1. What is a Triangular Pyramid Volume Calculator?

Definition: This calculator computes the volume of a triangular pyramid (tetrahedron) based on the dimensions of its triangular base and the pyramid's height (\( H \)). The volume depends on the area of the triangular base, calculated using different methods.

Purpose: Useful in geometry, architecture, and engineering for analyzing the volume of pyramid structures with a triangular base.

2. How Does the Calculator Work?

The calculations are based on the following formula:

Base Area Calculation:

• Base and Height: Area = \(\frac{1}{2} \times b \times h\)
• Three Sides (SSS): Area = \(\sqrt{s (s - a) (s - b) (s - c)}\), where \( s = \frac{a + b + c}{2} \) is the semi-perimeter

Unit Conversions:

• Input (Length): mm (×0.1), cm (×1), m (×100), in (×2.54), ft (×30.48) to cm
• Volume (from cm³): m³ (×0.000001), in³ (×0.0610237), ft³ (×0.0000353147), liters (×0.001)
• Area (from cm²): m² (×0.0001), in² (×0.155), ft² (×0.00107639)

Explanation: All dimensions are converted to cm, the base area is calculated based on the selected triangle type, and the volume is computed in cm³, then converted to other units.

3. Importance of Triangular Pyramid Volume Calculation

Details: Calculating the volume of a triangular pyramid is essential in architecture (e.g., designing pyramid roofs), engineering, and geometry education.

4. Using the Calculator

Tips: Select the triangle type (Base and Height or Three Sides), then enter the required dimensions (base length and triangle height, or three sides, and pyramid height) in mm, cm, m, in, or ft (all must be >0, with valid triangle inequalities for SSS). The result shows the volume in cm³, m³, in³, ft³, and liters, and the base area in cm², m², in², and ft² if calculated.

5. Example

Given (Base and Height Mode): Base length \( b = 6 \, \text{cm} \), triangle height \( h = 4 \, \text{cm} \), pyramid height \( H = 10 \, \text{cm} \).
Base Area Calculation: Area = \(\frac{1}{2} \times 6 \times 4 = 12 \, \text{cm}^2\).
Volume Calculation: \( V = \frac{1}{3} \times 12 \times 10 = 40 \, \text{cm}^3 \).
Given (Three Sides Mode): Sides \( a = 5 \, \text{cm} \), \( b = 5 \, \text{cm} \), \( c = 6 \, \text{cm} \), pyramid height \( H = 10 \, \text{cm} \).
Base Area Calculation: \( s = \frac{5 + 5 + 6}{2} = 8 \), Area = \(\sqrt{8 \times (8 - 5) \times (8 - 5) \times (8 - 6)} = \sqrt{8 \times 3 \times 3 \times 2} \approx 12 \, \text{cm}^2\).
Volume Calculation: \( V = \frac{1}{3} \times 12 \times 10 = 40 \, \text{cm}^3 \).
Conversions (Volume):
- \( \text{cm}^3 = 40 \),
- \( \text{m}^3 = 0.00004 \),
- \( \text{in}^3 \approx 2.441 \),
- \( \text{ft}^3 \approx 0.00141 \),
- \( \text{liters} = 0.04 \).
Conversions (Base Area):
- \( \text{cm}^2 = 12 \),
- \( \text{m}^2 = 0.0012 \),
- \( \text{in}^2 \approx 1.86 \),
- \( \text{ft}^2 \approx 0.0129 \).

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