Triangle Calculator
Results
Dimensions
- Side a
- Side b
- Side c
- Angle A
- Angle B
- Angle C
Properties
- Area
- Perimeter
- Inradius
- Circumradius
- Height a
- Height b
- Height c
Triangle Properties
Classification
- By Sides
- By Angles
Visual Representation
Angle Distribution
Side Lengths
What is a Triangle Calculator?
A triangle calculator is a powerful online tool designed to solve for the missing properties of a triangle, given a sufficient amount of initial information. Whether you have three sides, two sides and an angle, or two angles and a side, this calculator can determine the remaining sides, angles, area, perimeter, heights (altitudes), and other key geometric figures like the inradius and circumradius. It's an essential utility for students, engineers, architects, and anyone working with geometric shapes.
How to calculate area of a triangle?
The method for calculating a triangle's area depends on the information you have:
- Base and Height: The most common formula is $Area = \frac{1}{2} \times base \times height$. This is simple if you know the length of a side and its corresponding altitude.
- Three Sides (Heron’s Formula): When you know the lengths of all three sides (a, b, c), you can use Heron's formula. First, find the semi-perimeter $s = \frac{a+b+c}{2}$. Then, the area is $Area = \sqrt{s(s-a)(s-b)(s-c)}$.
- Two Sides and an Included Angle (Trigonometry): If you have two sides (a, b) and the angle (C) between them, the area can be found using trigonometry: $Area = \frac{1}{2}ab \sin(C)$.
Types of triangles
Triangles can be classified based on their sides and angles:
Classification by Sides:
- Equilateral Triangle: All three sides are equal in length, and all three internal angles are 60°.
- Isosceles Triangle: Two sides are equal in length. The angles opposite these sides are also equal.
- Scalene Triangle: All three sides have different lengths, and all three angles are different.
Classification by Angles:
- Right Triangle: One of the angles is exactly 90°. The side opposite the right angle is called the hypotenuse.
- Acute Triangle: All three internal angles are less than 90°.
- Obtuse Triangle: One of the angles is greater than 90°.
Applications of triangle calculations
Triangle calculations are fundamental in many fields:
- Engineering & Architecture: For designing stable structures like bridges, trusses, and domes. Calculating forces and stresses relies on trigonometric principles.
- Surveying & Navigation: Used in triangulation to determine distances and positions of points on Earth's surface or for GPS navigation.
- Computer Graphics & Video Games: 3D models are constructed from a mesh of triangles (polygons). Calculating their properties is crucial for rendering, lighting, and physics.
- Astronomy: To calculate distances to nearby stars using stellar parallax, which involves measuring the apparent shift of a star against a distant background.
Frequently Asked Questions (FAQ)
How do you calculate the area of a triangle with 3 sides?
You can calculate the area of a triangle with three sides (SSS) using Heron's formula. First, calculate the semi-perimeter (s), which is half the perimeter: $s = (a + b + c) / 2$. Then, the area is the square root of $s \times (s - a) \times (s - b) \times (s - c)$. Our calculator automates this process for you.
Can you solve a triangle with 2 sides and 1 angle?
Yes. If the angle is between the two known sides (Side-Angle-Side or SAS), the triangle is uniquely defined. You can use the Law of Cosines to find the third side and then the Law of Sines to find the remaining angles. If the angle is not between the sides (Side-Side-Angle or SSA), it's the 'ambiguous case' and might result in zero, one, or two possible triangles.
What is the difference between ASA and SAS?
ASA (Angle-Side-Angle) and SAS (Side-Angle-Side) are two different congruence criteria for triangles. In ASA, you know two angles and the side included between them. In SAS, you know two sides and the angle included between them. Both sets of information are sufficient to uniquely determine the size and shape of a triangle.
How accurate is Heron’s formula?
Heron's formula is mathematically exact. Its accuracy in practice depends only on the precision of the input side lengths and the computational precision of the device performing the calculation. For most applications, it is highly accurate.
Can a triangle have two right angles?
No, a triangle in Euclidean (flat) geometry cannot have two right angles. The sum of the interior angles of any triangle is always 180 degrees. If two angles were 90 degrees each, their sum would already be 180 degrees, leaving no room (0 degrees) for the third angle, which is impossible.
How do you classify a triangle by sides and angles?
A triangle is classified by its sides as Equilateral (all sides equal), Isosceles (two sides equal), or Scalene (no sides equal). By its angles, it's classified as Acute (all angles < 90°), Right (one angle = 90°), or Obtuse (one angle > 90°).
