Triangle Classification Calculator

Mode:

Side \( a \):

Side \( b \):

Side \( c \):

Angle \( \gamma \) (degrees):

Angle \( \alpha \) (degrees):

Angle \( \beta \) (degrees):

1. What is a Triangle Classification Calculator?

Definition: This calculator determines the angles and sides of a triangle based on different input methods.

Purpose: It helps classify triangles by their angles and sides, useful in geometry and engineering.

2. How Does the Calculator Work?

The calculator operates in three modes:

3 Sides Mode:

Angle \( \alpha \): \( \cos(\alpha) = \frac{b^2 + c^2 - a^2}{2bc} \)
Angle \( \beta \): \( \cos(\beta) = \frac{a^2 + c^2 - b^2}{2ac} \)
Angle \( \gamma \): \( \cos(\gamma) = \frac{a^2 + b^2 - c^2}{2ab} \)

Angle and 2 Sides Mode (where \( c \) is the longest side):

Angles \( \alpha \) and \( \beta \): If \( a = b \), then \( \alpha = \beta = \frac{180^\circ - \gamma}{2} \); else use Law of Sines \( \frac{\sin(\alpha)}{a} = \frac{\sin(\gamma)}{c} \)
Side \( c \): Using Law of Sines \( c = \frac{a \sin(\gamma)}{\sin(\alpha)} \)
Angle \( \gamma \): Given as input (angle between \( b \) and \( c \))

2 Angles Mode:

Angle \( \gamma \): \( \gamma = 180^\circ - \alpha - \beta \)

Unit Conversions:

Input Dimensions: m, cm (1 m = 100 cm), mm (1 m = 1000 mm), in (1 m = 39.3701 in), ft (1 m = 3.28084 ft), yd (1 m = 1.09361 yd)
Output Sides: m, cm, mm, in, ft, yd

Steps:

Select the mode (3 Sides, Angle and 2 Sides, or 2 Angles).
Input the required values with their units.
Convert all dimensions to meters for calculation.
Validate the inputs (e.g., triangle inequality for sides, angle sum for angles).
Calculate the outputs based on the mode's formulas, with angles formatted to 4 decimal places.

3. Importance of Triangle Classification

Classifying triangles is crucial for:

Geometry Education: Understanding triangle properties and trigonometric relationships.
Engineering Design: Analyzing structural components.
Architecture: Designing stable shapes.

4. Using the Calculator

Examples:

Example 1 (3 Sides Mode): For a triangle with \( a = 3 \, \text{cm} \), \( b = 4 \, \text{cm} \), \( c = 5 \, \text{cm} \):
- Convert: \( a = 0.03 \, \text{m} \), \( b = 0.04 \, \text{m} \), \( c = 0.05 \, \text{m} \)
- Angle \( \alpha \): \( \cos(\alpha) = \frac{0.04^2 + 0.05^2 - 0.03^2}{2 \times 0.04 \times 0.05} = 0.866 \), \( \alpha = 30^\circ \)
- Angle \( \beta \): \( \cos(\beta) = \frac{0.03^2 + 0.05^2 - 0.04^2}{2 \times 0.03 \times 0.05} = 0.5 \), \( \beta = 60^\circ \)
- Angle \( \gamma \): \( \cos(\gamma) = \frac{0.03^2 + 0.04^2 - 0.05^2}{2 \times 0.03 \times 0.04} = 0 \), \( \gamma = 90^\circ \)
Example 2 (Angle and 2 Sides Mode): For a triangle with \( a = 5 \, \text{cm} \), \( b = 5 \, \text{cm} \), \( \gamma = 130^\circ \):
- Convert: \( a = 0.05 \, \text{m} \), \( b = 0.05 \, \text{m} \)
- Angles \( \alpha \) and \( \beta \): Since \( a = b \), \( \alpha = \beta = \frac{180 - 130}{2} = 25^\circ \)
- Side \( c \): \( c = \frac{0.05 \times \sin(130^\circ)}{\sin(25^\circ)} \approx 0.09063 \, \text{m} \)
- Convert: \( c = 9.063 \, \text{cm} \)
Example 3 (2 Angles Mode): For a triangle with \( \alpha = 45^\circ \), \( \beta = 45^\circ \):
- Angle \( \gamma \): \( \gamma = 180 - 45 - 45 = 90^\circ \)

5. Frequently Asked Questions (FAQ)

Q: How can you classify a triangle by angles?
A: A triangle is acute if all angles are less than 90°, right if one angle is 90°, and obtuse if one angle is greater than 90°.

Q: Why is triangle classification important?
A: It helps in understanding triangle properties for applications in geometry, engineering, and design.