# Cube

A cube, one of the five platonic solids, is a three-dimensional solid with 6 congruent square faces. Thus it is a hexahedron (hex means six, hedron means face). It is also considered a special type of square prism. The diagram shows the shape of a cube.

Ice cubes, dice, and Rubik’s cube are some common examples of cube shaped objects in real life.

The parts of a cube are:

• 6 Faces – Opposite faces are parallel. Every face meets other four faces. The angles between any two faces are 90°.
• 8 Edges – Opposite edges are parallel
• 12 vertices – Each vertex is the meeting point of three faces and three edges.

A net of a cube can help us understand its shape from a 2-D view as shown.

## Formulas

### Surface Area

Surface Area or Total Surface Area (TSA) = 6a2, here a = edge

Lateral Surface Area (LSA) = 4a2 (this is inclusive in TSA)

So, TSA = LSA + 2a2 = 6a2

Let us solve an example to understand the concept better.

Find the lateral and total surface areas of a cube with a side of 4 in.

Solution:

As we know,
Lateral Surface Area (LSA) = 4a2, here a = 4 in
∴ LSA = 4 × (4)2
= 64 in2
Total Surface Area (LSA) = LSA + 2a2, here a = 4 in
∴ TSA = 64 + 2 × (4)2
= 96 in2

Find the surface area of a cube with a side length of 5 cm

Solution:

The side length is actually the length of the edge. So,
Total Surface Area (TSA) = 6a2, here a = 5 cm
= 6 × 52
= 150 cm2
Volume
Volume (V) = a3, here a = edge

Calculate the volume of a cube of 6 cm.

Solution:

As we know,
Volume (V) = a3, here a = 6 cm
∴ V = 63
= 216 cm3

### Diagonal

There are two types of diagonal in a cube – space diagonal, and body diagonal. The space or body diagonal of a cube is the main diagonal passing through the center in the inner space. The face diagonal is the diagonal on each of its faces.

The formulas are:

Face Diagonal = ${\sqrt{2}a}$

Space Diagonal (d) = ${\sqrt{3}a}$

Find the length of the main diagonal of a cube whose edge is 7 cm.

Solution:

As we know,
The main diagonal is the body diagonal. So, the length of the diagonal is:
Length of Space Diagonal (d) = ${\sqrt{3}a}$, here a = 7 cm
d = √3 × 7
≈ 12.12 cm

Finding the VOLUME of a cube when the DIAGONAL is known

Find the volume of a cube whose diagonal is 6 cm

Solution:

Here we will use an alternative formula for volume involving the diagonal.
V = ${\dfrac{\sqrt{3}}{9}d^{3}}$, here d = 6 cm
∴ V = √3 × 63/9
≈ 41.57 cm3

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