🔺 Pyramid Volume Calculator
Calculate volume, height, base area, or top area for pyramids and frustums
Enter Pyramid Dimensions
Square
Rectangle
Triangle
Custom Area
Pyramid • h=height
📊 Results Square Pyramid
Base Area (A₁)
100 cm²
Top Area (A₂)
0 cm²
Height (h)
15 cm
Volume
500 cm³
Slant Height
15.8 cm
Lateral Area
316 cm²
📝 Step-by-Step Solution
Given: Base = square (s = 10 cm), h = 15 cm
Base area: A = s² = 10² = 100 cm²
Pyramid volume formula: V = (1/3) × A × h
V = (1/3) × 100 × 15 = 500 cm³
Slant height: l = √(h² + (s/2)²) = √(225 + 25) = 15.81 cm
📐 Pyramid Formulas
Pyramid Volume: V = (1/3) × Base Area × Height
Frustum Volume: V = (h/3)(A₁ + A₂ + √(A₁×A₂))
Square Base Area: A = s²
Slant Height (square): l = √(h² + (s/2)²)
Lateral Surface: LA = (1/2) × Perimeter × Slant Height
Understanding Pyramids
🔺 Pyramid Shape
Base + triangular faces meeting at apex. Any polygon base works. Volume = 1/3 of prism with same base and height.
📐 The 1/3 Factor
Pyramid = 1/3 of prism volume. This holds for all pyramid shapes. Cone also uses 1/3 (circular pyramid).
🔻 Frustum
Truncated pyramid. Top cut off parallel to base. Has two parallel faces. Uses special formula with both areas.
📏 Slant Height
Distance from apex to base edge midpoint. Used for surface area. Different from perpendicular height.
Frequently Asked Questions
What is the volume formula for a pyramid?
V = (1/3) × Base Area × Height. Works for any polygon base. The
1/3 factor distinguishes pyramids from prisms.
Why is pyramid volume one-third of prism?
Geometric principle. Three identical pyramids can fill a prism.
Proven using calculus or Cavalieri's principle.
How do I find pyramid height from volume?
h = 3V / A. Rearrange V = (1/3)Ah. Multiply volume by 3, divide by
base area.
What is slant height vs height?
Height = perpendicular to base. Slant height = along the face from
apex to base edge. Slant > height always.
How do I calculate frustum volume?
V = (h/3)(A₁ + A₂ + √(A₁×A₂)). Where A₁ = base area, A₂ = top
area, h = height between them.
What is a frustum?
Truncated pyramid or cone. The portion remaining after cutting off
the top parallel to the base.
How do I find base area from volume?
A = 3V / h. Rearrange the formula. Need volume and height to
solve.
What base shapes can pyramids have?
Any polygon. Triangular, square, rectangular, pentagonal,
hexagonal, or irregular. Formula works for all.
How do I calculate surface area?
SA = Base Area + Lateral Area. Lateral area = (1/2) × Perimeter ×
Slant Height for regular pyramids.
Is a cone a type of pyramid?
Yes, circular pyramid. Cone = pyramid with circular base. Same V =
(1/3)πr²h formula structure.
What is a right vs oblique pyramid?
Right = apex directly above base center. Oblique = apex offset.
Volume formula works for both.
How tall is the Great Pyramid?
Originally ~146.5 meters. Square base ~230m. Volume ≈ 2.6 million
m³. Now ~138.5m due to erosion.
Does the 1/3 formula work for all pyramids?
Yes, any base shape. Regular or irregular, right or oblique. V =
(1/3) × Base Area × Height always works.
How do I find slant height?
For square pyramid: l = √(h² + (s/2)²). Pythagorean theorem with
height and half the base side.
What is lateral surface area?
Area of triangular faces only. Excludes the base. For regular
pyramid: LA = (1/2) × P × l.
How do I convert volume to liters?
1 L = 1000 cm³. Our calculator can output in liters, gallons, or
other units automatically.
What is apothem vs slant height?
Apothem = center to side midpoint of base. Slant height = apex to
base edge midpoint. Different measurements.