Find The Missing Dimension Of A Triangle Calculator

Calculate Missing Dimensions (SAS)

Enter two sides (a and b) and the angle between them (Angle C) to find the third side (c) and the other angles (A and B) of a triangle.

Side a	Side b	Side c	Angle A	Angle B	Angle C

Summary of triangle dimensions.

What is a Missing Triangle Dimension Calculator?

A Missing Triangle Dimension Calculator is a tool used to find unknown sides or angles of a triangle when you have enough information about its other dimensions. Triangles are fundamental shapes in geometry, and knowing all their dimensions is crucial in various fields like engineering, architecture, physics, and navigation. This specific calculator focuses on the SAS (Side-Angle-Side) case: if you know two sides and the angle between them, you can find the third side and the other two angles using the Law of Cosines and the Law of Sines.

Anyone dealing with geometric problems, from students learning trigonometry to professionals designing structures, can use a Missing Triangle Dimension Calculator. It automates complex calculations, saving time and reducing the risk of errors.

A common misconception is that you can solve any triangle with any three pieces of information. However, you need specific combinations (like SAS, SSS, ASA, AAS) to uniquely define a triangle and find missing dimensions. For instance, knowing only three angles (AAA) tells you the shape but not the size.

Missing Triangle Dimension Calculator: Formula and Mathematical Explanation

When you have two sides (a and b) and the included angle (C), you can use the following formulas:

1. Law of Cosines (to find side c):

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. To find side c:

c² = a² + b² - 2ab * cos(C)

So, c = sqrt(a² + b² - 2ab * cos(C))

Here, C is the angle in radians (degrees * π/180).

2. Law of Sines (to find Angle A or B after finding c):

Once side c is known, the Law of Sines can be used:

a / sin(A) = b / sin(B) = c / sin(C)

To find Angle A:

sin(A) = (a * sin(C)) / c

A = asin((a * sin(C)) / c) (result in radians, convert to degrees)

3. Sum of Angles (to find the last angle):

The sum of angles in any triangle is 180 degrees:

A + B + C = 180°

So, B = 180° - A - C (where A and C are in degrees).

4. Area of the Triangle (using SAS):

Area = 0.5 * a * b * sin(C)

Variables Table:

Variable	Meaning	Unit	Typical Range
a, b, c	Lengths of the sides of the triangle	Length units (e.g., cm, m, inches)	> 0
A, B, C	Angles opposite to sides a, b, c respectively	Degrees (or radians in formulas)	0° to 180° (0 to π radians)
Area	The area enclosed by the triangle	Square units	> 0

Variables used in triangle calculations.

Practical Examples (Real-World Use Cases)

Example 1: Surveying Land

A surveyor measures two sides of a triangular plot of land as 120 meters and 150 meters, and the angle between these two sides is 75 degrees. They need to find the length of the third side and the other angles.

• Side a = 120 m
• Side b = 150 m
• Angle C = 75°

Using the Missing Triangle Dimension Calculator (or the formulas):

• Side c ≈ 161.76 m
• Angle A ≈ 46.19°
• Angle B ≈ 58.81°
• Area ≈ 8693.33 m²

The third side is about 161.76 meters long.

Example 2: Navigation

A boat travels 20 miles on one bearing, then changes direction and travels 30 miles. If the angle between the two legs of the journey was 110 degrees, how far is the boat from its starting point in a straight line?

• Side a = 20 miles
• Side b = 30 miles
• Angle C = 110°

Using the Missing Triangle Dimension Calculator:

• Side c ≈ 41.68 miles
• Angle A ≈ 27.02°
• Angle B ≈ 42.98°
• Area ≈ 281.91 square miles

The boat is approximately 41.68 miles from its starting point.

How to Use This Missing Triangle Dimension Calculator

1. Enter Side a: Input the length of the first known side.
2. Enter Side b: Input the length of the second known side.
3. Enter Angle C: Input the angle (in degrees) between side a and side b. Ensure it's between 0 and 180.
4. View Results: The calculator will automatically display the length of side c, the measures of Angle A and Angle B, and the Area of the triangle as you type. The primary result, side c, is highlighted.
5. Check Formula: The formula used (Law of Cosines and Sines) is briefly explained below the results.
6. Reset: Click "Reset" to clear inputs and results to default values.
7. Copy Results: Click "Copy Results" to copy the inputs and calculated dimensions to your clipboard.
8. Chart and Table: The bar chart visually compares the side lengths, and the table summarizes all dimensions.

The Missing Triangle Dimension Calculator is particularly useful when you have the SAS (Side-Angle-Side) configuration. It helps you quickly determine the remaining dimensions without manual trigonometric calculations.

Key Factors That Affect Missing Triangle Dimension Calculator Results

1. Accuracy of Input Values: Small errors in the measured sides or angle can lead to larger errors in the calculated dimensions, especially with the Law of Sines for angles near 0 or 180 degrees.
2. Unit Consistency: Ensure both side lengths are in the same units. The output for the third side will be in the same unit. Angles are always in degrees for input here.
3. Angle Measurement: The angle input must be in degrees and must be the included angle between the two sides entered.
4. Triangle Inequality Theorem: For a valid triangle, the sum of the lengths of any two sides must be greater than the length of the third side. While our calculator starts with SAS (which usually forms a valid triangle if the angle is < 180), this is a fundamental principle.
5. Rounding: The precision of the results depends on the rounding applied. Our calculator typically shows two decimal places.
6. Valid Angle Range: The included angle C must be greater than 0 and less than 180 degrees. An angle of 0 or 180 would not form a triangle.

Understanding these factors helps in using the Missing Triangle Dimension Calculator effectively and interpreting its results correctly.

Frequently Asked Questions (FAQ)

What is the Law of Cosines?

The Law of Cosines is a formula relating the lengths of the sides of a triangle to the cosine of one of its angles: c² = a² + b² – 2ab cos(C).

What is the Law of Sines?

The Law of Sines relates the sides of a triangle to the sines of their opposite angles: a/sin(A) = b/sin(B) = c/sin(C).

Can I use this calculator if I have three sides (SSS)?

This specific calculator is optimized for SAS (Side-Angle-Side). To find angles from three sides, you would primarily use the Law of Cosines rearranged to solve for the angles. You'd need a different calculator mode or a general triangle calculator.

What if I have two angles and one side (ASA or AAS)?

You would first find the third angle (A+B+C=180) and then use the Law of Sines to find the missing sides. This calculator is for SAS. Look for an ASA/AAS triangle calculator.

Why does the angle C have to be between 0 and 180 degrees?

In a triangle, each angle must be greater than 0 and less than 180 degrees. The sum of all three must be exactly 180 degrees.

What units should I use for the sides?

You can use any unit of length (cm, meters, inches, feet, etc.), but be consistent for both side a and side b. The calculated side c will be in the same unit.

How is the area calculated?

When you have two sides and the included angle (SAS), the area is calculated as 0.5 * a * b * sin(C).

Is it possible to get no valid triangle?

With SAS, as long as the sides are positive and the angle is between 0 and 180, a unique triangle is formed. Issues arise more with SSS (if sides don't satisfy triangle inequality) or SSA (ambiguous case).