Pyramid Calculator
Calculate volume, lateral surface area, total surface area, base area and slant height of a right pyramid. Supports square and rectangular bases. Suitable for math homework, DIY construction and 3D modeling.
Square & rectangular Formula-based Step-friendly
Height is measured perpendicular from base to apex.
Pyramid formulas used
This tool uses the general pyramid volume formula and specific surface-area formulas for square and rectangular bases.
1. Volume of a pyramid
\( V = \frac{1}{3} B h \)
where \( B \) is the base area and \( h \) is the vertical height.
2. Square pyramid
Base area: \( B = a^2 \)
Slant height: \( \ell = \sqrt{ \left(\frac{a}{2}\right)^2 + h^2 } \)
Lateral area: \( A_L = 2 a \ell \)
Total area: \( A_T = a^2 + 2 a \ell \)
3. Rectangular pyramid
Base area: \( B = a b \)
Slant height along side a: \( \ell_a = \sqrt{ \left(\frac{b}{2}\right)^2 + h^2 } \)
Slant height along side b: \( \ell_b = \sqrt{ \left(\frac{a}{2}\right)^2 + h^2 } \)
Lateral area: \( A_L = a \ell_a + b \ell_b \)
Total area: \( A_T = ab + a \ell_a + b \ell_b \)
Example
Example: Square pyramid with base side 6 m and height 9 m.
- Base area \( B = 6^2 = 36 \text{ m}^2 \)
- Volume \( V = \frac{1}{3} \times 36 \times 9 = 108 \text{ m}^3 \)
- Slant height \( \ell = \sqrt{(6/2)^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90} \approx 9.487 \text{ m} \)
- Lateral area \( A_L = 2 \times 6 \times 9.487 \approx 113.84 \text{ m}^2 \)
- Total area \( A_T = 36 + 113.84 \approx 149.84 \text{ m}^2 \)
The calculator runs these steps automatically.