Rectangular Pyramid Volume Calculator

Base Length \( a \):

Base Width \( b \):

Height \( H \):

Volume

Volume (cm³): Volume (m³): Volume (in³): Volume (ft³): Volume (liters):

Surface Area

Surface Area (cm²): Surface Area (m²): Surface Area (in²): Surface Area (ft²):

Base Diagonal

Base Diagonal \( d \) (cm): Base Diagonal \( d \) (m): Base Diagonal \( d \) (in): Base Diagonal \( d \) (ft):

1. What is a Rectangular Pyramid Volume Calculator?

Definition: This calculator computes the volume, surface area, and base diagonal of a right rectangular pyramid based on its base length (\( a \)), base width (\( b \)), and height (\( H \)). A right rectangular pyramid is a three-dimensional shape with a rectangular base and an apex directly above the center of the base.

Purpose: Useful in geometry, architecture, and engineering for analyzing the properties of rectangular pyramids.

2. How Does the Calculator Work?

The calculations are based on the following formulas:

\[ V = \frac{abH}{3} \] \[ A = ab + a\sqrt{\left(\frac{b}{2}\right)^2 + H^2} + b\sqrt{\left(\frac{a}{2}\right)^2 + H^2} \] \[ d = \sqrt{a^2 + b^2} \]

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)
Surface Area (from cm²): m² (×0.0001), in² (×0.155), ft² (×0.00107639)
Diagonal (from cm): m (×0.01), in (×0.393701), ft (×0.0328084)

Explanation: All dimensions are converted to cm, the volume, surface area, and base diagonal are calculated, and then converted to other units.

3. Importance of Rectangular Pyramid Volume Calculation

Details: Calculating the volume and surface area of a rectangular pyramid is essential in architecture (e.g., designing pyramid-shaped structures), engineering, and geometry education.

4. Using the Calculator

Tips: Enter the base length \( a \), base width \( b \), and height \( H \) in mm, cm, m, in, or ft (all must be >0). The result shows the volume in cm³, m³, in³, ft³, and liters; the surface area in cm², m², in², and ft²; and the base diagonal in cm, m, in, and ft.

5. Example

Given: Base length \( a = 6 \, \text{cm} \), base width \( b = 8 \, \text{cm} \), height \( H = 10 \, \text{cm} \).
Volume Calculation: \( V = \frac{6 \times 8 \times 10}{3} = 160 \, \text{cm}^3 \).
Surface Area Calculation: \( A = (6 \times 8) + 6 \sqrt{\left(\frac{8}{2}\right)^2 + 10^2} + 8 \sqrt{\left(\frac{6}{2}\right)^2 + 10^2} = 48 + 6 \sqrt{116} + 8 \sqrt{109} \approx 48 + 64.621 + 83.556 \approx 196.177 \, \text{cm}^2 \).
Base Diagonal Calculation: \( d = \sqrt{6^2 + 8^2} = \sqrt{100} = 10 \, \text{cm} \).
Conversions (Volume):
- \( \text{cm}^3 = 160 \),
- \( \text{m}^3 = 0.00016 \),
- \( \text{in}^3 \approx 9.765 \),
- \( \text{ft}^3 \approx 0.00565 \),
- \( \text{liters} = 0.16 \).
Conversions (Surface Area):
- \( \text{cm}^2 \approx 196.177 \),
- \( \text{m}^2 \approx 0.0196 \),
- \( \text{in}^2 \approx 30.407 \),
- \( \text{ft}^2 \approx 0.211 \).
Conversions (Diagonal):
- \( \text{cm} = 10 \),
- \( \text{m} = 0.1 \),
- \( \text{in} \approx 3.937 \),
- \( \text{ft} \approx 0.328 \).

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