-Angle)

When two angles and the included side are known, the third angle is easily found, then use the Law of Sines to find remaining sides.

Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C)
Third angle: C = 180° – A – B

Example: Given angle A = 40°, side b = 10 units, and angle B = 60°:
• Angle C = 80°
• Side a ≈ 7.43 units
• Side c ≈ 11.39 units
• Area ≈ 42.30 square units

4. AAS (Angle-Angle-Side)

Similar to ASA, when two angles and a non-included side are known, you can find the third angle and then use the Law of Sines.

Example: Given angle A = 50°, angle B = 70°, and side a = 8 units:
• Angle C = 60°
• Side b ≈ 9.82 units
• Side c ≈ 9.05 units
• Area ≈ 34.76 square units

5. SSA (Side-Side-Angle) – Ambiguous Case

When two sides and a non-included angle are known, there may be zero, one, or two possible triangle solutions. This is known as the ambiguous case.

Note: SSA can produce multiple valid triangles. The calculator will identify when this occurs and provide the appropriate solution.

Key Triangle Formulas

Area Calculations

Heron's Formula (when three sides are known):
s = (a + b + c) / 2
Area = √[s(s-a)(s-b)(s-c)]

Using Side and Height:
Area = (base × height) / 2

Using Two Sides and Included Angle:
Area = (1/2) × a × b × sin(C)

Perimeter

Perimeter = a + b + c

Triangle Types

Triangles can be classified by their sides and angles:

By Sides:

• Equilateral: All three sides are equal (all angles are 60°)
• Isosceles: Two sides are equal (two angles are equal)
• Scalene: All sides are different lengths

By Angles:

• Acute: All angles are less than 90°
• Right: One angle equals exactly 90°
• Obtuse: One angle is greater than 90°

Practical Applications

Engineering and Construction

Engineers use triangle solving to design stable structures. For example, calculating the dimensions of roof trusses where the base might be 12 meters, the height 5 meters, and they need to find the length of the sloping beams and the angles at which they connect.

Navigation and Surveying

Surveyors use triangulation to measure distances. If they know the distance between two observation points (baseline) is 500 meters, and they measure angles to a distant landmark of 45° and 60°, they can calculate the exact distance to that landmark.

Physics and Engineering

When analyzing force vectors, engineers often need to solve triangles. For instance, if a force of 100 N is applied at 30° and another force of 150 N is applied at 120°, solving the resulting triangle helps find the resultant force and direction.

Triangle Inequality Theorem

Not all combinations of sides can form a valid triangle. The triangle inequality theorem states that the sum of any two sides must be greater than the third side:

Triangle Inequality:
a + b > c
b + c > a
a + c > b

Valid Triangle: Sides 3, 4, 5 → 3+4=7 > 5 ✓
Invalid Triangle: Sides 1, 2, 10 → 1+2=3 < 10 ✗

Special Right Triangles

45-45-90 Triangle

In this isosceles right triangle, if the legs are length x, the hypotenuse is x√2.

Example: If each leg = 5 units
Hypotenuse = 5√2 ≈ 7.07 units
Area = 12.5 square units

30-60-90 Triangle

In this special triangle, the sides are in the ratio 1 : √3 : 2.

Example: If the shortest side = 4 units
Medium side = 4√3 ≈ 6.93 units
Hypotenuse = 8 units
Area ≈ 13.86 square units

Advanced Concepts

Altitude and Median

The altitude is the perpendicular distance from a vertex to the opposite side. The median is a line segment from a vertex to the midpoint of the opposite side.

Altitude from vertex A to side a:
h = b × sin(C) = c × sin(B)

Median from vertex A:
m_a = (1/2)√(2b² + 2c² – a²)

Inscribed and Circumscribed Circles

Every triangle has an inscribed circle (incircle) and a circumscribed circle (circumcircle).

Inradius (radius of inscribed circle):
r = Area / s, where s = (a+b+c)/2

Circumradius (radius of circumscribed circle):
R = (abc) / (4 × Area)

Tips for Using the Triangle Solver

• Always ensure your input values can form a valid triangle
• Angles should be entered in degrees, not radians
• The sum of all angles in a triangle must equal 180°
• Use consistent units for all side measurements
• For SSA cases, check if multiple solutions exist
• Round intermediate calculations to avoid cumulative errors
• Verify results using the triangle inequality theorem

Common Mistakes to Avoid

• Using wrong angle units: Ensure you're using degrees, not radians
• Violating triangle inequality: Check that side lengths can actually form a triangle
• Ignoring ambiguous cases: SSA may have two valid solutions
• Mixing units: Use consistent units throughout calculations
• Assuming integer results: Triangle measurements often involve decimals

Frequently Asked Questions

What is the minimum information needed to solve a triangle?

You need at least three pieces of information, including at least one side length. Common combinations are: three sides (SSS), two sides and an angle (SAS or SSA), two angles and a side (ASA or AAS).

Can three angles alone determine a triangle?

No. While three angles determine the shape of a triangle, they don't determine its size. You need at least one side length to find the actual dimensions.

What happens if my inputs don't form a valid triangle?

The calculator will detect invalid inputs (such as sides that violate the triangle inequality theorem or angles that don't sum to 180°) and display an error message.

How accurate are the calculations?

The calculator uses precise trigonometric functions and provides results rounded to two decimal places, which is sufficient for most practical applications.

Can I solve a triangle with only two sides?

No. Two sides alone are insufficient. You need either the third side or at least one angle to uniquely determine the triangle.

Conclusion

The triangle solver calculator is an invaluable tool for anyone working with triangular shapes and measurements. Whether you're a student learning trigonometry, an engineer designing structures, a surveyor measuring land, or simply curious about triangle properties, this calculator provides quick, accurate solutions to all your triangle-solving needs. By understanding the underlying principles and formulas, you can better appreciate how triangles work and apply this knowledge to real-world problems.