Maharashtra State Board · Class 10 · Mathematics · Geometry Chapter 7

Mensuration — Surface Areas and Volumes — Formula Sheet

Board Formulas

11 formulas

1.Cylinder★
$V = π r^{2} h, C S A = 2 π r h, T S A = 2 π r (r + h)$
$r$ : Radius of the circular base · $h$ : Height of the cylinder
TSA includes the two circular bases; CSA is the curved lateral surface only.
2.Cone★
$V = \frac{1}{3} π r^{2} h, C S A = π r l, T S A = π r (r + l)$
$l$ : Slant height · $h$ : Perpendicular height
Always find slant height l before computing cone surface area.
3.Slant Height of Cone
$l = r^{2} + h^{2}$
Apply Pythagoras to the right triangle with legs r and h.
4.Sphere
$V = \frac{4}{3} π r^{3}, S A = 4 π r^{2}$
A sphere has no flat base — surface area formula covers the entire closed surface.
5.Hemisphere★
$V = \frac{2}{3} π r^{3}, C S A = 2 π r^{2}, T S A = 3 π r^{2}$
TSA of hemisphere = curved surface (2πr²) + flat circular base (πr²).
6.Frustum of a Cone (Volume)
$V = \frac{π h}{3} (R^{2} + R r + r^{2})$
$R$ : Radius of larger (bottom) base · $r$ : Radius of smaller (top) base · $h$ : Height between the two bases
A frustum is the piece of a cone cut off by a plane parallel to the base.
7.Frustum — Slant Height and CSA
$l = h^{2} + (R - r)^{2}, C S A = π (R + r) l$
TSA = π(R + r)l + πR² + πr² (adds two circular bases).
8.Cube
$V = a^{3}, T S A = 6 a^{2}$
a = edge length. Diagonal = a√3.
9.Cuboid
$V = l bh, T S A = 2 (l b + bh + h l)$
l, b, h are length, breadth, height. Diagonal = √(l² + b² + h²).
10.Volume Conservation (Recasting)★
$V_{old} = V_{new}$
When a solid is melted and recast, total volume is preserved (no wastage stated).
11.Combined Solid Surface Area
$S A_{total} = \sum exposed surfaces only$
Do NOT include internal surfaces where two solids meet.