Advanced triangle solver with real-time calculations, visual diagrams, and comprehensive analysis. Supports all triangle types and calculation methods.
Enter triangle parameters to see results
Provide at least 3 values (combination of sides and angles)
Master triangle calculations with this comprehensive educational resource covering all fundamental concepts, advanced theorems, and practical applications.
A triangle is one of the most fundamental shapes in geometry, consisting of three vertices connected by three line segments called sides. This simple yet powerful geometric figure forms the backbone of countless mathematical concepts, engineering applications, and real-world problem-solving scenarios. Understanding triangles is essential for mastering geometry, trigonometry, and advanced mathematics.
All three sides are equal in length (a = b = c)
Two sides are equal (e.g., a = b ≠ c)
All sides are different lengths (a ≠ b ≠ c)
All angles are less than 90°
One angle equals exactly 90°
One angle is greater than 90°
The Law of Sines establishes a fundamental relationship between the sides and angles of any triangle. This powerful theorem states that the ratio of each side length to the sine of its opposite angle remains constant throughout the triangle.
Where R is the circumradius (radius of the circumscribed circle)
Given: Triangle ABC with angle A = 30°, angle B = 45°, and side a = 10 units. Find side b.
The Law of Cosines is a generalization of the Pythagorean theorem that applies to all triangles, not just right triangles. It relates the lengths of the sides of a triangle to the cosine of one of its angles.
Alternative forms: a² = b² + c² - 2bc·cos(A) and b² = a² + c² - 2ac·cos(B)
The Pythagorean theorem is perhaps the most famous theorem in mathematics, establishing the relationship between the sides of a right triangle. This fundamental principle has been known for over 2,500 years and forms the basis for distance calculations in coordinate geometry.
Find hypotenuse when legs are known: c = √(a² + b²)
Find leg when hypotenuse and other leg are known: a = √(c² - b²)
Check if triangle is right-angled by testing if a² + b² = c²
Calculating the area of a triangle is fundamental to geometry and has numerous practical applications. Different scenarios require different formulas, and understanding when to use each method is crucial for efficient problem-solving.
Best for: When height is easily measurable
Applications: Basic geometry, construction planning
Note: Height must be perpendicular to the base
Example: Base = 10 cm, Height = 6 cm → Area = 30 cm²
Best for: SAS triangle configurations
Applications: Engineering, navigation, surveying
Note: Angle must be between the two known sides
Example: a = 8, b = 6, C = 30° → Area = 12 units²
Where: s = (a+b+c)/2 (semiperimeter)
Best for: When all three sides are known (SSS)
Applications: Land surveying, architectural design
Advantage: No angle measurements required
Where: r = inradius, s = semiperimeter
Best for: Circle-triangle relationships
Applications: Optimization problems, packing theory
Related: r = Area/s
Useful when circumradius is known or needs to be found
Alternative when side and two angles are known
A median of a triangle is a line segment connecting a vertex to the midpoint of the opposite side. Every triangle has three medians, and they possess remarkable properties that are fundamental to understanding triangle geometry.
Where m_a, m_b, m_c are medians to sides a, b, c respectively
An altitude of a triangle is the perpendicular line segment from a vertex to the opposite side (or extension of the opposite side). The three altitudes of a triangle are concurrent at a point called the orthocenter, which has different locations depending on the triangle type.
Where h_a, h_b, h_c are altitudes to sides a, b, c respectively
For triangle with vertices A(x₁,y₁), B(x₂,y₂), C(x₃,y₃), the orthocenter coordinates can be calculated using the intersection of altitude equations. This is essential in coordinate geometry and analytical applications.
Every triangle has two important associated circles: the circumcircle (circumscribed circle) that passes through all three vertices, and the incircle (inscribed circle) that is tangent to all three sides. These circles reveal deep geometric relationships and have practical applications.
Relates distance between circumcenter and incenter
Passes through nine important points: midpoints of sides, feet of altitudes, and midpoints of segments from vertices to orthocenter.
Triangle mathematics extends far beyond academic exercises, forming the foundation for countless practical applications across engineering, science, technology, and everyday problem-solving scenarios.
Modern computer graphics and 3D modeling heavily rely on triangle mathematics for rendering, animation, and simulation. Understanding these applications helps bridge theoretical knowledge with cutting-edge technology.
With this comprehensive understanding of triangle mathematics, you're equipped to solve complex geometric problems, apply these concepts in real-world scenarios, and appreciate the elegant relationships that govern one of geometry's most fundamental shapes. Practice with our calculator to reinforce these concepts and develop mathematical intuition.
Common questions and detailed answers about triangle calculations, geometry, and our calculator
You need at least 3 pieces of information about a triangle to solve it completely. This can be:
The SSA (Side-Side-Angle) case is called the "ambiguous case" because when you know two sides and an angle opposite one of them, there might be two different triangles that satisfy these conditions. This happens when the angle is acute and the side opposite to it is shorter than the other given side. Our calculator automatically detects and handles these scenarios.
Degrees and radians are two different ways to measure angles:
Our calculator uses double-precision floating-point arithmetic, providing accuracy to approximately 15-16 decimal places. Results are displayed to 4 decimal places by default, which is more than sufficient for most practical applications. For engineering and scientific work, this level of precision exceeds typical measurement accuracy.
These are special line segments in triangles with important properties:
Absolutely! This calculator is designed for both educational and professional use. It's suitable for construction planning, surveying, engineering design, navigation, architecture, and any field requiring precise triangle calculations. The advanced features like circumradius and inradius calculations make it valuable for specialized applications.
The visualization tab displays an accurate, scaled representation of your triangle with labeled sides, angles, and vertices. This helps you verify that your inputs make geometric sense and provides visual context for the numerical results. The triangle is automatically scaled to fit the display while maintaining correct proportions.
The calculator automatically validates your inputs and provides helpful error messages. Common issues include violating the triangle inequality (sum of two sides must be greater than the third side) or having angles that don't sum to 180°. Pay attention to the validation messages for guidance on correcting inputs.
Yes! Use the "Save Result" button to store calculations in your session history. The history tab keeps track of your last 10 calculations with timestamps and triangle classifications. You can also copy individual values to your clipboard using the copy buttons next to each result.
Note: History is stored locally in your browser session and will be cleared when you close the tab or refresh the page.
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