Ever wonder how much space your water bottle actually holds or how much air fits inside a basketball? That’s where mensuration comes into play! Mensuration is just a fancy word in math and geometry that’s all about measuring lengths, areas, and volumes of different shapes. When it comes to flat, 2D shapes like rectangles or circles, we focus on perimeter and area. But when we move into 3D shapes—like cubes, cylinders, cones, or spheres—we’re talking about lateral area, total surface area, and most importantly, volume.

In this guide, we’re going to break down exactly what volume is and how to calculate it for all sorts of shapes, from cuboids and cones to spheres and hemispheres. Trust us, knowing these formulas is a total game-changer, especially if you’re prepping for big tests like the SAT, ACT, PSAT/NMSQT, GED, GRE, GMAT, AP Exams, PERT, Accuplacer, or even the MCAT. Plus, we’ve included some solved questions to give you plenty of practice and boost your confidence. Ready to master volume once and for all? Let’s dive in!

Volume

Volume in mathematics is defined as the space occupied within the boundaries of an entity in 3D space. In other words, it is the capacity of the object in the 3D space. For example, if a jar holds 700 ml of milk, then its volume is said to be 700 ml. Also, the amount of water required to fill a water bottle or a water tank is its volume.

Thus finding the volume of an object further helps us to specify the amount needed to fill that particular object, like the amount of water needed to refill a bottle, a jar, a bucket, an aquarium or a water tank.

Units of Volume

The volume of any given solid is calculated in cubic units. For instance, if measurements of the objects are given in meters, then the unit of the volume will be cubic meters. Cubic meter i.e. \(\text{m}^3\) is the S.I. unit of volume.

However, the other units of volume are liter, milliliter and gallons. In addition to this units like cubic inches (\(\text{in}^3\)), cubic feet (\(\text{ft}^3\)), cubic centimeter (\(\text{cm}^3\)), and cubic yards (\(\text{yd}^3\)) are also used to measure the volume of objects.

Volume of 3D-Shapes

Every object has a nature of occupying space. Now that we know what volume is and its unit, let us understand the volume formula for different known geometric shapes in the 3D space.

Volume of a Cube

A cube is a three-dimensional shape having six faces such that each of the faces connects with the other four faces. All the sides of a cube are of equal length. A cube can be built by stacking a specified number of squares one above another. The figure below resembles the same.

If we consider the measure of the sides of a cube = a

Then the volume of a cube=\(a^{3}\)

Rubik’s cubes, dice, sugar cubes, etc are some real-life examples of a cube shape.

Volume of a Cuboid

A cuboid is considered the 3D representation of a rectangle. The sides are of different lengths and are named length, breadth, and height. A cuboid can be constructed by stacking a specified number of rectangles of defined length, breadth and height one above another. The figure below represents the same.

If h denotes the height, l denotes the length and b denotes the breadth of the cuboid, then the volume of the cuboid=l × b × h.

Commonly found shapes of cuboids are books, bricks, mattresses, shoe boxes, etc.

Volume of a Cone

A cone is a 3D shape having only one vertex point and a circular base. The image below depicts the same.

If h denotes the height, l specifies the slant height and r is the radius of a cone then the volume of a cone is calculated using the formula \( \left(\frac{1}{3}\right)\pi r^{2}h\).

Some of the real-life examples of a cone are; a funnel, ice-cream cone, party hats, Christmas tree, etc.

Volume of Frustum of Cone

The frustum of a cone is a section or part of the cone without its vertex. That is when the given cone is divided into two pieces with a plane that is parallel to the base of the cone. The figure below shows the same.

The formula for the volume of frustum of cone = \(\frac{\pi h}{3}\left[\frac{\left(R^{3}-r^{3}\right)}{r}\right]\)

(OR)

Volume of frustum of cone = \(\frac{\pi h}{3}(R^2+Rr+r^{2 })\)

Learn about the various concept of Geometry here.

Volume of a Cylinder

A cylinder is another 3D shape with circular bases and a specific height isolating the two bases. The length between these two bases is termed the height of the cylinder. It can be considered as a tube-like structure having two parallel circular bases bound by a curved surface at a specified length from the center.

If r denotes the radius of the circular bases and h denotes the height of the cylinder, then the volume of cylinder formula=\( \pi r^{2} h\)

Similar to the way we saw; cubes can be constructed using squares, and cuboids can be built using rectangles, a cylinder can be created using a defined number of circles of the same size one above the other. Water bottles, pencils, pens, buckets, candles, cans, etc are some real-life examples of a cylinder.

Also read about Area of a Circle here.

Volume of Hollow Cylinder

A cylinder that is hollow or empty from the inside and holds some thickness at the peripheral is called a hollow cylinder. For such a cylinder there is some contrast between the internal and external radius. Consider the diagram below to understand the same.

Here if ‘r’, denote the inner radius of the base and ‘R’ denote the outer radius of the base plus ‘h’ denotes the height of the hollow right circular cylinder then the volume of the hollow cylinder = \(\pi h\left(R^{2}-r^{2 }\right)\)

Tubes, canals, circular buildings, straws, etc are some real-life examples of a hollow cylinder.

Know more about the different types of Quadrilateral.

Volume of a Sphere

A sphere is another three-dimensional shape that is commonly used in mathematics. The figure below resembles the same. If r denotes the radius then the volume of a sphere or the volume of a solid sphere is given by the formula;\(\left(\frac{4}{3}\right)\pi r^{3}\)

Basketballs, football, tennis ball, oranges, globes, etc are some real-life examples of the sphere.

Volume of Hollow Sphere

A hollow sphere is a sphere that is hollow from the inside in such a way that a wall of equal thickness is created within the internal space of the ball. In a hollow sphere, there are two radii one for the outer sphere and another for the inner sphere.

If R is the radius of the outer sphere and r is the radius of the inner sphere then the volume of the hollow sphere= Volume of the outer sphere – Volume of the inner sphere

\(V=\frac{4}{3}\pi R^3-\frac{4}{3}\pi r^3=\frac{4}{3}\pi\left(R^3-r^3\right)\)

Also, learn about Circumference of a Circle with this article.

Volume of Hemisphere

When we cut a sphere by a plane passing exactly via the centre, then the shape obtained is named a hemisphere. That is we can say that a hemisphere is a 3D shape that is exactly half of a sphere. If r denotes the radius of a hemisphere then the volume of hemisphere formula is \( \left(\frac{2}{3}\right)\pi r^{3}\).

Some of the real-life examples of hemisphere shapes are igloo, bowl, mushroom, tea cup, hat, ice-cream scoop, etc.

Volume of a Pyramid

A pyramid is a 3-dimensional figure made with a bottom of a polygon and triangular faces all linked together. Depending upon the base there are different types of pyramids like square pyramid, triangular pyramid, rectangular pyramid, pentagonal pyramid, hexagonal pyramid, etc.

This further implies that if the base of a pyramid is a rectangle, then the respective pyramid is called a rectangular pyramid. The figure below shows the same.

The volume of a pyramid is given by the formula; \(\frac{1}{3}\times B\times h\)

Here, B denotes the area of the base(square pyramid, triangular pyramid, rectangular pyramid, pentagonal pyramid, hexagonal pyramid, etc.) and h is the height.

Learn the Various Properties of Rectangles.

Volume of a Prism

A prism is a three-dimensional shape such that the bases are identical in shape and face each other i.e. the bases are parallel to each other. The bases of a prism can be a triangle, rectangle, square, or any other type of polygon, however, the other faces of the prism are parallelograms or rectangles. The image below shows some types of prisms.

The volume of Prism = Base Area × Height

The prisms are considered to be polyhedron objects i.e. these objects hold multiple flat faces. The statement further says that a prism cannot have any side that is curved for example shapes like a cone, cylinder or sphere are not prisms.

Know more about the Mensuration 3D here.

List of Volume Formulas

The table below will give you a detailed list of volume formulas for all the possible 3-D shapes for a quick revision.

Learn the Area of Polygon.

Volume of Combination of Shapes

In the previous headers, we saw the concept and volume formula for different 3-dimensional shapes. Let us now learn the volume of the combination of shapes. When two different solid shapes merge to form a different shape altogether, this leads to a diverse level of measurement. We will learn three such combinational shapes; capsule, ice cream cone and tents.

Volume of a Capsule

A capsule is made up of a sphere that has been cut into two half via the center and the two ends are then separated by a cylinder as shown in the figure below.

It can also be understood as two hemispheres joined through a cylinder.

The volume of a capsule is equal to the addition of a volume of the sphere having radius r(or the volume of two hemispheres) + the volume of the cylindrical portion.

\(V=\ \left(\frac{4}{3}\right)\pi r^3+\pi r^2h=\pi r^2\left(\frac{4r}{3}+h\right)\)

Volume of an Ice Cream Cone

An ice cream cone is a combination of two 3d shapes that is a cone and a hemisphere. Hence, the volume of ice cream = Volume of hemisphere + Volume of a cone.

\(V=\frac{3}{2}\pi r^{3}+\frac{1}{3}\pi r^{2}h\)

Volume of a Circus Tent

A circus tent is a combination of two 3d shapes that is a cylinder and a cone. The arrangement is such that a cone is placed on top of a cylinder. A hut also has a tent-like structure.

The volume of a circus tent is equal to the addition of a volume of the cone + the volume of the cylindrical portion.

\(V=\frac{1}{3}\pi r^2h+\pi r^2h\)

Solved Examples on Volume

Topics of menstruation like 2D and 3D shapes can be best understood by practising as many numbers of questions as possible. As such topic mostly comprises formulas. Thus knowing all such formulas lets us go through some solved questions.

Solved Example 1: Estimate the volume of a football where the radius is 7 cm (take the value of \(\pi=\frac{22}{7}\) )?

Solution: Given the radius of the footballs= 7 cm.

As football resembles the shape of a solid sphere, using the formula for the same.

V=\( \left(\frac{4}{3}\right)\pi r^{3}\)

Substituting the values we get;

\(V=\left(\frac{4}{3}\right)\pi r^3=\frac{4}{3}\times\frac{22}{7}\times7\times7\times7\)

On further simplification we get;

Volume of a football =1437.33 \(\text{cm}^3\)

Solved Example 2: A cylindrical water tank has a base radius of 14 cm and a height of 20 cm. Determine the volume of water that the tank can hold (take the value of \(\pi=\frac{22}{7}\) ).

Solution: Given the radius and height of the cylindrical water tank as 14 cm and 20 cm respectively.

Volume of a cylinder formula=\( \pi r^{2} h\)

\( V=\frac{22}{7}\times14\times14\times20\text{ cm}^3\)

\(V=12320\text{ cm}^3\)

\(\left(1\text{cm}^3=1\text{ml}\right)\)

V=12320 ml.

Solved Example 3: A conical Christmas tree is to be made using organic material. If the height of the Christmas tree is 15 inches and the diameter of the base is 8 inches. Calculate the amount of organic material used to build the tree (take the value of \(\pi=\frac{22}{7}\) ).

Solution: The dimensions for the Christmas tree are;

Height=15 inches

Base = 8 inches, radius= 4 inches.

Volume of a cone=\( \left(\frac{1}{3}\right)\pi r^{2}h\)

\(V=\left(\frac{1}{3}\right)\pi r^2h=\frac{1}{3}\times\frac{22}{7}\times4\times4\times15\ \text{inches}^3\)

On simplification we get;

\(V=251.428\ \text{inches}^3\)

Solved Example 4: A structure is built by using a cubic block of the length of 3 cm each. If the structure is made of 21 such blocks then calculate the volume occupied by the structure.

Solution: Given the measure of the side of the cubic block of =3 cm

First calculates the volume of one cube;

Volume of a cube=\( a^{3}\)

Here, a is the length of the edge of the given cube.

Substituting the value we obtain, the Volume of one cube=

\( V=3^3\ \text{cm}^3=3\times3\times3\ \text{cm}^3\)

\( V=27\ \text{cm}^3\)

There are 21 such blocks need to build the structure. Therefore, the volume occupied by the complete structure= \( 21\times27\text{ cm}^3=567\text{ cm}^3\)

Solved Example 5: Choose the suitable options for the given 3D shapes.

3D Shapes.

(a)Hemisphere

(b)Prism

(c)Cone

(d)Sphere

Formulas.

(1) Base Area × height

(2)\( \left(\frac{1}{3}\right)\pi r^{2}h\)

(3)\( \left(\frac{2}{3}\right)\pi r^{3}\)

(4)\( \left(\frac{4}{3}\right)\pi r^{3}\)

Solution: Considering the shapes and the options the correct match is as followhs:

Option (a) will go with option (3) as the volume of hemisphere=\( \left(\frac{2}{3}\right)\pi r^{3}\)

Option (b) will go with option (1) as the volume of Prism=Base Area × height

Option (b) will go with option (1) as the volume of cone=\( \left(\frac{1}{3}\right)\pi r^{2}h\)

Option (d) will go with option (4) as the volume of sphere= \( \left(\frac{4}{3}\right)\pi r^{3}\).

Solved Example 6: Fill in the blanks for the statements with suitable answers.

(1) Cubic meters (\(m^{3}\)), cubic centimeters (\(cm^{3}\)), liters are____ units for the measurement of volume.

(2) cubic foot, fluid ounce, cubic inch, pints and gallons are____ units for the measurement of volume.

(3) 1 \(cm^{3}\) ____ ml

Solution: Let us start answering the statements one by one.

(1) Cubic meters (\(m^{3}\)), cubic centimeters (\(cm^{3}\)), liters are metric units for the measurement of volume.

(2) cubic foot, fluid ounce, cubic inch, pints and gallons are US Standard units for the measurement of volume.

(3) 1 \(cm^{3}\) 1 ml

As;

1 l = 1000 \(cm^{3}\)

Also, 1 l = 1000 ml

On comparing the two we get; 1000 \(cm^{3}\) = 1000 ml

Thus, 1 \(cm^{3}\) = 1 ml

Conclusion

Mensuration might sound fancy, but it’s really all about measuring the space different shapes take up. Knowing how to calculate volume is super handy—whether you’re filling up a water tank, estimating materials, or prepping for exams like the SAT or GRE. From cubes and cylinders to spheres and pyramids, these formulas make it easy to break down any 3D shape. So keep practicing, and soon enough, you’ll be a pro at solving volume problems with confidence and speed!