Pick a solid, enter its measurements and get the total surface area, with the base, curved side and other faces listed separately so you can leave out any you don't need. Results are also given in square metres and square feet.
Choose the solid.
Choose the unit your measurements are in.
Enter the measurements shown on the diagram.
Read the total and the separate faces.
Sphere: SA = 4 × π × r²
Cylinder: SA = 2 × π × r × (r + h)
Cone: SA = π × r × (r + √(r² + h²))
Cube: SA = 6 × a²
Rectangular tank: SA = 2 × (lw + lh + wh)
Conical frustum: SA = π(R² + r²) + π(R + r)√((R − r)² + h²)
Capsule: SA = 4 × π × r² + 2 × π × r × h
Spherical cap: SA = 2 × π × R × h + π × r²
Square pyramid: SA = a² + 2a√((a ÷ 2)² + h²)
Hemisphere: SA = 3 × π × r²
Ellipsoid: SA ≈ 4π((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ) ÷ 3)^(1/p), p = 1.6075
Surface area is the total area of the outside of a solid: every face, flat or curved, added together. The surface area calculator finds it for eleven solids, including the sphere, cylinder, cone, cube, rectangular tank, capsule, frustum and square pyramid. You choose the solid, the unit and the sizes marked on the drawing.
The result is split into its parts, such as the two circular ends and the curved side of a cylinder, because real jobs rarely need the whole thing. Painting an open tank leaves out the top, a label covers only the curved side, and a tent has no floor. Having the parts listed lets you add only the faces you need. Totals are also shown in square metres and square feet for buying paint, sheet metal, tiles or cloth.
1. Identify the solid and measure its sizes in one unit.
2. List its faces: flat ends, flat sides and curved surfaces.
3. Find the area of each face with its own formula, for example π r² for a circular end.
4. For a cone or pyramid, first find the slant height using Pythagoras.
5. Add the faces you need. Total surface area includes every face.
6. Write the answer in square units such as cm² or m².
A curved surface can often be flattened without stretching. The side of a cylinder unrolls into a rectangle whose width is the circumference 2π r and whose height is h, so its area is 2π r h. The side of a cone unrolls into a sector of a circle whose radius is the slant height, giving π r l. This is why slant height, not vertical height, appears in the curved area of cones, frustums and pyramids.
School textbooks use three names. Curved surface area, or CSA, is only the curved part. Lateral surface area is the sides without the top and bottom, which for a cylinder is the same as the curved part. Total surface area, or TSA, adds the ends. A closed cylinder therefore has TSA 2π r (r + h), while an open-top tank has one circle less. Reading which one a question asks for prevents most mistakes.
The surface of a sphere cannot be laid flat without distortion, which is why every flat map of the Earth stretches some regions. Its area is still simple: 4π r², exactly four times the area of a circle with the same radius, and equal to the curved surface of the cylinder that just encloses it. An ellipsoid has no simple exact formula, so the calculator uses Knud Thomsen's approximation, which is accurate to about one percent.
Lakshmi wants to paint the outside of a closed cylindrical water tank with radius 0.7 m and height 1.5 m, and needs the area to buy paint.
Cylinder: formula: SA = 2 × π × r × (r + h)
Two circular ends: 2 × π × 0.7² = 3.0788 m²
Curved side: 2 × π × 0.7 × 1.5 = 6.5973 m²
Add the parts: 3.0788 + 6.5973 = 9.6761 m²
Answer: Total surface area 9.6761 m²; Two circular ends 3.0788 m²; Curved side 6.5973 m²
Using the vertical height instead of the slant height for the curved surface of a cone or pyramid.
Giving the curved surface area when the question asks for the total, or the other way round.
Counting the base of an open container or a tent.
Using the diameter in place of the radius.
Writing the answer in cubic units instead of square units.
Estimating paint, primer or waterproofing for tanks, pillars and domes.
Working out sheet metal, cardboard or cloth needed to make a container, box or tent.
Sizing labels and wrappers for cans and bottles.
Calculating heat loss or cooling, which depends on surface area.
Mensuration problems on CSA and TSA in Class 9 and 10.
What is the difference between lateral and total surface area?
Lateral (curved) surface area leaves out the ends. Total surface area includes them. The calculator shows both parts.
How much paint do I need for a tank?
Find the surface area in square metres, leave out any face you won't paint, then divide by the coverage on the tin, often about 10 m² per litre per coat.
Why is the ellipsoid answer approximate?
A general ellipsoid has no simple exact formula. Knud Thomsen's approximation used here is within about 1.1% of the true value.
What is slant height?
The distance along the sloping side of a cone or pyramid from the base edge to the tip. It is found with Pythagoras from the radius (or half the edge) and the height.