What is the formula for the volume of a pyramid?

The volume of any pyramid is one third of the base area multiplied by the perpendicular height: V = (1/3) x Base Area x h. For a square-base pyramid with side a and height h, this gives V = (1/3) x a^2 x h. For a rectangular base with length l and width w, it is V = (1/3) x l x w x h. For a triangular base, the base area is (1/2) x base x height of the triangle.

How do you find the slant height of a pyramid?

The slant height (l) runs from the apex down to the midpoint of a base edge. For a square-base pyramid with base side a and perpendicular height h, slant height = sqrt(h^2 + (a/2)^2). For a rectangular pyramid, there are two different slant heights: one for the length side, sqrt(h^2 + (w/2)^2), and one for the width side, sqrt(h^2 + (l/2)^2). The slant height is needed to calculate the lateral surface area.

What is the difference between lateral surface area and total surface area of a pyramid?

Lateral surface area is the combined area of the triangular faces only, not including the base. Total surface area adds the base area to the lateral area. For a square-base pyramid: lateral area = 2 x a x slant height, and total surface area = a^2 + 2 x a x slant height. If you are painting the outside of a solid pyramid (base included), use total surface area. If you are calculating just the sloping faces (for example, for a tent or a roof), use lateral surface area.

Pyramid Volume and Surface Area Calculator

Calculate the volume and total surface area of a pyramid with a square, rectangular, or triangular base. Enter the base dimensions and perpendicular height to get instant results with full working.

Calculate.co.nz is proud to be partnered with Premium Homes, a recognised leader in eco-friendly, sustainable, and energy-efficient homebuilding. With a dedicated team and award-winning experience, they create homes that prioritise health, comfort, and long-term performance. Their founders, Andrew and Kelly, set out to raise the standard of residential construction in New Zealand by combining practical building expertise with a clear commitment to doing things better for homeowners.

Standard geometry Formulas based on standard Euclidean geometry for right pyramids with a flat polygonal base and apex directly above the centroid.

1. Pyramid Type

Base shape

Base side length (a)

units

Base width (b)

units

Triangle base (b)

units

Triangle height (perpendicular, t)

units

2. Pyramid Height

Perpendicular height (h)

units

Please enter positive numbers for all dimensions.

Unit label (optional, for display only)

Results

Volume
-
cubic units

Total Surface Area

square units

Lateral Surface Area

square units

Slant Height

units

Dimensions

Base shape-

Base area-

Perpendicular height (h)-

Slant height (l)-

Volume-

Surface Areas

Base area-

Lateral surface area-

Total surface area-

Worked Example

Pyramid Formulas

A right pyramid has a flat polygonal base and an apex directly above the centre of the base. The perpendicular height (h) is the straight-line distance from the apex to the base. The slant height (l) runs from the apex to the midpoint of a base edge along the face of the pyramid.

Base Type	Base Area	Slant Height	Volume	Total Surface Area
Square (side a)	a^2	sqrt(h^2 + (a/2)^2)	(1/3) x a^2 x h	a^2 + 2al
Rectangle (l x w)	l x w	Two values: sqrt(h^2+(w/2)^2) and sqrt(h^2+(l/2)^2)	(1/3) x l x w x h	lw + l x sl_w + w x sl_l
Right triangle (base b, height t)	(1/2) x b x t	Varies per face	(1/3) x (1/2) x b x t x h	Base area + sum of three lateral faces

How Volume Is Calculated

All pyramids share the same volume formula: V = (1/3) x Base Area x h. This holds regardless of the base shape. The factor of one third arises because a pyramid fills exactly one third of the volume of a prism with the same base and height. You can verify this by filling a pyramid-shaped container three times to fill an equivalent prism.

How Surface Area Is Calculated

The total surface area of a pyramid equals the base area plus the combined area of all the triangular lateral faces. Each lateral face is a triangle with a base equal to one edge of the pyramid's base and a height equal to the slant height (l). For a square-base pyramid with side a, there are four identical lateral faces, each with area (1/2) x a x l, giving a lateral area of 2al. Total surface area = a^2 + 2al.

For a rectangular-base pyramid, the two pairs of lateral faces have different slant heights (one slant height for the length direction, another for the width direction). The calculator handles both.

Square Pyramid Worked Example

For a square-base pyramid with base side a = 5 units and height h = 4 units:

Slant height: l = sqrt(4^2 + (5/2)^2) = sqrt(16 + 6.25) = sqrt(22.25) = 4.717 units
Volume: V = (1/3) x 5^2 x 4 = (1/3) x 100 = 33.33 cubic units
Lateral surface area: 2 x 5 x 4.717 = 47.17 square units
Total surface area: 25 + 47.17 = 72.17 square units

These results match the calculator's defaults.

Practical Uses

Pyramid calculations appear in architecture (roof design, monuments), civil engineering (earthworks and stockpile volumes), and manufacturing (cone and hopper design). The Great Pyramid of Giza has a square base of approximately 230.4 m per side and an original height of about 146.5 m, giving a volume of roughly 2.58 million cubic metres.

Related Calculators

Maths and Stats Calculators: full collection of geometry and maths tools.
Cube Surface Area and Volume Calculator: volume and surface area for a cube.
Area of a Triangle Calculator: find triangle area from base and height.
Area of a Triangle (Heron) Calculator: triangle area from three side lengths.
Annulus Calculator: area and perimeter of a ring shape.

Sources and method: Standard Euclidean geometry formulas for right pyramids. Weisstein, Eric W. "Pyramid." MathWorld. Coxeter, H.S.M. Introduction to Geometry, 2nd ed. (1969).

This calculator applies to right pyramids where the apex is directly above the centroid of the base. Oblique pyramids (where the apex is offset) use the same volume formula but have different lateral face areas not covered here.

If you've found a bug, or would like to contact us, or learn more about James Graham and Calculate.co.nz.

Calculate.co.nz is partnered with Interest.co.nz for New Zealand's highest quality calculators and financial analysis.

Calculate.co.nz is the sister site of CalculatorHub.com, the world's largest calculator website by tool count.

All calculators and tools are provided for educational and indicative purposes only and do not constitute financial advice.

Calculate.co.nz is proudly part of the Realtor.co.nz group, New Zealand's leading property transaction literacy platform, helping Kiwis understand the home buying and selling process from start to finish. Whether you're a first home buyer navigating your first property purchase, an investor evaluating your next acquisition, or a homeowner planning to sell, Realtor.co.nz provides clear, independent, and trustworthy guidance on every step of the New Zealand property transaction journey.

Calculate.co.nz is also partnered with Health Based Building and Premium Homes to promote informed choices that lead to better long-term outcomes for Kiwi households.

Calculate.co.nz is hosted in Auckland via SiteHost new Zealand.

All content on this website, including calculators, tools, source code, and design, is protected under the Copyright Act 1994 (New Zealand). No part of this site may be reproduced, copied, distributed, stored, or used in any form without prior written permission from the owner.

About & trust: Why Calculate is NZ's most comprehensive · By the Numbers · How we compare · Editorial standards · How we keep data current · NZ finance glossary · Research & data · Financial literacy NZ · About · Privacy policy · Terms of use

Reviewed and maintained. Last reviewed 2026-07-02 and checked on a twice-monthly cycle against IRD, RBNZ and Stats NZ. How we keep data current.