Surface Areas and Volumes Cheat Sheet

Formulae

Cube

\text{TSA} = 6a^{2}, \quad \text{LSA} = 4a^{2}, \quad V = a^{3}

a = edge of the cube. TSA = total surface area, LSA = lateral surface area, V = volume.

Cuboid

\text{TSA} = 2(lb + bh + hl), \quad \text{LSA} = 2h(l+b), \quad V = lbh

l = length, b = breadth, h = height.

Cylinder

\text{CSA} = 2\pi rh, \quad \text{TSA} = 2\pi r(r+h), \quad V = \pi r^{2}h

r = radius of base, h = height. CSA = curved surface area.

Cone

l = \sqrt{r^{2}+h^{2}}, \quad \text{CSA} = \pi rl, \quad \text{TSA} = \pi r(l+r), \quad V = \frac{1}{3}\pi r^{2}h

r = base radius, h = height, l = slant height.

Sphere

\text{SA} = 4\pi r^{2}, \quad V = \frac{4}{3}\pi r^{3}

r = radius. A sphere has only one (curved) surface.

Hemisphere

\text{CSA} = 2\pi r^{2}, \quad \text{TSA} = 3\pi r^{2}, \quad V = \frac{2}{3}\pi r^{3}

r = radius. TSA includes the flat circular base as well as the curved surface.

Frustum of a cone

l = \sqrt{h^{2}+(r_{1}-r_{2})^{2}}, \quad \text{CSA} = \pi(r_{1}+r_{2})l, \quad V = \frac{1}{3}\pi h(r_{1}^{2}+r_{2}^{2}+r_{1}r_{2})

r₁ = larger radius, r₂ = smaller radius, h = height, l = slant height.

Surface area: The total area of the surface of a solid.
Lateral surface area: The area of the curved or side surface only, excluding the bases.
Total surface area: The lateral surface area plus the areas of the top and bottom faces.
Volume: The amount of space enclosed by a solid; measured in cubic units.
Slant height: The distance from the vertex of a cone to a point on the edge of its base.
Frustum: The part of a cone left when a smaller cone is cut off by a plane parallel to the base.

Solids are measured by surface area (square units) and volume (cubic units).
For a cone of radius r, height h and slant height l, l² = r² + h².
A sphere has no flat surface, so its surface area is 4πr².
A hemisphere has one flat circular face, so its total surface area is 3πr².
When a solid is melted and recast into another shape, its volume remains the same.
When a solid is dipped completely into a liquid, the rise in level gives the volume of the solid.
For objects made by joining two solids, the total surface area is the sum of the curved surfaces minus the area hidden at the join.
The volume of a cone is one-third of the volume of a cylinder with the same base and height.