CBSE · Class 6-12

Class 10 · Mathematics (Standard)

Unit 12 · Class 10Mathematics (Standard)

Surface Areas and Volumes

Formulas for combinations of solids: cube, cuboid, cylinder, cone, sphere and hemisphere.

Lessons

5 concepts

1
Standard Solid Formulas
Cube (edge $a$ ): TSA $= 6a^2$ , volume $= a^3$ . Cuboid ( $l\times b\times h$ ): TSA $= 2(lb + bh + hl)$ , vol $= lbh$ . Cylinder ( $r, h$ ): CSA $= 2\pi rh$ , TSA $= 2\pi r(r+h)$ , vol $= \pi r^2 h$ . Sphere ( $r$ ): SA $= 4\pi r^2$ …
2
Cone and Slant Height
Cone (radius $r$ , height $h$ , slant $\ell$ ): $\ell = \sqrt{r^2 + h^2}$ ; CSA $= \pi r\ell$ , TSA $= \pi r(r + \ell)$ , vol $= \tfrac{1}{3}\pi r^2 h$ . Notice cone volume is one-third of a cylinder with same base and height.
3
Hemisphere
Hemisphere (radius $r$ ): curved SA $= 2\pi r^2$ ; TSA (with base) $= 3\pi r^2$ ; volume $= \tfrac{2}{3}\pi r^3$ . Half of a sphere.
4
Combining Solids
Real objects (like a capsule, an ice-cream cone with scoop, or a tent) are combinations of standard solids. When computing surface area of a combination, add only the outer surfaces (not the internal joining faces); for …
5
Conversion of Solids
If one solid is melted and recast into another, the volumes are equal: $V_{\text{initial}} = V_{\text{final}}$ . Use this to find dimensions of the recast solid. Similarly, water displacement problems equate the added vol…

Skills to practise

Skill · medium

Surface Areas and Volumes — Practice