Pyramid Calculator
Calculate pyramid properties including volume, surface area, slant height, and slope angle for any base shape. Includes famous Egyptian pyramid presets.
V = (1/3) x Base Area x HeightPyramid Properties
Volume
500 cm^3
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All Pyramid Properties
Volume (V)
500 cm^3
V = (1/3) x Base x h
Base Area
100 cm^2
Slant Height
15.8114 cm
s = sqrt(h^2 + a^2)
Lateral Surface Area
316.2278 cm^2
Total Surface Area
416.2278 cm^2
Side Face Slope
71.57 deg
arctan(h / apothem)
Apothem
5 cm
Center to base edge
Pyramid Formulas
Volume Formula
- V = (1/3) x Base Area x h
- Square: V = (1/3) x s^2 x h
- Rectangle: V = (1/3) x l x w x h
- Triangle: V = (1/3) x (sqrt(3)/4) x s^2 x h
Surface Area Formulas
- Lateral = (1/2) x Perimeter x Slant
- Total = Base Area + Lateral
- Slant = sqrt(h^2 + apothem^2)
Why 1/3? The volume of a pyramid is exactly 1/3 of the volume of a prism with the same base and height. This was proven by ancient Greek mathematicians using a method called exhaustion.
Pyramid Properties
Volume
500 cm^3
?How to Calculate Pyramid Properties
Pyramid formulas: Volume = (1/3) x Base Area x Height. Slant Height = sqrt(h^2 + a^2) where a is the apothem (distance from base center to edge). Total Surface Area = Base Area + Lateral Surface Area. A pyramid has exactly 1/3 the volume of a prism with the same base and height. For a square pyramid: V = (1/3) x s^2 x h.
What is a Pyramid?
A pyramid is a three-dimensional shape with a polygon base and triangular faces that meet at a single point called the apex. The height is the perpendicular distance from the base to the apex. Pyramids are named by their base shape: square pyramid (4 sides), triangular pyramid (tetrahedron, 3 sides), pentagonal pyramid (5 sides), etc. The ancient Egyptian pyramids are famous examples of square-based pyramids.
Key Facts About Pyramids
- Volume = (1/3) x Base Area x Height for ALL pyramids
- Square pyramid V = (1/3) x s^2 x h
- Slant height s = sqrt(h^2 + apothem^2)
- A pyramid has 1/3 the volume of a prism with same base and height
- The Great Pyramid of Giza has a base of 230.4m and original height of 146.6m
- Regular pyramids have congruent triangular faces
- Side face slope angle = arctan(slant height / half base)
- Pyramid frustum volume = (h/3)(A1 + A2 + sqrt(A1 x A2))
Quick Answer
Pyramid formulas: Volume = (1/3) x Base Area x Height. Slant Height = sqrt(h^2 + a^2) where a is the apothem (distance from base center to edge). Total Surface Area = Base Area + Lateral Surface Area. A pyramid has exactly 1/3 the volume of a prism with the same base and height. For a square pyramid: V = (1/3) x s^2 x h.
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Frequently Asked Questions
The volume of any pyramid is V = (1/3) x Base Area x Height. For a square pyramid with side s and height h: V = (1/3)s^2h. For a rectangular pyramid: V = (1/3)lwh. This 1/3 factor applies to all pyramids regardless of base shape.
Slant height is the distance from the apex to the midpoint of a base edge, measured along a face. For a square pyramid: slant = sqrt(h^2 + (s/2)^2), where h is height and s is base side. The apothem (distance from center to edge midpoint) is s/2 for a square.
The Great Pyramid originally stood 146.6 meters (481 feet) tall with a base of 230.4 meters (756 feet). The slope angle is 51.84 degrees. It was built around 2560 BC and contains approximately 2.3 million stone blocks.
A frustum is a pyramid with the top cut off parallel to the base, like a truncated pyramid. Its volume is V = (h/3)(A1 + A2 + sqrt(A1 x A2)), where A1 and A2 are the areas of the two parallel faces and h is the height between them.
This can be proven using calculus or Cavalieri's principle. Intuitively, three congruent pyramids can be assembled to form a prism. Ancient mathematicians like Democritus and Eudoxus established this ratio around 400 BC.
Height (h) is the perpendicular distance from the base to the apex, measured straight up through the center. Slant height (s) is measured along a triangular face from the apex to the midpoint of a base edge. Slant height is always longer than height.
Total surface area = Base Area + Lateral Surface Area. For a regular pyramid with n sides: Lateral Area = (1/2) x perimeter x slant height. For a square pyramid: Lateral = 2 x s x slant height, where s is the base side.
The apothem is the distance from the center of the base to the midpoint of any base edge. For a square with side s, apothem = s/2. For a regular hexagon with side s, apothem = s x sqrt(3)/2. The apothem is used to calculate slant height.
Last updated: 2025-01-15
Pyramid Properties
Volume
500 cm^3