Surface Area Calculator
CSA, LSA and TSA of 14 solids with formulas, slant-height working and diagrams.
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Total surface area in other units
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About the Surface Area Calculator
Choose a solid, type its sizes and get every surface area it has: the curved surface area (CSA) of round solids, the lateral surface area (LSA) of boxes, pyramids and prisms, the total surface area (TSA), and the areas of the ends or base — each with its formula, your numbers substituted step by step, and a labelled diagram. Where a slant height is needed, as for cones, frustums and pyramids, the Pythagoras step that finds it is shown too; you can also start from the slant height.
It covers cubes, cuboids, cylinders, hollow cylinders, cones, frustums, spheres, hemispheres, spherical caps, pyramids (square, rectangular or regular-polygon base), prisms, capsules, ellipsoids and tori. With whole-number or fractional sizes the answers are also exact in terms of π (TSA = 24π cm²), π can be 22/7 or 3.14 as many textbooks ask, and the main area is converted to other units.
How to use it
- Choose the solid and the unit your sizes are in.
- Type the sizes labelled on the diagram — radius or diameter, height or slant height, as you know them.
- Read the total surface area and, in the tiles, the curved or lateral area and the area of the ends; the steps show each formula with your numbers, and exact π forms when your inputs allow.
- Set π to 22/7 or 3.14 if the question says so, then copy or download the working. The table converts the main area to other units.
Examples
r = 7 cm, slant height l = 25 cm
CSA = 550 cm², TSA = 704 cm²
CSA = πrl = (22/7) × 7 × 25 and TSA = πr(l + r) = 22 × 32. The height is √(25² − 7²) = 24 cm.
r = 7 cm, h = 10 cm
CSA = 440 cm², TSA = 748 cm²
CSA = 2πrh; TSA = 2πr(r + h) = 2 × (22/7) × 7 × 17.
r = 7 cm
CSA = 308 cm², TSA = 462 cm²
A solid hemisphere has its flat circle too: TSA = 2πr² + πr² = 3πr².
R = 8, r = 4, h = 3
l = 5, CSA = 60π ≈ 188.4956, TSA = 140π ≈ 439.823
l = √(3² + (8 − 4)²) = 5; CSA = π(R + r)l; TSA adds πR² + πr².
base edge a = 6, height h = 4
slant height 5, LSA = 60, TSA = 96
s = √(4² + 3²) = 5; LSA = 4 × ½ × 6 × 5; TSA adds the base 6² = 36.
R = 6, r = 2
S = 48π² ≈ 473.741
By Pappus’s theorem S = (2πr)(2πR) = 4π²Rr.
Common uses
- Checking NCERT Class 9 and 10 surface-area exercises, including the π = 22/7 convention.
- Working out the sheet metal, fabric or paint needed to cover a tank, cone, dome or box (add an allowance for overlaps and waste).
- Finding the slant height of a cone, frustum or pyramid from its height, or the height from the slant height.
- Comparing the curved and total surface of a container with and without its lid or base.
Curved, lateral and total surface area
The curved surface area (CSA) of a round solid is the area of its curved part only — the side of a cylinder, the slanted surface of a cone, the dome of a hemisphere. For solids with flat faces the same idea is called the lateral surface area (LSA): the four walls of a cuboid, the triangular faces of a pyramid, the rectangles of a prism. The total surface area (TSA) adds the flat ends, bases or lids. An open tank or a cone-shaped tent uses the CSA; a closed box or solid uses the TSA. This follows the terms used in NCERT textbooks.
The formulas
- Cube: LSA = 4a², TSA = 6a². Cuboid: LSA = 2h(l + b), TSA = 2(lb + bh + hl).
- Cylinder: CSA = 2πrh, TSA = 2πr(r + h). Hollow cylinder: TSA = 2πRh + 2πrh + 2π(R² − r²).
- Cone: l = √(r² + h²), CSA = πrl, TSA = πr(l + r). Frustum: l = √(h² + (R − r)²), CSA = π(R + r)l, TSA = π(R + r)l + πR² + πr².
- Sphere: S = 4πr². Hemisphere: CSA = 2πr², TSA = 3πr². Spherical cap: CSA = 2πrh.
- Pyramid: LSA = ½ × base perimeter × slant height (two slant heights for a rectangular base); TSA = LSA + base. Prism: LSA = base perimeter × length, TSA = LSA + 2 × base.
- Torus: S = 4π²Rr. Ellipsoid: no elementary formula — computed exactly from elliptic integrals (DLMF 19.33.2).
Slant height
The slant height is measured along the surface, from the apex (or the top edge) straight down to the base edge. It is the hypotenuse of a right triangle: for a cone l² = r² + h²; for a frustum l² = h² + (R − r)²; for a pyramid on a square of edge a, s² = h² + (a/2)², measured from the apex to the middle of a base edge — not to a corner, which is the longer lateral edge.
Limitations
- The figures are ideal solids. For real objects, add allowances for seams, overlaps and waste, and measure the surface you actually need to cover.
- A pyramid or prism known only by its base area has no surface area here — it depends on the base shape.
- Results are rounded to the decimal places you choose, and the unit table keeps at least six significant digits (calculations carry about 15). Sizes must be positive and at most 10¹² in the chosen unit.
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Frequently asked questions
What is the difference between CSA and TSA?
The curved (or lateral) surface area leaves out the flat parts; the total surface area includes them. For a cylinder of radius 7 cm and height 10 cm, CSA = 2πrh = 440 cm² (with π = 22/7) is just the side, while TSA = 2πr(r + h) = 748 cm² adds the two circular ends of 154 cm² each.
How do I find the slant height of a cone?
Use Pythagoras with the radius and the height: l = √(r² + h²). A cone with r = 3 cm and h = 4 cm has l = 5 cm. If you know the slant height but not the height, choose “Slant height” and the calculator works back to h = √(l² − r²).
Does the total surface area of a hemisphere include the flat face?
Yes. A solid hemisphere has a curved surface of 2πr² and a flat circular face of πr², so TSA = 3πr². A hollow hemispherical bowl has only the curved surface (its inside and outside are counted separately if needed).
How do I find the surface area of a pyramid?
Add the areas of its triangular faces and its base. For a regular pyramid each face is ½ × base edge × slant height, so LSA = ½ × perimeter × slant height. A square pyramid with base edge 6 and height 4 has slant height √(4² + 3²) = 5, LSA = 60 and TSA = 60 + 36 = 96.
Should I use π = 22/7 or 3.14?
Use whatever your question specifies; many NCERT problems use 22/7, which gives whole numbers when the radius is a multiple of 7. The exact answer in terms of π (such as 24π cm²) is shown either way.
Why is there no formula for the surface area of an ellipsoid?
Unlike its volume (⁴⁄₃πabc), an ellipsoid’s surface area cannot be written with elementary functions; it needs elliptic integrals. The calculator evaluates those exactly (to about 15 digits) and also shows Knud Thomsen’s well-known approximation, which is within about 1.1 %.