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Surface Area Calculator: Total Surface Area of 3D Shapes
Select a solid, enter its dimensions and get the total surface area, split into lateral (side) and base parts where that applies. Useful for painting, wrapping, coating and heat-transfer estimates.
Surface area formulas used on this page
| Solid | Total surface area |
|---|---|
| Cube | 6 a² |
| Box | 2(lw + lh + wh) |
| Closed cylinder | 2π r (r + h) |
| Sphere | 4 π r² |
| Solid hemisphere | 3 π r² |
| Cone | π r (r + l), with l = √(r² + h²) |
| Rectangular pyramid | lw + l √(h² + (w/2)²) + w √(h² + (l/2)²) |
| Cone frustum | π(R² + r²) + π(R + r) s |
Height versus slant height
Cones, pyramids and frustums have two different heights. The vertical height h runs straight from the base to the tip. The slant height runs along the sloping face, and it is the number that enters the side-area formula. The two are linked by the Pythagorean theorem: for a cone, l² = r² + h². A cone with radius 3 and height 4 has slant height 5, a total area of π × 3 × (3 + 5) = 24π, which is about 75.40, and a lateral area of 15π. If you measured a slant length on the object itself, convert to vertical height first, or treat the vertical height as unknown and solve for it with the Pythagorean relation.
What surface area is used for
- Coatings: paint, plating or insulation is bought by area. Add an allowance for waste, overlaps and multiple coats.
- Wrapping and packaging: the surface area of a box is the minimum material before folds and flaps.
- Heat and chemistry: heat loss and reaction rates often scale with exposed surface. A sphere has the smallest surface for a given volume, which is why small round shapes lose heat slowly.
For the volume of the same solids, use the volume calculator.
Open or closed? Common mistakes
The totals here include every face. An open tank, a cup, a tent without a floor or a pipe are missing one or both bases, so subtract the base areas shown in the breakdown or use the lateral value alone. A hollow pipe also has an inner surface, so its coating area is the outer and inner lateral areas added together. Another trap is mixing radius and diameter: doubling the radius quadruples a sphere surface area. Finally, results apply to ideal smooth geometric solids. Rough, ribbed or perforated surfaces have more area than the formula suggests.
Worked example
A cone has a base radius of 3 cm and a vertical height of 4 cm. Find the area of its whole surface, base included.
Inputs
| Solid | Cone (with base) |
|---|---|
| Length unit | cm |
| Height (perpendicular) | 4 |
| Radius (bottom radius for a frustum) | 3 |
Result
Total area of every face, including the bases.
| Working | πr(r + l) with slant height l = √(r² + h²) = 5 |
|---|---|
| Lateral (side) area | 47.1239 cm² |
| Base area(s) combined | 28.2743 cm² |
| Slant height | 5 |
Accuracy note: Results use standard double-precision arithmetic, so very large or very small values may be rounded. Check important results by another method.
Frequently asked questions
How do I find the surface area of a cylinder?
Add the curved side, 2 × π × radius × height, to the two circular ends, 2 × π × radius squared. Together that is 2πr(r + h) for a closed cylinder.
What is lateral surface area?
Lateral area is the area of the sides only, leaving out the top and bottom bases. For a cylinder it is the rectangle you get by unrolling the curved wall.
How is slant height different from height?
Height is measured straight up and down at a right angle to the base. Slant height is measured along a sloping face, so it is always longer than the height.
Why is the surface area of a sphere four times the area of its circle?
The result 4πr² equals four times the area of a great circle with radius r. Archimedes proved this relationship over two thousand years ago.
Do I include the base in the answer?
The headline total includes every face. For open shapes such as a cup or a tent, use the lateral value or subtract the missing base areas listed in the breakdown.
Related calculators
References and sources
- Standard solid geometry formulas for surface and lateral area; Archimedes, On the Sphere and Cylinder, for the sphere area result.