Volume Formula and Surface Area of Every Solid

What volume means in maths and physics

Here is the volume definition in plain words: volume is the amount of space a solid takes up. Think of how many unit cubes, each 1 cm on every edge, you could pack inside it. That is why volume is counted in cubic units: $\mathrm{cm^3}$, $\mathrm{m^3}$. The SI unit used in physics is the cubic metre, and for liquids we use litres and millilitres.

Surface area is different: it is the total area of the outer faces, measured in square units, and tells you how much paint, paper or sheet metal you need. Questions say "total surface area" when every face counts, and "curved" or "lateral" surface area when the top and bottom are left out (an open pipe, the walls of a room).

In physics, the volume of an irregular stone is found by dropping it in a measuring cylinder and reading the rise in water level. Density is mass divided by volume, so the same formulas appear in science numericals too.

Volume of cube formula and cuboid formula

Cube

Every edge of a cube is the same length $a$, so the volume is side times side times side, $a^3$ (some notes write it as s x s x s). The six faces are squares, so TSA is $6a^2$; leave out the top and bottom and the four walls give $4a^2$. The long diagonal from one corner to the opposite one is $\sqrt{3}\,a$.

Cuboid (rectangular box)

A cuboid, or rectangular prism, has length $l$, breadth $b$ and height $h$. Its volume is $lbh$. The surface area of a rectangular box has three pairs of equal faces, giving $2(lb + bh + hl)$. The lateral surface area of a cuboid means the four side walls only: $2h(l + b)$. That is the area you plaster when you paint the walls of a room but not the floor and ceiling.

There is no "volume of a square": a square is flat and has only area. If a question says that, it means a cube.

Volume of cylinder formula and surface area

A cylinder is a circle pushed straight up through a height $h$. Its base area is $\pi r^2$, so $V = \pi r^2 h$. Unroll the curved side and you get a rectangle whose width is the base circumference $2\pi r$ and whose height is $h$: that is the CSA, $2\pi rh$. Add two circular ends for the TSA, $2\pi r(r + h)$.

A hollow pipe with outer radius $R$ and inner radius $r$ holds metal of volume $\pi(R^2 - r^2)h$.

Volume of cone formula and slant height

A cone with the same base and height as a cylinder holds exactly one third as much. You can test this with water: three full cones fill the cylinder. So $V = \tfrac{1}{3}\pi r^2 h$.

The curved surface uses the slant height $l$, not the vertical height. The radius, the height and the slant height form a right triangle, so $l = \sqrt{r^2 + h^2}$. Then CSA is $\pi rl$ and TSA is $\pi r(l + r)$.

Sphere volume formula and hemisphere

A sphere of radius $r$ has volume $\tfrac{4}{3}\pi r^3$ and surface area $4\pi r^2$, which is four times the area of its biggest circle. Both formulas come from calculus or from Archimedes' comparison with a cylinder, and you are expected to memorise them in class 9.

A hemisphere is half a sphere. Its volume is $\tfrac{2}{3}\pi r^3$ and its curved surface is $2\pi r^2$. A solid hemisphere also has a flat circular face, so its TSA is $2\pi r^2 + \pi r^2 = 3\pi r^2$. A hollow bowl has only the curved part.

Note that a circle is flat, so it has no volume. A "volume of the circle formula" search usually means the sphere.

Pyramid volume formula, prism and frustum

Pyramid

Any pyramid, whatever the shape of its base, has $V = \tfrac{1}{3} \times \text{base area} \times h$. For a square base of side $a$, that is $\tfrac{1}{3}a^2 h$. The lateral surface is made of triangles; each has the base edge as its base and the slant height of the face as its height.

Prism

A prism keeps the same cross-section all along its length. Its volume is that cross-section area times the length. The lateral surface of a prism is a set of rectangles, one per edge of the base, so LSA $=$ base perimeter $\times h$, and TSA adds the two end faces. A triangular prism with a right triangle base of legs 3 cm and 4 cm (hypotenuse 5 cm) and length 10 cm has volume $6 \times 10 = 60\ \mathrm{cm^3}$, LSA $12 \times 10 = 120\ \mathrm{cm^2}$ and TSA $132\ \mathrm{cm^2}$.

Frustum of a cone

Slice the top off a cone parallel to the base, and the bucket-shaped piece left is a frustum. With end radii $R$ and $r$, height $h$ and slant height $l = \sqrt{h^2 + (R - r)^2}$:

For $R = 10$, $r = 4$, $h = 8$ (cm): $l = 10$, $V = 416\pi \approx 1306.24\ \mathrm{cm^3}$ with $\pi = 3.14$, CSA $= 140\pi \approx 439.6\ \mathrm{cm^2}$, and TSA, adding both circles, $= 256\pi \approx 803.84\ \mathrm{cm^2}$. The frustum is not in every board's current syllabus, so check yours; entrance tests still ask it.

Units, litres and capacity

Since volume uses three lengths, a unit change gets cubed. One metre is 100 cm, so one cubic metre is $100^3 = 10^6$ cubic centimetres.

Combination of solids and melting problems

Class 10 loves shapes built from two solids. For the surface area, add only the parts you can see from outside; the joined faces are hidden. For the volume, simply add (or subtract, if one is scooped out).

Surface area and volume formulas for class 9 and class 10

Middle school mensuration starts with the cube, cuboid and cylinder. The class 9 chapter on surface areas and volumes covers the cone, sphere and hemisphere, and class 10 uses these solids in combination problems. Every formula on this page fits on the one-page sheet; print the one-page PDF offered with this page and keep it in your geometry notebook.

The flat-shape formulas these solids rely on, such as the circle, the area of equilateral triangle formula and the area of quadrilateral formula, are on the area formulas page. Base perimeters for prisms come from the perimeter page, and the tools for deriving the sphere formula by calculus are on the integration formulas page.