Volume plays thirds
Any pyramid’s volume is ⅓ of a prism with the same base and height: V = (1/3)·B·h. Cones obey the same one‑third rule.
Prism sibling
Pyramid Calculator: Volume, Surface Area, Height & Slant Height
Diagram & Inputs
Tip: Provide a and n, plus at least one of h, ℓ, S, or V. The calculator fills in the rest and flags inconsistencies.
Results
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About This Pyramid Calculator
Release Updates
v1.1 (May 19, 2026)
- Added lateral surface area as a dedicated result alongside total surface area and volume.
- Expanded the calculator guide with pyramid volume, surface area, square pyramid, triangular pyramid, and slant height sections.
- Added a pyramid formulas table and worked examples for common calculation types.
- Improved validation so the base edge must be greater than zero before solving.
- Expanded FAQ answers, square pyramid surface area, height, slant height, and reverse solving.
Pyramid Volume Calculator
The volume of a pyramid is V = (1/3)Bh, where B is the base area and h is the vertical height.
Surface Area of a Pyramid Calculator
The total surface area is the base area plus the lateral surface area. For a regular pyramid, S = B + 1/2 · p · ℓ.
Square Pyramid Calculator
For a square pyramid with side length a and height h, the volume is V = a²h/3 and the slant height is ℓ = √(h² + (a/2)²).
Triangular Pyramid Calculator
For a regular triangular pyramid, the base is an equilateral triangle. The calculator uses the regular-polygon base area formula.
How to Find Slant Height of a Pyramid
The slant height is found using ℓ² = h² + r², where r is the inradius of the base.
Pyramid Formulas
| Quantity | Formula |
|---|---|
| Volume | V = (1/3)Bh |
| Total surface area | S = B + 1/2 · p · ℓ |
| Lateral surface area | L = 1/2 · p · ℓ |
| Base area, regular n-gon | B = n · a² / (4 · tan(π/n)) |
| Slant height | ℓ = √(h² + r²) |
| Base inradius | r = a / (2 · tan(π/n)) |
Worked Pyramid Calculator Examples
Example: Find the Volume of a Square Pyramid
Side length = 6 cm, height = 10 cm. The square base area is 6² = 36 cm², so Volume = 1/3 × 36 × 10 = 120 cm³.
Example: Find the Surface Area of a Square Pyramid
Side length = 8 cm, slant height = 5 cm. The surface area is 8² + 2 × 8 × 5 = 144 cm².
Example: Find the Volume of a Triangular Pyramid
For a regular triangular base with side 6 cm and height 9 cm, the base area is 6²√3/4 ≈ 15.59 cm². The volume is 1/3 × 15.59 × 9 ≈ 46.77 cm³.
Example: Find the Slant Height of a Pyramid
For a square pyramid with side length 8 cm and height 3 cm, the base inradius is r = 8/2 = 4 cm. The slant height is ℓ = √(3² + 4²) = 5 cm.
Example: Find Height from Volume
Given a square pyramid with side length 5 cm and volume 100 cm³, the base area is B = 25 cm². Solve backward with h = 3V/B = 300/25 = 12 cm.
How the Pyramid Calculator Works
This pyramid calculator helps you find missing dimensions for a regular pyramid—a pyramid whose base is a regular polygon (triangle, square, pentagon, and so on). If you know the base edge length and how many sides the base has, the calculator can work out the height, slant height, surface area, and volume using standard geometry formulas. It is a fast way to solve homework problems, check designs, or plan materials without digging through a textbook.
Key ideas in simple terms
A pyramid has a flat base and triangular faces that meet at the top. The height is the straight vertical distance from the base to the tip. The slant height runs along a triangular face. The surface area adds up the base area and the areas of the triangular faces. The volume measures how much space the pyramid encloses. These pieces are connected, so if you provide a few of them, the rest can be calculated.
How to use the pyramid calculator
- Enter the base edge length
aand select the number of sides n for the base. - Provide at least one more value: the height
h, slant heightℓ, total surface areaS, or volumeV. - Choose a length unit and decimal precision.
- Click Calculate to see all remaining dimensions and checks for consistency.
Formulas used (for reference)
- Base perimeter:
p = n·a - Base inradius:
r = a / (2·tan(π/n)) - Base area:
B = n·a² / (4·tan(π/n)) - Lateral surface area:
L = ½·p·ℓ - Surface area:
S = B + ½·p·ℓ - Volume:
V = (1/3)·B·h - Relation:
ℓ² = h² + r² - From S:
ℓ = 2(S - B)/p(requiresaandn) - From V:
h = 3V/B(requiresaandn)
Where it’s useful
Regular pyramids show up in architecture, packaging, art, and 3D modeling. A square pyramid is common in roof designs, while triangular pyramids (tetrahedrons) appear in structural frames and engineering models. If you are building a scale model, designing a decorative pyramid, or estimating material for a project, the surface area and volume outputs provide practical guidance.
Units: a, h, and ℓ use a length unit (for example, cm); S uses
the squared unit (such as cm²); and V uses the cubed unit (such as cm³).
The calculator runs locally in your browser, so your inputs remain private.
5 Fun Facts about Pyramids
Slant vs true height
Slant height ℓ and vertical height h meet via ℓ² = h² + r², where r is the inradius of the base polygon—Pythagoras hiding in the side face.
Pythagoras link
Square-base shortcut
For a square pyramid, each face is an isosceles triangle. Its altitude is √(ℓ² − (a/2)²), so total area splits neatly into four equal faces plus the base.
Face geometry
Egyptian slope
The Great Pyramid’s original slope was close to a 14:11 rise-run (~51.8°). Its height-to-base ratio almost matches a circle’s radius to half-circumference.
Historic curiosity
n-gon continuum
Change the base: triangle, pentagon, decagon—regular polygons all work. As n grows large, a regular pyramid approaches a cone.
Shape spectrum
Pyramid Calculator: FAQs
How do you calculate the volume of a pyramid?
Multiply the base area by the height, then divide by 3: V = (1/3)Bh.
How do you calculate the surface area of a square pyramid?
Use S = a² + 2aℓ, where a is the base side length and ℓ is the slant height.
What is the difference between height and slant height?
Height is the vertical distance from the base to the apex. Slant height is measured along a triangular face.
Can this calculator solve for height from volume?
Yes. If the base edge and volume are known, the calculator uses h = 3V/B.
Can this calculator solve for slant height from surface area?
Yes. For a regular pyramid, ℓ = 2(S - B)/p.
Which inputs are valid to solve a pyramid?
Provide a and n plus one of h, ℓ, S, or V. Two or more values are fine; the tool checks consistency.
What formulas are used?
B = n·a²/(4·tan(π/n)), S = B + ½·n·a·ℓ, V = (1/3)·B·h, and ℓ² = h² + r² with r = a/(2·tan(π/n)).
Does the calculator keep my data private?
Yes. Computation is entirely client-side; nothing is uploaded.
Can I change units or decimal places?
Yes. Choose a length unit for a, h, and ℓ. Surface area uses the squared unit; volume uses the cubed unit.