Surface Area of Cube: Lateral and Total Surface Area

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NEO Thu, Feb 03, 2022 12:16 PM UTC

This pdf includes the following topics:-
Surface Area of a Cube
The lateral surface area of a cube
Examples with step-by-step explanation.

1. Surface Area of a Cube
Cube: In our everyday life, we come across objects like a dice,
Rubik’s cube, Sugar cube, and Ice cube, etc. These objects are in the
shape of a cube. All these objects are made of six square plane
Dice Rubik's cube
Sugar cubes Ice cube
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2. A cube is a three-dimensional shape whose length, breadth and
height are all equal. The cube has six surfaces called faces. Each
face of a cube is a square, two adjacent faces of a cube meet in a
line segment called edge and all of a cube's corners (called vertices)
are right angles. Ultimately, a cube has the shape of a square box.
A B
C
G
B
F H
E D
A cube has six faces (FHDE, FAGE, ABCG BCHD, ABFH & GCDE), eight
vertices (A, B, C, D, E, F, G & H), twelve edges (AB, BC, CG, GA, BH,
HD, DC, DE, FE, FH, AF, & GE).
In the case of a cube, its length, breadth, and height are equal.
Then, side of cube = Length = Breadth = Height
So, the figure with these dimensions would be like the shape
shown below. Here, side of cube = a.
a
a 5 a
a a a a
a 1 a 2 a 3 a 4 a
a a a a
a 6 a
a
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a

3. So, the sum of the areas of the six squares is:
Area of square 1 = (𝐚 × 𝐚) + Area of square 2 = (𝐚 × 𝐚) + Area of
square 3 = (𝐚 × 𝐚) + Area of square 4 = (𝐚 × 𝐚) + Area of square 5
= (𝐚 × 𝐚) + Area of square 6 = (𝐚 × 𝐚).
⇒Surface Area of Cube = 𝟐(𝐚 × 𝐚) + 𝟐(𝐚 × 𝐚) + 𝟐(𝐚 × 𝐚).
= 𝟐[(𝐚 × 𝐚) + (𝐚 × 𝐚) + (𝐚 × 𝐚)].
= 𝟐(𝐚𝟐 + 𝐚𝟐 + 𝐚𝟐 ).
= 𝟐(𝟑𝐚𝟐 ).
= 6 𝐚𝟐 .
Where a is the length of edges of the cube.
Surface Area of Cube = 6 𝐚𝟐
Example: Find the surface area of a Rubik’s cube whose edge is 6
6 cm 6 cm
6 cm
Solution: Clearly, Rubik’s cube is in the form of a cube.
Here, edge of Rubik’s cube = a = 6 cm.
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4. Therefore, the surface area of Rubik’s cube = 6𝐚𝟐 .
= 6(𝟔𝟐 )𝐜𝐦𝟐 .
= 6 × 36 𝐜𝐦𝟐 .
= 216 𝐜𝐦𝟐 .
Hence, the surface area of Rubik's cube = 216 𝐜𝐦𝟐 .
Lateral Surface Area of a Cube: If out of the six faces of a cube, we
only find the sum of the areas of four faces leaving the bottom and
top faces. This sum is called the lateral surface area of the cube.
Consider a cube of the side as ‘a’ which is shown in the figure
below. A B
C
G
B
a
H
F
a
E a D
Lateral surface area of cube
= Area of face HBCD + Area of face CDEG + Area of face GEFA + Area
of face ABHF.
= (𝐚 × 𝐚) + (𝐚 × 𝐚) + (𝐚 × 𝐚) + (𝐚 × 𝐚).
= 𝐚𝟐 + 𝐚𝟐 + 𝐚𝟐 +𝐚𝟐 = 𝟒𝐚𝟐 .
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5. Lateral surface area of a cube = 4𝐚𝟐
Example 1: Find the lateral surface area of a dice whose edge is 5
cm.
5 cm
Solution: Clearly, dice is in the form of a cube.
Here, length of edge of dice = a = 5 cm.
Therefore, Lateral surface area of the dice = 4𝐚𝟐 .
= 4(𝟓𝟐 )𝐜𝐦𝟐 .
= 4 × 25 𝐜𝐦𝟐 .
= 100 𝐜𝐦𝟐 .
Hence, the lateral surface area of the dice = 100 𝐜𝐦𝟐 .
Example 2:Five cubes each of side 5 cm are joined end to end. Find
the surface area of the resulting cuboid.
5
cm
5 cm
5 cm 5 cm 5 cm 5 cm 5 cm
Solution: The dimensions of the cuboid so formed are as under :
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6. 𝒍 = length = (5+5+5+5+5) cm = 25 cm,
𝒃 = breadth = 5 cm and 𝒉 = height = 5 cm.
So, surface area of the cuboid = 𝟐(𝒍𝒃 + 𝒃𝒉 + 𝒍𝒉).
= 2(𝟐𝟓 𝐜𝐦 × 𝟓 𝐜𝐦 + 𝟓 𝐜𝐦 × 𝟓 𝐜𝐦 + 𝟐𝟓 𝐜𝐦 × 𝟓 𝐜𝐦).
= 2(𝟏𝟐𝟓 𝐜𝐦𝟐 + 𝟐𝟓 𝐜𝐦𝟐 + 𝟏𝟐𝟓 𝐜𝐦𝟐 ).
= 2(𝟐𝟕𝟓 𝐜𝐦𝟐 ).
= 𝟓𝟓𝟎 𝐜𝐦𝟐 .
Hence, Surface area of the cuboid formed = 550 𝐜𝐦𝟐 .
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