Pyramid Calculator

What is a Square Pyramid?

A square pyramid is a three-dimensional geometric shape that has a square base and four triangular faces that meet at a single point called the apex. It is one of the Johnson solids and a special case of a pyramid where the base is a regular polygon (a square). The Great Pyramid of Giza is one of the most famous examples of a square pyramid.

This calculator helps you compute all essential properties of a square pyramid: volume (V), lateral surface area (L), base surface area (B), total surface area (A), slant height (s), lateral edge length (e), base perimeter (P), side face slope (m), and side face angle (θ). You can enter any two known values and the tool will automatically calculate all remaining properties.

Square Pyramid Formulas

The following formulas are used to calculate the properties of a square pyramid where a is the side length of the base and h is the height from the base to the apex:

Volume: V = (1/3)a²h

The volume of a pyramid is one-third the area of the base times the height. This is because a pyramid is essentially a cone with a polygonal base.

Slant Height: s = √(h² + (a/2)²)

The slant height is the distance from the apex to the midpoint of any base edge, calculated using the Pythagorean theorem. It represents the height of each triangular face.

Lateral Surface Area: L = 2as = a√(a² + 4h²)

This is the total area of all four triangular faces. Each triangular face has an area of (1/2) × a × s, and there are four of them.

Base Surface Area: B = a²

This is simply the area of the square base.

Total Surface Area: A = L + B = a(a + √(a² + 4h²))

The total surface area combines the lateral surface area and the base area.

Lateral Edge Length: e = √(h² + a²/2)

The lateral edge is the distance from the apex to a corner of the base.

Side Face Slope: m = h / (a/2)

The slope of the triangular faces, calculated as rise over run.

Side Face Angle: θ = tan&supmin;¹(h / (a/2))

The angle between the base plane and each triangular face.

How to Use the Pyramid Calculator

Using this square pyramid geometry calculator is simple:

1. Select Known Values: Choose which two pyramid properties you know from the dropdown menu. Options include side & height, side & surface area, side & volume, side & slant height, height & volume, or height & slant height.
2. Enter Values: Input your known measurements. All values must be positive numbers.
3. View Results: The calculator instantly computes all remaining properties including volume, surface areas, slant height, edge length, and angles.

Example Calculation

Example: Find the volume, surface area, and slant height of a square pyramid with base side length a = 6 m and height h = 10 m.

• Slant Height: s = √(10² + (6/2)²) = √(100 + 9) = √109 = 10.440 m
• Volume: V = (1/3) × 36 × 10 = 120 m³
• Base Area: B = 36 m²
• Lateral Area: L = 2 × 6 × 10.440 = 125.28 m²
• Total Surface Area: A = 36 + 125.28 = 161.28 m²