Ever wonder how much paint you’d need to cover a box or how much water fits inside a bottle? That’s where surface area and volume come in! Every real-world object around you—whether it’s a cube, sphere, cone, or cylinder—has three dimensions: length, width, and height. The outer covering of these shapes is their surface area, and the space inside them is their volume.

For 2D shapes like squares, circles, or triangles, we only talk about area—since they’re flat, there’s no volume to measure. But for 3D shapes, knowing both surface area and volume is super useful, especially if you're gearing up for tests like the SAT, ACT, PSAT/NMSQT, GED, GRE, GMAT, AP Exams, PERT, Accuplacer, or even the MCAT. You’ll often be asked to calculate these values!

In this guide, we'll break down the essential surface area and volume formulas, complete with easy-to-follow images and a handy formula chart. Plus, we’ll walk you through solved examples to help you master the concepts and feel confident tackling related questions on test day. Let’s dive in and make sense of surface area and volume once and for all!

Surface Area and Volume

The space which is occupied by a two-dimensional flat surface is known as the area. It is measured in square units. The area which is occupied by a three-dimensional object by its outer surface is known as the surface area of the object. That is also measured in square units.

There are two types of surface area which are as follows.

• Total Surface Area
• Lateral Surface Area

Total Surface Area

The Total Surface area of a three-dimensional object can be described as “the total area covered by an object consists of its area of the base as well as the area of the curved part. If the object has a curved surface and base, then the total area will be the sum of the curved surface area and the area of the base.

Lateral Surface Area

The lateral surface area of an object can be described as “the area of an object which is covered by the curved surface area of an object. Lateral surface area is also known as curved lateral surface area. For shapes such as a cone, it is often known as the lateral surface area.

Volume

The total amount of space that a three-dimensional object occupies is known as volume. That is measured in cubic units. The two-dimensional object doesn’t have volume but has area.

Surface Area and Volume Formulas

The following are the surface area formulas of respective 3-D shapes.

CUBOID

Let the cuboid’s length = l, breadth = b, and height = h units. Then,

The volume of the cuboid = \( (l\times b\times h) \) cubic units.

The surface area of the cuboid = \( 2(l\times b + b\times h +l\times h)\) sq.units.

CUBE

Let each edge of a cube is of length a. Then,

The volume of the cube = \( a^{3} \) cubic units.

The surface area of the cube = \( 6\times a^{2} \) sq. units.

CYLINDER

Let the radius of the cylinder’s base = r and Height = h. Then,

The volume of the cylinder =\( \pi\times r^{2}\times h \) cubic units.

The curved surface area of the cylinder = \( 2\pi\times r\times h \) sq. units.

Total surface area of the cylinder =\( 2\pi\times r\times (h+r) \) sq. units.

CONE

Let radius of base of the cone = r and Height = h. Then,

The cone’s slant height, l = \( \sqrt{h^{2}+r^{2}} \)  units.

The cone’s volume = \(\frac{\pi}{3}\times r^{2}\times h\) cubic units.

The cone’s curved surface area = \( \pi\times r\times l \) sq. units.

Total cone’s surface area = \( \pi\times r\times l+\pi\times r^{2} \) sq. units.

SPHERE

Let the radius of the sphere is r. Then,

The volume of the sphere = \( \frac{4}{3}\pi\times r^{2}\times h \) cubic units.

The surface area of the sphere = \( 4\pi\times r^{2} \) sq. units.

HEMISPHERE

Let the radius of a hemisphere is r. Then,

The volume of the hemisphere = \( \frac{2}{3}\pi\times r^{2}\times h \) cubic units.

The curved surface area of the hemisphere =\( 2\pi\times r^{2} \) sq. units.

Total surface area of the hemisphere = \( 3\pi\times r^{2} \) sq. units.

Solved Examples of Surface Area and Volume Formulas

Example 1: What is the surface area of a cuboid with length, width and height equal to 4 c m, 2 cm and 5 cm, respectively?

A1: Given, the dimensions of cuboid are:

length, l = 4 cm

width, b = 2 cm

height, h = 5 cm

Surface area of cuboid =  \( 2(l\times b + b\times h +l\times h)\)

= \( 2(2\times 4 + 5\times 4+ 5\times 2)\)

= 76 square cm.

Thus, the surface area of cuboid is 76 square cm.

Example 2: What is the volume of a cylinder whose base radii are 2, cm and height is 30 cm?

A2: Given,

Radius of bases, r = 2 cm

Height of cylinder = 30 cm

The volume of cylinder =\( \pi\times r^{2}\times h \)

=\( \pi\times 2^{2}\times 30 \)

= 376.99 cubic cm.

Answer: Thus, the volume of the cylinder is 376.99 cubic cm.

Conclusion 

To sum it all up—surface area and volume formulas might seem tricky at first, but once you break them down, they’re totally manageable! Whether you’re prepping for the SAT, ACT, GED, or any other big test, knowing these formulas will give you a solid edge. Plus, they’re super useful for real-life situations, too. Keep practicing, stay confident, and soon enough, you’ll be acing these questions without breaking a sweat!