Surface Area Calculator
Calculate the surface area of common 3D shapes.
Cube
Sphere
Cylinder
Cone
Quick Examples
Surface Area Formulas
| Shape | Formula | Notes |
|---|---|---|
| Cube | SA = 6s^2 | 6 square faces |
| Sphere | SA = 4 pi r^2 | Curved surface only |
| Cylinder | SA = 2 pi r^2 + 2 pi r h | Top + bottom + side |
| Cone | SA = pi r^2 + pi r s | Base + curved (s = slant) |
| Rectangular Prism | SA = 2(lw + lh + wh) | All 6 faces |
What is Surface Area?
Surface area is the total area of all surfaces (faces) of a 3D object. It's measured in square units like square meters (m2) or square feet (ft2).
Lateral vs Total Surface Area
Lateral surface area: Only the sides (not including top/bottom)
Total surface area: All surfaces including top and bottom
How to Use This Surface Area Calculator
- Choose the shape: click Cube, Sphere, Cylinder, or Cone at the top of the tool.
- Enter the dimensions the shape asks for. For a cone, enter the slant height (see below), not the vertical height.
- Click "Calculate Surface Area": the answer comes back in your input unit, squared.
What Is Surface Area?
Surface area is the total area of the "skin" covering a 3D object — every flat face and curved surface added together. Volume tells you how much fits inside; surface area tells you how much there is to paint, wrap, coat, or plate on the outside. It is always measured in square units (cm², m², ft²) because area is fundamentally two-dimensional.
Total vs Lateral Surface Area: When to Use Each
For cylinders, cones, and pyramids the two concepts split apart:
| Type | What it includes | Use it for |
|---|---|---|
| Total | Every face: sides plus top and bottom | Shrink-wrapping, full paint jobs, plating |
| Lateral | Sides only — no top or bottom | Can labels, column wraps, open-top containers |
Soda can example: a can with radius 3.3 cm and height 12 cm. Lateral area = 2 × π × 3.3 × 12 ≈ 248.81 cm² (the label area — what manufacturers print on). Total = 2 × π × 3.3² + 248.81 ≈ 317.24 cm² (the full skin, lids included).
Surface Area Formulas Explained
The pattern is always the same: add up the area of every face. "Unfolding" a shape into its flat net shows where each term comes from:
- Cube: SA = 6s². Six identical square faces.
- Rectangular prism: SA = 2(lw + lh + wh). Three pairs of identical rectangles — front/back, top/bottom, left/right.
- Cylinder (closed): SA = 2πr² + 2πrh. Two circular caps plus the curved side, which unrolls into a rectangle 2πr wide and h tall.
- Sphere: SA = 4πr². Exactly four times the area of its great circle — a fact Archimedes proved without calculus.
- Cone (closed): SA = πr² + πrℓ, where ℓ is the slant height. Circular base plus the curved side, which unrolls into a sector of a circle with radius ℓ.
Slant Height for Cones
The cone's curved side is measured along the slant height ℓ — the distance from the apex to the base edge along the outside — not the vertical height. Get it with the Pythagorean theorem:
Worked example: a cone with radius 4 and height 12 has slant height ℓ = √(16 + 144) = √160 ≈ 12.65. Lateral area = π × 4 × 12.65 ≈ 158.95. Total = π × 16 + 158.95 ≈ 209.22.
Worked Examples
- Cube, s = 5: SA = 6 × 25 = 150 square units.
- Box, 12 × 8 × 4: SA = 2 × (96 + 48 + 32) = 352 square units — the minimum gift wrap needed before overlap.
- Sphere, r = 11: SA = 4 × π × 121 ≈ 1520.53 square units (about the leather on a basketball).
- Cylinder, r = 3, h = 10: SA = 2 × π × 9 + 2 × π × 3 × 10 ≈ 56.55 + 188.50 = 245.04 square units.
Where Surface Area Calculations Matter
- Painting and coating: paint coverage is rated in area per gallon. A cylindrical column 3 ft in diameter and 10 ft tall has lateral area 2 × π × 1.5 × 10 ≈ 94.25 ft². At 350 ft² per gallon, that is about 0.27 gallons per coat — buy two quarts to be safe.
- Packaging: wrap, foil, and shrink film are sized by surface area; add 10–15% for overlap and waste.
- Heat transfer: radiators, heat sinks, and cooling fins are designed to maximize surface area in a small volume.
- Chemistry: powdered sugar dissolves faster than cubes because powder exposes far more surface area per gram.
The Cube-Square Law
Doubling every dimension makes surface area 4× larger but volume 8× larger. So as objects grow, their surface-to-volume ratio shrinks — which is why large animals retain heat better than small ones, and why cells stay tiny instead of growing into blobs.
Common Mistakes
- Using vertical height instead of slant height for a cone's curved side. The formula needs ℓ = √(r² + h²).
- Forgetting one circular cap: 2πr² covers both ends of a closed cylinder; πr² alone is just one.
- Using lateral area when total is needed (or vice versa). Check whether the top and bottom count for your job.
- Entering the diameter as the radius.
- Dropping the squared unit: surface area is always cm², m², or ft² — never plain cm.
Tip: when estimating paint or wrap, compute the surface area first, then divide by the coverage rate — and always round the material order up.
Surface Area of a Square Pyramid
A square pyramid has a square base plus four triangular faces. The triangles use the slant height — from a base edge midpoint to the apex:
Worked example: base 6, height 9: ℓ = √(9 + 81) = √90 ≈ 9.49. SA = 36 + 2 × 6 × 9.49 ≈ 149.84.
Hemispheres and Open Containers
Hemisphere: curved area 2πr² plus the flat base πr² gives total 3πr². A dome with radius 5: SA = 3 × π × 25 ≈ 235.62.
Open-top cylinder: drop one cap: SA = πr² + 2πrh. An open tank with radius 3 and height 10: SA = π × 9 + 2 × π × 3 × 10 ≈ 28.27 + 188.50 = 216.77.