Surface Area Calculator
Find the total, lateral, and base surface area of 10 common 3D solids. Choose your shape, enter its dimensions in any unit, and the calculator returns a full area breakdown with the formula and step-by-step working.
Formula
Worked example
Cylinder with r = 3 cm and h = 8 cm: lateral area = 2π × 3 × 8 = 48π ≈ 150.80 cm². Two circular ends = 2π × 3² = 18π ≈ 56.55 cm². Total = 66π ≈ 207.35 cm². The side wall accounts for about 72.7% of the total area.
What surface area measures
Surface area is the combined area of every outer face of a three-dimensional solid, measured in square units such as square centimetres, square inches, or square metres. Unlike volume, which fills the interior, surface area measures only the boundary that wraps around the object. It tells you how much material you need to paint, wrap, coat, or manufacture the outer shell of a solid, which is why it appears in packaging engineering, heat-transfer calculations, construction estimating, and biology (the surface-to-volume ratio governs how efficiently cells and organisms exchange heat and nutrients).
Lateral area versus base area
Most reference books report a single total surface area, but splitting it into lateral area (the side or curved surface) and base area (the flat top and bottom faces) is often more useful in practice. When painting the outside of a cylinder, for example, you might want to prime only the cylindrical side and leave the cut ends bare: that is the lateral area. When tiling the floor under a pyramid, you need only the base area. This calculator reports both components alongside the total so you can apply the number that matches your project. For a sphere and capsule the split between "lateral" and "base" is a matter of convention; here the full curved surface is treated as the lateral component.
Formulas for every shape
A sphere is a single continuous curved surface: A = 4πr². A hemisphere adds a flat circular base to half a sphere: A = 2πr² + πr² = 3πr². A cube has six equal square faces: A = 6a². A cylinder wraps a rectangle around two circles: lateral area 2πrh plus two ends 2πr². A cone has a circular base πr² and a lateral surface πrl where l = √(r² + h²) is the slant height. A rectangular box has three pairs of matching faces: A = 2(lw + lh + wh). A capsule consists of a cylindrical side 2πrh plus two hemispherical ends that together equal a full sphere 4πr². A square pyramid has a square base a² plus four congruent triangular faces whose total is 2al, where l = √((a/2)² + h²). A triangular prism has two triangular ends (area found by Heron's formula from the three side lengths) plus three rectangular side faces with combined area equal to the perimeter of the triangle times the prism length. A conical frustum (a cone with its top sliced off) has lateral area π(r₁ + r₂)l and two circular ends π(r₁² + r₂²), where l = √(h² + (r₁ - r₂)²) is the slant height.
Unit consistency and scaling
Switch the unit selector to work in metres, centimetres, millimetres, feet, inches, or yards. Every dimension must be in the same unit before the formula is applied, so the result always carries the matching square unit. Because area scales with the square of the linear dimension, a conversion factor must be squared: 1 foot equals 12 inches, so 1 square foot equals 144 square inches. Changing the size of a solid by a factor k multiplies its surface area by k² and its volume by k³. This difference, called the surface-to-volume ratio, decreases as objects grow larger and is the reason large animals lose heat more slowly per unit of mass than small ones.
Surface area formulas at a glance
| Solid | Total surface area | Lateral area | Base area |
|---|---|---|---|
| Sphere | 4πr² | 4πr² (all curved) | 0 |
| Hemisphere | 3πr² | 2πr² (dome) | πr² |
| Cube | 6a² | 4a² | 2a² |
| Cylinder | 2πr(r + h) | 2πrh | 2πr² |
| Cone | πr(r + l) | πrl | πr² |
| Rectangular box | 2(lw + lh + wh) | 2h(l + w) | 2lw |
| Capsule | 2πr(2r + h) | 2πrh | 4πr² |
| Square pyramid | a² + 2al | 2al | a² |
| Triangular prism | 2 × tri + perimeter × L | perimeter × L | 2 × tri |
| Conical frustum | π(r₁²+r₂²+(r₁+r₂)l) | π(r₁+r₂)l | π(r₁²+r₂²) |
r = radius, a = edge or base side, h = height, l = slant height, L = prism length, r₁/r₂ = frustum radii.
Frequently asked questions
What is the difference between total surface area and lateral surface area?
Total surface area includes every face of the solid: the curved or rectangular sides AND the flat base(s) and top(s). Lateral surface area counts only the side surfaces, excluding the base faces. For a cylinder, for example, the lateral area is the unrolled rectangle 2πrh, while the total adds two circular ends to give 2πr(r + h). This calculator reports both so you can pick the figure that matches your project.
Why does a cone use slant height instead of vertical height?
The curved surface of a cone slopes from the base rim up to the apex, so its actual length is the slant height l = √(r² + h²), calculated from the radius and vertical height with the Pythagorean theorem. Using the vertical height h alone would understate the lateral surface and give the wrong area. The calculator computes l automatically from the radius and height you enter.
Does the calculator include the ends of a cylinder or cone?
Yes, the total surface area includes every face. For a cylinder that means both circular ends plus the lateral side. For a cone it means the circular base plus the lateral surface. If you need only the curved side (for example to estimate paint for the side of a tank without the ends), read the "Lateral area" output instead.
How do I calculate the surface area of a triangular prism?
A triangular prism has two triangular ends and three rectangular side faces. Enter the three side lengths of the triangular cross-section and the prism length. The calculator uses Heron's formula, s = (a + b + c) / 2 then area = √(s(s-a)(s-b)(s-c)), to find the triangle area, doubles it for both ends, and adds the perimeter of the triangle times the prism length for the three rectangular faces.
What is a conical frustum and how is its surface area calculated?
A conical frustum is a cone with the pointed top sliced off by a plane parallel to the base, leaving two circular faces of different radii r₁ and r₂. Its lateral surface area is π(r₁ + r₂) × l, where l = √(h² + (r₁ - r₂)²) is the slant height of the sloped side. The two circular ends add π(r₁² + r₂²). Setting r₂ = 0 reduces the frustum to a full cone.
How does changing the unit affect the result?
Selecting a different unit (such as switching from centimetres to inches) rescales every dimension and returns the area in the matching square unit. Because area is a two-dimensional measure, the conversion factor is squared: 1 inch = 2.54 cm, so 1 in² = 6.4516 cm². The underlying computation always works in metres and then converts the result, so you can enter dimensions in any supported unit without risk of mixing them.