The Square Pyramid Surface Area Calculator calculates the total surface area of a pyramid with a square base by adding the base (a²) to the four triangular faces (2as).
Square Pyramid Surface Area Calculator
Total, lateral and base area of a square pyramid from base side and height or slant height.
You can enter the base side with either the vertical height or the slant height. The calculator converts one to the other using the Pythagorean theorem and also shows the lateral area and base area separately.
Formula
A = a² + 2as
Slant height from height: s = √(h² + (a ÷ 2)²)
Height from slant height: h = √(s² − (a ÷ 2)²)
Where:
- A = total surface area
- a = base side length
- s = slant height (apex to the middle of a base edge)
- h = vertical height (apex to the center of the base)
- Lateral area L = 2as; base area B = a²
How the Calculation Works
Step 1: If you entered the vertical height, find the slant height: s = √(h² + (a/2)²).
Step 2: Find the area of the four triangles: 4 × ½ × a × s = 2as.
Step 3: Add the square base a² to get the total surface area.
Variables Explained
| Variable | Meaning | Unit |
|---|---|---|
| a | Base side | cm, m, in, ft |
| h | Vertical height | cm, m, in, ft |
| s | Slant height | cm, m, in, ft |
| A | Total surface area | cm², m², in², ft² |
Example Calculation
Using the vertical height
Given: base side = 6 cm, height = 4 cm
Slant height: s = √(4² + 3²) = √25 = 5 cm
Calculation: A = 6² + 2 × 6 × 5 = 36 + 60
Result: A = 96 cm² (lateral area 60 cm², base 36 cm²)
Using the slant height
Given: base side = 10 ft, slant height = 13 ft
Calculation: A = 10² + 2 × 10 × 13 = 100 + 260
Result: A = 360 ft²; height = √(13² − 5²) = 12 ft
Common Area Conversion Table
| Unit | Equivalent |
|---|---|
| 1 m² | 10,000 cm² |
| 1 ft² | 144 in² |
| 1 m² | 10.7639 ft² |
Shape Properties
A square pyramid has 5 faces (1 square and 4 identical triangles), 8 edges and 5 vertices. The volume is V = a²h ÷ 3 — one third of a box with the same base and height.
Dimensions Explained
Three different “heights”
- Vertical height (h): straight down from the apex to the base center.
- Slant height (s): down the middle of a face to the midpoint of a base edge.
- Lateral edge: from the apex to a corner of the base — the longest of the three, and not used here.
Frequently Asked Questions
How do you find the surface area of a square pyramid?
A = a² + 2as: the square base plus four triangles of base a and height s.
How do you find the surface area of a square pyramid with only the height?
Find the slant height first, s = √(h² + (a/2)²), then use A = a² + 2as.
What is the lateral area of a square pyramid?
The four triangular faces only: L = 2as.
How do you find the slant height of a square pyramid?
Use the right triangle formed by the height and half the base side: s = √(h² + (a/2)²).
How much material do I need to cover a pyramid roof?
Use the lateral area (no base). Add an allowance for overlaps and cuts, which depends on the roofing material.