The Square Surface Area Calculator calculates the area of a flat square by squaring its side (A = a²), or from its diagonal using A = d² ÷ 2. It also shows the perimeter and diagonal.
Square Surface Area Calculator
Area of a flat square, or the surface area of a square-based prism or cube.
If you need the surface area of a 3D square shape, the square prism mode adds the two square ends and four rectangular sides: A = 2a² + 4ah. Enter your measurements below.
Formula
Flat square: A = a², P = 4a, d = a√2
From the diagonal: A = d² ÷ 2, a = d ÷ √2
Square prism: A = 2a² + 4ah
Where:
- A = area or surface area; P = perimeter
- a = side length; d = diagonal; h = prism height
A cube is a square prism with h = a, so its surface area is 6a².
How the Calculation Works
Step 1: For a flat square, multiply the side by itself.
Step 2: From a diagonal, square the diagonal and halve it (the diagonal splits the square into two right triangles).
Step 3: For a square prism, add two square ends (2a²) and four rectangles (4 × a × h).
Variables Explained
| Variable | Meaning | Unit |
|---|---|---|
| a | Side length | cm, m, in, ft |
| d | Diagonal (corner to corner) | cm, m, in, ft |
| h | Height of the prism | cm, m, in, ft |
| A | Area / surface area | cm², m², in², ft² |
Example Calculation
Flat square
Given: side = 5 cm
Result: A = 5² = 25 cm²; P = 20 cm; d ≈ 7.07 cm
From the diagonal
Given: diagonal = 10 in
Result: A = 10² ÷ 2 = 50 in²; side ≈ 7.07 in
Square prism
Given: base side = 5 cm, height = 10 cm
Calculation: A = 2 × 25 + 4 × 5 × 10 = 50 + 200
Result: A = 250 cm²
Common Area Conversion Table
| Unit | Equivalent |
|---|---|
| 1 m² | 10,000 cm² |
| 1 ft² | 144 in² |
| 1 yd² | 9 ft² |
| 1 m² | 10.7639 ft² |
Shape Properties
A square has four equal sides and four right angles. Its diagonal is always side × √2 (about 1.414 × side), and doubling the side makes the area four times larger.
Dimensions Explained
Square vs cube vs square prism
- Square: flat, area = a².
- Cube: 3D with all edges equal, surface area = 6a².
- Square prism: square ends with a different height, surface area = 2a² + 4ah.
Frequently Asked Questions
What is the formula for the area of a square?
Area = side × side (a²). A 5 cm square has an area of 25 cm².
How do you find the area of a square from its diagonal?
Square the diagonal and divide by 2: A = d² ÷ 2.
What is the surface area of a square?
A flat square has area a². If you mean a cube, its surface area is 6a².
How do you find the side of a square from its area?
Take the square root: a = √A. An area of 49 ft² means 7 ft sides.
How many square feet is a 12 by 12 room?
12 × 12 = 144 ft².