Surface Area Calculator
This professional-grade surface area calculator helps students, engineers, and makers compute total and lateral surface areas for common 3D shapes. It supports multiple units, instant validation, and a clear, accessible interface designed for accuracy and speed.
Calculator
Results
Details
Data Source and Methodology
Authoritative source: Weisstein, Eric W. “Surface Area.” MathWorld—A Wolfram Web Resource (accessed 2025). https://mathworld.wolfram.com/SurfaceArea.html.
All calculations strictly follow the formulas and definitions provided by this source.
The Formulas Explained
Cube (edge a): S = 6a^2
\[ S_{\text{cube}} = 6a^2 \]
Rectangular prism (l, w, h):
\[ S_{\text{box}} = 2(lw + lh + wh) \]
Sphere (radius r):
\[ S_{\text{sphere}} = 4\pi r^2 \]
Cylinder (radius r, height h):
\[
S_{\text{lateral}} = 2\pi r h,\quad
S_{\text{total, closed}} = 2\pi r(h + r),\quad
S_{\text{open-top}} = 2\pi r h + \pi r^2,\quad
S_{\text{open-both}} = 2\pi r h
\]
Cone (right circular; radius r, height h, slant s = √(r² + h²)):
\[
S_{\text{lateral}} = \pi r s,\quad
S_{\text{total (with base)}} = \pi r s + \pi r^2
\]
Square pyramid (base edge b, height h, slant s = √((b/2)² + h²)):
\[
S_{\text{lateral}} = 2 b s,\quad
S_{\text{total}} = 2 b s + b^2
\]
Triangular prism (base sides a,b,c; length L; base area A via Heron):
\[
p = \tfrac{1}{2}(a+b+c),\quad
A = \sqrt{p(p-a)(p-b)(p-c)},\quad
S_{\text{lateral}} = (a+b+c)L,\quad
S_{\text{total}} = (a+b+c)L + 2A
\]
Glossary of Variables
- a, b, c: side lengths (length)
- l, w, h: length, width, height (length)
- r: radius (length)
- s: slant height (length)
- L: prism length (length)
- S: surface area (area)
- lateral surface area: the area of the sides, excluding bases
- base area: the combined area of base faces
How It Works: A Step-by-Step Example
Example: Closed cylinder with radius r = 4 cm and height h = 10 cm.
- Choose Cylinder and set unit to centimeters.
- Enter r = 4, h = 10 and Ends = Closed (both).
- Use the total surface area formula:
S = 2πr(h + r). - Substitute: S = 2π × 4 × (10 + 4) = 112π ≈ 351.858 cm².
- The calculator also reports lateral area 2πrh = 80π ≈ 251.327 cm² and base area 2πr² = 32π ≈ 100.531 cm².
FAQ
Can I exclude one or both bases for cylinders?
Yes. Use the Ends option: Closed (both), Open-top, Open-bottom, or Open-both to control which base faces are included.
Does the cone calculation require slant height?
No. Enter radius and vertical height; the tool computes slant height using s = √(r² + h²).
Which units are supported?
mm, cm, m, in, ft, yd. Results are shown in your chosen unit squared and in square meters for reference.
How precise are the results?
Double-precision arithmetic is used internally. You control the displayed precision (0–6 decimals) without affecting internal accuracy.
What validations are enforced?
All dimensions must be positive. Triangular prism sides must satisfy the triangle inequality. Inline messages explain any issue and how to fix it.
Is this suitable for education and engineering?
Yes. The tool follows well-established formulas, offers a step-by-step breakdown, and is optimized for clarity and accessibility.