Surface Area Calculator
Choose a solid, enter its measurements in any length unit, and get its lateral and total surface area as exact values — in terms of π or a square root where that applies — with the decimal, the formula with your numbers put in, the net of the solid with each face's area marked, a labeled figure, and the area in other units. Free, no sign-up; it all runs in your browser.
Results
48π cm²
Total surface area
150.8 cm²
Total as a decimal
30π cm²
Lateral (curved) area
Other measurements
| Measurement | Exact | Decimal |
|---|---|---|
| Base area | 9π cm² | 28.27 cm² |
The working
B = π × r²
LA = 2 × π × r × h
SA = LA + 2B
B = π × r² = π × 3² = 9π
LA = 2 × π × r × h = 2 × π × 3 × 5 = 30π
SA = LA + 2B = 30π + 2 × 9π = 48π
The net
Unfold the solid and the surface lies flat: add up the areas of the pieces to get the total.
Figure
The same area in other units
| Unit | Area |
|---|---|
| mm² | 15,079.64 |
| cm² | 150.7964 |
| m² | 0.01507964 |
| km² | 1.508E-8 |
| in² | 23.3735 |
| ft² | 0.1623159 |
| yd² | 0.01803511 |
| mi² | 5.8223E-9 |
| ha | 0.000001507964 |
| ac | 0.000003726261 |
How to Use the Surface Area Calculator
- 1. Choose the solid. Cube, rectangular prism, triangular prism, cylinder, cone, sphere, hemisphere, pyramid with a rectangular base, cone frustum or ellipsoid.
- 2. Enter the measurements. Whole numbers, decimals and fractions such as 3/4 are read exactly. Use one unit throughout, or tick the box to give each measurement its own unit and choose the unit of the answer.
- 3. Read the areas. The total surface area comes first, in exact form such as 48π cm², with the lateral area beside it and a decimal rounded to the places you choose. Base areas and slant heights are listed below.
- 4. Follow the net. The net unfolds the solid into flat pieces, each marked with its area expression; the total is their sum. The working shows every formula with your numbers in it.
Lateral and Total Surface Area
The lateral area is the area of the sides only — the faces that are not bases. The total surface area adds the base or bases: a closed can has two circular ends, a cone has one, and a sphere has no separate bases at all. Whether to include the bases depends on the question: the label around a can is lateral area, the metal needed to make it is total.
| Solid | Lateral area | Total surface area |
|---|---|---|
| Cube | 4a² | 6a² |
| Rectangular prism | 2h(l + w) | 2(lw + lh + wh) |
| Triangular prism | (a + b + c) × L | Lateral area + 2B, where B is the triangle's area |
| Cylinder | 2πrh | 2πrh + 2πr² = 2πr(r + h) |
| Cone | πrl, where l = √(r² + h²) | πrl + πr² = πr(r + l) |
| Sphere | none (one curved surface) | 4πr² |
| Hemisphere | 2πr² (curved part) | 3πr² (curved part plus the flat circle) |
| Pyramid, rectangular base | l × s₁ + w × s₂, with s₁ = √(h² + (w/2)²) and s₂ = √(h² + (l/2)²) | Lateral area + l × w |
| Cone frustum | π(R + r) × l, where l = √((R − r)² + h²) | Lateral area + πR² + πr² |
| Ellipsoid | none | No closed form — approximate |
The slant height l of a cone or frustum is the length along the surface, not the vertical height h. The calculator finds it from the Pythagorean theorem and keeps it exact, so a cone of radius 3 and height 4 has slant height exactly 5.
Worked Examples
- Cylinder. Radius 3 and height 5: lateral 2π × 3 × 5 = 30π, two bases 2 × 9π = 18π, total 48π ≈ 150.80.
- Cone. Radius 3 and height 4: l = 5, lateral π × 3 × 5 = 15π, base 9π, total 24π ≈ 75.40.
- Sphere and hemisphere. Radius 3: sphere 4π × 9 = 36π ≈ 113.10. The hemisphere has half of that on its curved side, 18π, plus the flat circle 9π: 27π ≈ 84.82.
- Square pyramid. A 6 by 6 base and height 4: each face has slant height √(16 + 9) = 5, so lateral area is 4 × ½ × 6 × 5 = 60, base 36, total 96.
- Cone frustum. Bottom radius 4, top radius 2, height 3: l = √13, lateral π × 6 × √13 = 6π√13 ≈ 67.97, bases 16π + 4π = 20π, total 20π + 6π√13 ≈ 130.79.
What a Net Shows
A net is the solid cut along its edges and laid flat. A box unfolds into six rectangles, a prism into rectangles plus two end polygons, a pyramid into a base with triangles around it. Round solids unroll too: the side of a cylinder becomes a rectangle 2πr wide and h tall, and the side of a cone becomes a circular sector of radius l whose arc has length 2πr, so its angle is 360° × r / l — 216° for r = 3 and l = 5. A sphere cannot be unrolled without stretching, which is why every flat map of the Earth distorts something.
Common Mistakes
- Using the vertical height of a cone, pyramid or frustum where the lateral formula needs the slant height.
- Counting the bases when the question asks for lateral area, or leaving them out of a total.
- Entering a diameter where a radius is wanted.
- Forgetting that a hemisphere's total includes the flat circle: 3πr², not 2πr².
Frequently Asked Questions
- What is the difference between lateral and total surface area?
- Lateral surface area counts only the sides of a solid; total surface area also counts its base or bases. For a cylinder of radius 3 and height 5 the lateral area is 30π and the total, with both circular ends, is 48π.
- How do you find the surface area of a cylinder?
- Add the curved side and the two ends: SA = 2πrh + 2πr². The side unrolls into a rectangle with width equal to the circumference, 2πr, and height h. Each end is a circle of area πr².
- How do you find the surface area of a cone?
- First find the slant height l = √(r² + h²), then add the curved side πrl to the base πr²: SA = πr(r + l). A common error is using the vertical height h in place of l in the first term.
- What is the surface area of a sphere?
- SA = 4πr², which is four times the area of a great circle πr². Archimedes proved this result. A sphere of radius 3 has surface area 36π, about 113.10 square units.
- How do you find the surface area of a rectangular prism?
- Add the areas of the three pairs of opposite faces: SA = 2(lw + lh + wh). A box measuring 6 by 4 by 3 has 2(24 + 18 + 12) = 108 square units, of which the four side faces make up 60.
- What is a net in geometry?
- A net is a flat pattern that folds into a solid. Because the pieces are flat, the surface area of the solid equals the combined area of its net, which is why finding surface area often starts with drawing it.
- Is the surface area of an ellipsoid exact here?
- No. A general ellipsoid has no closed-form surface area; the exact value needs an elliptic integral. The calculator uses Knud Thomsen's approximation, 4π × ((ap bp + ap cp + bp cp) / 3)1/p with p ≈ 1.6075, which is within about 1% and exact for a sphere. The volume formula, by contrast, is exact.