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Pyramid Calculator

Calculate pyramid volume, total surface area, lateral area, and slant heights for square and rectangular pyramids with step-by-step steps.

Pyramid Measurements
Volume
48
Total Surface Area
96
Lateral Area
60
Slant Height
5
Deduction Steps:
Base: length = 6, width = 6; Height = 4
Base Area: A_base = l × w = 6 × 6 = 36
Volume: V = ⅓ × A_base × h = ⅓ × 36 × 4 = 48
Slant Height (along length): s_l = √(h² + (w/2)²) = 5
Slant Height (along width): s_w = √(h² + (l/2)²) = 5
Lateral Surface Area: A_lat = (l × s_l) + (w × s_w) = 60
Total Surface Area: A_total = A_base + A_lat = 96
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What Is a Pyramid Calculator?

A pyramid is a polyhedron formed by connecting a polygonal base and a point called the apex. Each base edge and the apex form a triangular lateral face.

How to Use This Calculator

  1. Enter base length, base width, and vertical height.
  2. View the calculated Volume, Slant Heights, Lateral Area, and Total Surface Area.
  3. Review the step-by-step mathematical deductions.

Pyramid Volume and Surface Area Formulas

V = \frac{1}{3}A_{\text{base}} h = \frac{1}{3}lwh, \quad s_l = \sqrt{h^2 + (w/2)^2}, \quad s_w = \sqrt{h^2 + (l/2)^2}

Volume is one-third base area times vertical height. Lateral surface area is the sum of the four triangular side faces calculated from slant heights.

Worked Example

Scenario: Find volume and surface area of a square pyramid with base 6×6 and height 4

Base Area: A_base = 6 × 6 = 36

Volume: V = ⅓ × 36 × 4 = 48

Slant height: s = √(4² + 3²) = √(16 + 9) = 5

Lateral area: 4 triangles = 4 × (½ × 6 × 5) = 60

Total Surface Area: A_total = 36 + 60 = 96

Result: Volume = 48, Total Surface Area = 96, Slant Height = 5

Tips & Key Notes

  • A pyramid has exactly one-third the volume of a prism with the same base and height.
  • For a square pyramid, all four triangular lateral faces are congruent isosceles triangles.
  • The slant height is measured from the center of a base edge up the face to the apex.

Frequently Asked Questions

What is the formula for the volume of a pyramid?

V = (1/3) × Base Area × Height.

How do you calculate the slant height of a pyramid?

Apply the Pythagorean theorem to the right triangle inside: slant height = √(height² + (half_base)²).

How many faces and vertices does a square pyramid have?

A square pyramid has 5 faces (1 square base + 4 triangular sides), 8 edges, and 5 vertices.

What is a tetrahedron?

A tetrahedron is a triangular pyramid where all four faces (including the base) are triangles.

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