Introduction to Mensuration
Formulae for surface areas and volumes of 3D objects.
Mensuration — Introduction to Surface Areas and Volumes
Mensuration is the branch of Class 10 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) that deals with measuring the surface area and volume of solid shapes. While earlier classes focused on flat, two-dimensional figures like triangles and circles, this chapter takes the next step into three dimensions — boxes, cylinders, cones, spheres, and hemispheres.
This introductory lesson lays the groundwork for the entire chapter by presenting the standard formulas for six fundamental solids. Every exercise that follows — including combinations of two or more solids joined together — builds directly on these six formula sets, so getting comfortable with them now will make the rest of the chapter far easier.
1. Cuboid
A cuboid is a solid with six rectangular faces — think of a matchbox, a brick, or a book. It has three different measurements: length (l), breadth (b) and height (h).
Lateral surface area = 2h(l + b)Total surface area = 2(lh + bh + lb)Volume = l × b × h2. Cube
A cube is simply a special cuboid where all three dimensions are equal — length, breadth and height are all the same value, l. A dice or a Rubik's cube is a perfect everyday example.
Lateral surface area = 4l²Total surface area = 6l²Volume = l³3. Cylinder
A cylinder has two flat circular ends (radius r) joined by a curved surface, with height h between them — like a tin can, a pipe, or a water tank.
Lateral surface area = 2πrhTotal surface area = 2πr(h + r)Volume = πr²h4. Cone
A cone has a single circular base (radius r) that narrows smoothly up to a single point called the apex, at height h above the base — like an ice-cream cone or a birthday party hat. The distance from the apex to any point on the base's edge is called the slant height, l.
Lateral surface area = πrlTotal surface area = πr(l + r)Volume = ⅓ πr²hSlant height: l² = r² + h²5. Sphere
A sphere is a perfectly round solid, like a ball — every single point on its surface is the same distance, r, from its centre. Unlike the other solids, a sphere has no flat faces or edges at all, so it has only one surface area formula (there is no separate "lateral" and "total" version).
Surface area = 4πr²Volume = (4/3) πr³6. Hemisphere
A hemisphere is exactly half of a sphere, cut straight through its centre — like a bowl, a dome, or half an orange. It has one flat circular face (the cut) and one curved face, both with radius r.
Lateral (curved) surface area = 2πr²Total surface area = 3πr²Volume = (2/3) πr³All Six Formulas at a Glance
Keeping all the formulas side by side makes it much easier to spot the patterns — notice how every curved-surface formula contains π, while the cuboid and cube (made only of flat rectangular faces) never do.
| Solid | Lateral / Curved S.A. | Total Surface Area | Volume |
|---|---|---|---|
| Cuboid | 2h(l + b) | 2(lh + bh + lb) | l × b × h |
| Cube | 4l² | 6l² | l³ |
| Cylinder | 2πrh | 2πr(h + r) | πr²h |
| Cone | πrl | πr(l + r) | ⅓ πr²h |
| Sphere | — | 4πr² | (4/3) πr³ |
| Hemisphere | 2πr² | 3πr² | (2/3) πr³ |
Common Mistakes to Avoid
- Mixing up lateral and total surface area: Lateral (or curved) surface area excludes the flat top/bottom faces — always re-read the question to check whether it asks for "curved surface area," "lateral surface area," or "total surface area."
- Using height instead of slant height in a cone (or vice versa): πrl (using slant height l) gives the curved surface area of a cone, while the volume formula ⅓πr²h uses the vertical height h — these are not interchangeable.
- Forgetting the flat face area for a hemisphere: A hemisphere has a circular flat face in addition to its curved surface — when a question asks for total surface area, both 2πr² and πr² must be added.
- Using the wrong value of π: Use π = 22/7 when the radius is a multiple of 7 (since it cancels neatly), and π = 3.14 otherwise, unless the question specifies which one to use.
- Mismatched units: If the radius is given in cm but the height is given in m, convert both to the same unit before substituting into any formula — this is an easy mark to lose.
What This Lesson Prepares You For
Every exercise in the Mensuration chapter builds on the six formula sets covered here, especially when two or more solids are joined together to form a single combined shape. The circle-related formulas you have already studied — particularly the area of a circle used inside the cylinder, cone, sphere and hemisphere formulas above — come directly from the sector and segment area work in Exercise 9.3, so revisiting that lesson is useful if the πr² term ever feels unfamiliar.
Going forward, look out for questions that ask you to find the surface area or volume of a single solid directly from these formulas, as well as more advanced questions that combine two solids (such as a hemispherical bowl with a cylindrical rim, or a cone mounted on a cylinder) — these combined problems simply require adding or subtracting the appropriate formulas from the table above.