Find Values of Triangle Calculator – Solve Any Triangle

Find Values of Triangle Calculator

Triangle Solver

Select known values (3 total):

Enter the lengths of the three sides:

Side a:

Side b:

Side c:

Enter two sides and the angle between them:

Side 1 (e.g., b):

Angle (degrees, e.g., A):

Side 2 (e.g., c):

Assuming angle is between Side 1 and Side 2.

Enter two angles and the side between them:

Angle 1 (degrees):

Side (between angles):

Angle 2 (degrees):

Enter two angles and a side NOT between them:

Angle 1 (degrees):

Angle 2 (degrees):

Side (not between):

Enter two sides and an angle NOT between them (ambiguous case):

Side 1 (e.g., a):

Side 2 (e.g., b):

Angle (degrees, e.g., B):

SSA can have 0, 1, or 2 solutions. The calculator will attempt to find them.

What is a Find Values of Triangle Calculator?

A Find Values of Triangle Calculator, also known as a triangle solver, is a tool used to determine the unknown sides, angles, area, perimeter, and other properties of a triangle based on a minimum of three known values. Whether you have three sides (SSS), two sides and the included angle (SAS), two angles and a side (ASA or AAS), or even the ambiguous case of two sides and a non-included angle (SSA), this calculator can find the missing pieces.

This tool is invaluable for students learning geometry and trigonometry, engineers, architects, surveyors, and anyone who needs to solve triangle-related problems. It eliminates manual calculations using the Law of Sines, Law of Cosines, and other trigonometric formulas, providing quick and accurate results.

Common misconceptions include thinking any three values will define a unique triangle (SSA can yield two) or that all triangles are simple right-angled triangles.

Triangle Formulas and Mathematical Explanation

To find the values of a triangle, we use several fundamental laws and formulas:

Law of Sines: Relates the sides of a triangle to the sines of its opposite angles: a/sin(A) = b/sin(B) = c/sin(C).
Law of Cosines: Relates the lengths of the sides to the cosine of one of its angles: c² = a² + b² – 2ab cos(C). It's used to find a third side given two sides and the included angle, or to find angles given three sides.
Sum of Angles: The sum of the interior angles of any triangle is always 180 degrees: A + B + C = 180°.
Area Formulas:
- Given SAS: Area = 0.5 * a * b * sin(C)
- Given SSS (Heron's Formula): Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 is the semi-perimeter.
- Given base and height: Area = 0.5 * base * height
Perimeter: P = a + b + c
Inradius (r): Radius of the inscribed circle, r = Area / s
Circumradius (R): Radius of the circumscribed circle, R = abc / (4 * Area)
Heights (h_a, h_b, h_c): h_a = 2 * Area / a, etc.

The Find Values of Triangle Calculator implements these formulas based on the user's input.

Variables Table

Variable	Meaning	Unit	Typical Range
a, b, c	Lengths of the sides opposite angles A, B, C	Length units (e.g., m, cm)	> 0
A, B, C	Interior angles of the triangle	Degrees	> 0 and < 180
Area	The space enclosed by the triangle	Square length units	> 0
Perimeter	The sum of the side lengths	Length units	> 0
s	Semi-perimeter	Length units	> 0
r	Inradius	Length units	> 0
R	Circumradius	Length units	> 0
h_a, h_b, h_c	Heights relative to sides a, b, c	Length units	> 0

Practical Examples (Real-World Use Cases)

Let's see how the Find Values of Triangle Calculator works:

Example 1: SSS (Side, Side, Side)

You know three sides: a = 7, b = 10, c = 12.

Inputs: Side a=7, Side b=10, Side c=12

The calculator uses the Law of Cosines to find the angles, then calculates the area and perimeter.

Outputs (approximate): Angle A ≈ 35.43°, Angle B ≈ 56.25°, Angle C ≈ 88.32°, Area ≈ 34.99, Perimeter = 29.

Example 2: SAS (Side, Angle, Side)

You have two sides and the included angle: Side b = 8, Angle C = 60°, Side a = 5.

Inputs: Side a=5, Angle C=60, Side b=8

The calculator uses the Law of Cosines to find side c, then the Law of Sines for other angles, area, and perimeter.

Outputs (approximate): Side c ≈ 7, Angle A ≈ 38.21°, Angle B ≈ 81.79°, Area ≈ 17.32, Perimeter = 20.

How to Use This Find Values of Triangle Calculator

Select Input Type: Choose the combination of values you know (SSS, SAS, ASA, AAS, or SSA) from the dropdown menu.
Enter Known Values: Input the lengths of the sides and/or measures of the angles (in degrees) into the corresponding fields that appear based on your selection. Ensure the values are positive and angles are less than 180°.
Calculate: Click the "Calculate" button.
View Results: The calculator will display the unknown sides, angles, area, perimeter, triangle type, inradius, circumradius, and heights. For SSA, it may indicate 0, 1, or 2 solutions.
Interpret Results: Check the calculated values. The primary result highlights a key finding, like the area or a missing side/angle. The intermediate results give a full breakdown.
Use Chart and Table: The bar chart visualizes the side lengths and angles, and the table summarizes all inputs and outputs for easy reference.
Copy or Reset: Use "Copy Results" to copy the data, or "Reset" to clear inputs for a new calculation with our Find Values of Triangle Calculator.

Key Factors That Affect Find Values of Triangle Calculator Results

Accuracy of Input: Small errors in input values, especially angles, can lead to significant differences in calculated results.
Units: Ensure all side lengths are in the same units. The calculator treats them as generic units, but consistency is key for area and perimeter interpretation.
Angle Units: Angles must be entered in degrees.
Triangle Inequality (SSS): For three sides to form a triangle, the sum of any two sides must be greater than the third side. The calculator checks this.
Angle Sum (ASA/AAS): The sum of two given angles must be less than 180 degrees.
SSA Ambiguity: The Side-Side-Angle case can be ambiguous, leading to zero, one, or two possible triangles. The calculator attempts to identify these based on the Law of Sines. Understanding the conditions for SSA is important.
Rounding: The results are rounded to a certain number of decimal places, which might introduce very minor differences if you compare with manual calculations using full precision.

Frequently Asked Questions (FAQ)

Q1: What is the Law of Sines used for?: A1: The Law of Sines is used to find unknown sides or angles when you know either two angles and one side (ASA or AAS), or two sides and a non-included angle (SSA).
Q2: What is the Law of Cosines used for?: A2: The Law of Cosines is used to find the third side of a triangle when you know two sides and the included angle (SAS), or to find the angles of a triangle when you know all three sides (SSS).
Q3: What if the three sides I enter don't form a triangle?: A3: If the sides violate the triangle inequality theorem (the sum of any two sides must be greater than the third), the Find Values of Triangle Calculator will indicate that no triangle can be formed with those side lengths.
Q4: What is the SSA (Side-Side-Angle) case, and why is it ambiguous?: A4: SSA is when you know two sides and an angle that is NOT between those two sides. It's ambiguous because there might be zero, one, or two possible triangles that fit the given information, depending on the length of the side opposite the given angle relative to the other given side and the height of the potential triangle.
Q5: How does the calculator handle the SSA case?: A5: Our Find Values of Triangle Calculator attempts to find both possible solutions in the SSA case if they exist and will inform you if there are 0, 1, or 2 solutions.
Q6: Can this calculator solve right-angled triangles?: A6: Yes, if you know it's a right-angled triangle, you can input 90 degrees as one of the angles along with two other pieces of information (sides or another angle) and use the appropriate mode (e.g., SAS, ASA, AAS).
Q7: What units should I use for sides?: A7: You can use any unit (cm, meters, inches, etc.), but be consistent. The area will be in square units of whatever unit you used for the sides.
Q8: What does "inradius" and "circumradius" mean?: A8: The inradius is the radius of the largest circle that can be inscribed within the triangle, touching all three sides. The circumradius is the radius of the circle that passes through all three vertices of the triangle.

Related Tools and Internal Resources

Right Triangle Calculator: A specialized calculator for triangles with a 90-degree angle.
Pythagorean Theorem Calculator: Calculate the missing side of a right triangle.
Area Calculator: Calculate the area of various shapes, including triangles.
Geometry Calculators: A collection of calculators for various geometric problems.
Angle Converter: Convert between different angle units (degrees, radians).
Trigonometry Formulas: Learn more about the formulas used in triangle calculations.

Find Values Of Triangle Calculator