Pyramid (square based) - area and volume
Online calculator — enter the values and get the result instantly, with the formula and a worked example.
A = Area
V = volume
a = side "a"
h = height
s = slant triangle height
BA = basement area
LA = lateral area
A square-based pyramid is a three-dimensional solid whose base is a square and whose four triangular faces rise from the base edges to meet at a single point called the apex. It therefore has 5 faces, 8 edges and 5 vertices, and when the apex sits directly above the centre of the square it is called a right pyramid, which is perfectly symmetrical. Two heights matter for this shape: the vertical height, measured straight from the centre of the base to the apex, and the slant height, measured along a triangular face from the apex to the midpoint of a base edge. Its volume equals one third of the base area multiplied by the vertical height, so a pyramid always holds a third of the prism that shares the same base and height. Its surface area is the sum of the square base and the four identical triangular faces. Square-based pyramids appear throughout architecture and history, most famously in the pyramids of Egypt, and are still used today in rooftops, monuments and packaging design.
Volume V
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Surface area S
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Slant height s
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| Base area BA | – |
| Lateral surface LA | – |
Drag a base corner (a) or the apex (h) Re-center
Worked example
The side of the base of the pyramid is 4 cm, the height of the pyramid is 6 cm. Let's calculate the area and volume of the pyramid.
Surface area of the pyramid:
First, we calculate the area of the base as SC = a * a = 4 * 4 = 16 cm²
To calculate the lateral area, we need the slant height, which we calculate using the Pythagorean theorem c² = a² + b². Instead of a², we use (a/2)², so 4/2 = 2, and instead of b², we use the height 6. Then c² = 2² + 6² = 40
c = √40 = 6.32.
The slant height is thus 6.32 cm.
The lateral area is then calculated as SC = 4 * ((a * s) / 2) = 4 * ((4 * 6.32) / 2) = 50.56 cm²
The total area of the pyramid is then the sum of the base and the lateral area, thus 16 + 50.56 = 66.56 cm²
Volume of the pyramid:
Using the formula V = (SZ * v) / 3, we substitute the values V = (16 * 6) / 3 = 32 cm³
