Finding Dimensions Of A Triangle Calculator

Use this triangle dimensions calculator to find the sides, angles, area, and perimeter of a triangle based on known values.

Triangle Dimensions Summary

Dimension	Value	Unit
Side a	5.00	units
Side b	5.00	units
Side c	5.00	units
Angle A	60.00	degrees
Angle B	60.00	degrees
Angle C	60.00	degrees
Area	10.83	sq units
Perimeter	15.00	units
Type	Equilateral, Acute	–

Bar chart comparing side lengths and angles (in degrees).

What is a Triangle Dimensions Calculator?

A triangle dimensions calculator is a tool used to determine the unknown dimensions (sides and angles), area, and perimeter of a triangle based on a sufficient number of known dimensions. You typically need at least three pieces of information (like three sides, or two sides and an angle, or two angles and a side) to define a unique triangle or a limited set of triangles. Our triangle dimensions calculator supports SSS (Side-Side-Side), SAS (Side-Angle-Side), and ASA (Angle-Side-Angle) input methods to find the missing values.

This calculator is useful for students learning trigonometry, engineers, architects, and anyone needing to solve for triangle properties. It automates complex calculations like the Law of Sines and Law of Cosines, as well as area formulas.

Common misconceptions include thinking any three values will define a triangle (e.g., three angles don't define side lengths) or that SSA (Side-Side-Angle) always gives one unique triangle (it can be ambiguous).

Triangle Dimensions Formula and Mathematical Explanation

The triangle dimensions calculator uses several fundamental trigonometric formulas:

1. Given SSS (Side-Side-Side):

If you know sides a, b, and c, you first check the Triangle Inequality Theorem: a+b > c, a+c > b, and b+c > a. If true, you can find the angles using the Law of Cosines:

• cos(A) = (b² + c² – a²) / 2bc
• cos(B) = (a² + c² – b²) / 2ac
• cos(C) = (a² + b² – c²) / 2ab

The angles A, B, C are found by taking the arccos of these values. The sum A+B+C should be 180°.

Area (Heron's Formula): s = (a+b+c)/2, Area = √[s(s-a)(s-b)(s-c)]

2. Given SAS (Side-Angle-Side):

If you know sides b, c, and the included angle A, you find side a using the Law of Cosines:

• a² = b² + c² – 2bc * cos(A)

Then, you can find other angles using the Law of Sines (a/sin(A) = b/sin(B)) or Law of Cosines again. B = arccos((a² + c² – b²) / 2ac), C = 180 – A – B.

Area = 0.5 * b * c * sin(A)

3. Given ASA (Angle-Side-Angle):

If you know angles A, B, and the included side c, first find angle C: C = 180 – A – B. Then use the Law of Sines to find sides a and b:

• a / sin(A) = c / sin(C) => a = c * sin(A) / sin(C)
• b / sin(B) = c / sin(C) => b = c * sin(B) / sin(C)

Area = 0.5 * c * b * sin(A) or using side c and angles: Area = (c² * sin(A) * sin(B)) / (2 * sin(C))

Perimeter is always a + b + c.

Variables Used

Variable	Meaning	Unit	Typical Range
a, b, c	Lengths of the sides opposite angles A, B, C respectively	units (e.g., cm, m, inches)	> 0
A, B, C	Angles at vertices A, B, C respectively	degrees	0° – 180° (sum = 180°)
s	Semi-perimeter (a+b+c)/2	units	> 0
Area	Area of the triangle	square units	≥ 0
Perimeter	Sum of side lengths (a+b+c)	units	> 0

Practical Examples (Real-World Use Cases)

Example 1: SSS

You have a triangular piece of land with sides 30m, 40m, and 50m. Using the triangle dimensions calculator with SSS:

• Inputs: a=30, b=40, c=50
• Outputs: Angle A ≈ 36.87°, Angle B ≈ 53.13°, Angle C = 90° (Right-angled triangle), Area = 600 sq m, Perimeter = 120m.

Example 2: SAS

You want to fence a garden area. You know two sides are 15m and 20m, and the angle between them is 70°. Using the triangle dimensions calculator with SAS:

• Inputs: b=15, A=70, c=20
• Outputs: Side a ≈ 20.3m, Angle B ≈ 46.2°, Angle C ≈ 63.8°, Area ≈ 140.95 sq m, Perimeter ≈ 55.3m.

How to Use This Triangle Dimensions Calculator

1. Select Input Type: Choose SSS, SAS, or ASA based on the values you know.
2. Enter Known Values: Input the side lengths and/or angles into the corresponding fields that appear. Ensure angles are in degrees.
3. Calculate: The calculator updates results in real-time as you type, or you can click "Calculate".
4. Review Results: The calculator displays the unknown sides, angles, area, perimeter, and triangle type. A table and chart also summarize the dimensions. The "Primary Result" highlights the area.
5. Use Reset/Copy: "Reset" restores default values, "Copy Results" copies key information to your clipboard.

Understanding the results helps in various fields, from geometry homework to construction projects. Check if the "Calculation Status" is OK or indicates an error (like invalid triangle inequality). The triangle dimensions calculator is a powerful tool for these tasks.

Key Factors That Affect Triangle Dimension Results

1. Input Accuracy: Small errors in input side lengths or angles can lead to significant differences in calculated dimensions, especially with the Law of Sines/Cosines.
2. Input Type (SSS, SAS, ASA): The combination of known values determines the method and sensitivity. SSS requires the triangle inequality to be met.
3. Angle Units: Ensure angles are input in degrees, as the calculator converts them to radians for trigonometric functions.
4. Triangle Inequality (SSS): For SSS, the sum of any two sides must be greater than the third side. If not, a triangle cannot be formed. Our triangle dimensions calculator checks this.
5. Sum of Angles (ASA): For ASA, the two known angles must sum to less than 180 degrees.
6. Ambiguous Case (SSA): While not directly implemented here to avoid confusion, knowing two sides and a non-included angle (SSA) can result in zero, one, or two possible triangles. This triangle dimensions calculator focuses on unambiguous cases SSS, SAS, ASA.
7. Rounding: The displayed results are rounded, which might lead to very slight discrepancies if you manually sum rounded angles to 180°.

Frequently Asked Questions (FAQ)

1. What is the minimum information needed to define a triangle?

You generally need three independent pieces of information, such as SSS, SAS, or ASA. Three angles (AAA) are not enough to determine side lengths (they define similarity). Our triangle dimensions calculator uses SSS, SAS, and ASA.

2. Can I use the calculator for a right-angled triangle?

Yes, if you know it's right-angled and have two other pieces of info (like two sides, or one side and one acute angle), you can often deduce SSS, SAS, or ASA conditions. For instance, if you know two legs (a, b) and it's right-angled at C (90°), it's SAS (a, 90°, b).

3. What if the SSS inputs don't form a triangle?

The calculator will indicate an error if the triangle inequality (a+b>c, etc.) is not met.

4. How does the calculator find the area?

It uses Heron's formula for SSS (Area = √[s(s-a)(s-b)(s-c)]) or the formula Area = 0.5 * b * c * sin(A) for SAS, and adjusts for ASA.

5. What is the Law of Sines and Law of Cosines?

The Law of Sines relates sides to the sines of opposite angles (a/sinA = b/sinB = c/sinC). The Law of Cosines relates the three sides to one angle (c² = a² + b² – 2ab cosC).

6. Does this calculator handle the SSA (Side-Side-Angle) case?

This specific triangle dimensions calculator does not explicitly handle the SSA case due to its potential ambiguity (0, 1, or 2 solutions). It focuses on SSS, SAS, and ASA, which generally yield unique triangles.

7. How are triangle types (Equilateral, Isosceles, Scalene; Acute, Obtuse, Right) determined?

By comparing side lengths (a=b=c, a=b or b=c or a=c, all different) and angles (all < 90°, one = 90°, one > 90°).

8. Why is the sum of angles sometimes slightly off 180° in the results?

This can be due to rounding of the calculated angle values. The internal calculations are more precise.

Related Tools and Internal Resources

• Right Triangle Calculator: A specialized tool for right-angled triangles.
• Pythagorean Theorem Calculator: Calculate the sides of a right triangle using a² + b² = c².
• Area Calculator: Calculate areas of various shapes, including triangles with different inputs.
• Angle Converter: Convert between degrees and radians.
• Law of Sines Calculator: Focuses specifically on the Law of Sines.
• Law of Cosines Calculator: Focuses specifically on the Law of Cosines.