Volume of Cone, Cylinder and Sphere: Formulas, Surface Areas and Ratios

These solids share one measurement, the radius r of the circular base or of the sphere, and the cylinder and cone also have a height h. Most questions use π = 22/7 with a radius that is a multiple of 7, so the 7 cancels and the arithmetic stays whole.

For a cone, l is the slant height, the distance from the tip to the edge of the base. It is the hypotenuse of a right triangle with sides r and h, so l = √(r² + h²).

Its curved surface area is 2 × 22/7 × 7 × 10 = 440 cm². This is the area of the label on a can, which unrolls into a rectangle of length 2πr and height h.

Most cone questions need the slant height first. Look for familiar right-triangle triples such as 3–4–5, 5–12–13, 7–24–25 and 20–21–29.

Its curved surface area is πrl = 22/7 × 7 × 25 = 550 cm².

Its volume is ⅓ × 22/7 × 49 × 24 = 1,232 cm³.

Its volume is ⁴⁄₃ × 22/7 × 9,261 = 38,808 cm³.

When a cone, a hemisphere and a cylinder share the same radius r, and the cone and cylinder have height h = r, their volumes are ⅓πr³, ⅔πr³ and πr³.

Changing the radius scales the solids quickly. Areas grow with r² and volumes with r³, so doubling the radius of a sphere multiplies its surface area by 4.

When one solid is melted and recast into another, the volume does not change. Write both volumes, set them equal, and solve for the unknown measurement. π cancels from both sides, so you rarely need its value.

A pipe or a hollow cylinder has an outer radius R and an inner radius r. The metal in it is the outer cylinder minus the hole: π(R² − r²)h. Factorising R² − r² as (R + r)(R − r) often makes the arithmetic easier.

Combined solids, such as a cone on top of a hemisphere or a cylinder with hemispherical ends, are handled piece by piece. Add the volumes of the parts. For surface area, add only the faces that remain on the outside: the joined circles are hidden and must not be counted.

• ✘ Using the height h instead of the slant height l in the cone's surface area.
• ✘ Forgetting the flat face in a solid hemisphere's total surface area.
• ✘ Taking the diameter as the radius. Halve it first.
• ✘ Scaling volumes by r² instead of r³ when the radius changes.

Flat-faced solids are on cube and cuboid. The right-triangle triples used here also appear in square roots questions, and the formula list is in the Quant cheat code.

Should I use π = 22/7 or 3.14?

Use whatever the question states. Most SSC questions choose radii that are multiples of 7 so that 22/7 cancels neatly. If the question gives no value and the radius is a multiple of 7, 22/7 is usually the intended choice.

Why is the hemisphere's total surface area 3πr²?

The curved part is half of the sphere's 4πr², which is 2πr². A solid hemisphere also has a flat circular face of area πr². Together they give 3πr². A hollow bowl has only the curved 2πr².

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

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