Triangle Calculator
Our advanced triangle calculator helps you find all properties of a triangle including sides, angles, area, perimeter, and more. Simply enter at least 3 values (including at least one side) and let our calculator do the rest.
Triangle Input
Please provide 3 values including at least one side to the following 6 fields, and click the "Calculate" button. When radians are selected as the angle unit, it can take values such as pi/2, pi/4, etc.
Triangle Visualization
Triangle Results
Triangle Knowledge Quiz
Test your understanding of triangles with these real-world scenario questions!
Great job!
You've completed the Triangle Knowledge Quiz.
About Triangles
A triangle is a polygon that has three vertices. A vertex is a point where two or more curves, lines, or edges meet; in the case of a triangle, the three vertices are joined by three line segments called edges. A triangle is usually referred to by its vertices. Hence, a triangle with vertices a, b, and c is typically denoted as Δabc.
Types of Triangles
Triangles can be classified based on the length of their sides:
- Equilateral Triangle: All three sides have equal lengths
- Isosceles Triangle: Two sides have equal lengths
- Scalene Triangle: None of the sides have equal lengths
Triangles can also be classified based on their internal angles:
- Right Triangle: One of the angles is 90°
- Obtuse Triangle: One of the angles is greater than 90°
- Acute Triangle: All of the angles are less than 90°
Triangle Properties
Here are some important properties of triangles:
- The interior angles of a triangle always add up to 180°
- The sum of the lengths of any two sides of a triangle is always larger than the length of the third side
- It is not possible for a triangle to have more than one vertex with an internal angle greater than or equal to 90°
Triangle Formulas
Area of a Triangle
There are multiple formulas for calculating the area of a triangle:
- Using base and height: Area = ½ × base × height
- Using two sides and the included angle: Area = ½ × a × b × sin(C)
- Using three sides (Heron's formula): Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2
Pythagorean Theorem
For right triangles: a² + b² = c², where c is the hypotenuse.
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)
Law of Cosines
a² = b² + c² - 2bc × cos(A)
Median, Inradius, and Circumradius
Median
The median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. The three medians intersect at the centroid of the triangle.
Inradius
The inradius is the radius of the largest circle that fits inside the triangle. It can be calculated as: inradius = Area / s, where s is the semiperimeter.
Circumradius
The circumradius is the radius of the circle that passes through all three vertices of the triangle. It can be calculated as: circumradius = a / (2 × sin(A)).
Frequently Asked Questions
Tips & Tricks
Understanding Triangle Notation
In triangle notation, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. This convention is used consistently in our calculator.
Working with Special Triangles
For special triangles like 30-60-90 or 45-45-90 right triangles, you can use known ratios to quickly verify your calculations. For example, in a 45-45-90 triangle, the two legs are equal, and the hypotenuse is √2 times the length of a leg.
Checking Your Results
After calculating, always verify that the angles add up to 180° and that the triangle inequality holds (the sum of any two sides is greater than the third side).
Practical Applications
Triangle calculations are used in many fields including construction, engineering, navigation, astronomy, and computer graphics. Understanding how to work with triangles can be valuable in many real-world situations.
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