Math - Geometry Lesson Plans : Surface Area of Cuboids & Cubes

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Mensuration - Formulae for Surface Area of Cuboids & Cubes

A cuboid is a rectangular solid which has six rectangular faces.
A cube is a rectangular solid which has six square faces.

In the figure alongside of the cuboid, length = AB = CD = EF = GH.
In the figure alongside of the cuboid, breadth = AD = BC = EH = FG.
In the figure alongside of the cuboid, height = AE = BF = CG = DH.

In the figure alongside of the cuboid, diagonal = √

AB^{2} + BC^{2} + BF^{2}

Diagonal of a cuboid = √

Length^{2} + Breadth^{2} + Height^{2}

Example
Find the diagonal (in m upto two decimal places) of a cuboid 8 m long, 4 m broad and 2 m high. Solution.

Diagonal of a cuboid = √

Length^{2} + Breadth^{2} + Height^{2}

= √

8^{2} + 4^{2} + 2^{2}

= 9.17 m.

Total surface area of a cuboid = 2 (Length × Breadth + Breadth × Height + Length × Height)
In the figure alongside of the cuboid, total surface area = 2 (AB × BC + BC × BF + AB × BF).

Example
Find the total surface area (in m^{2}) of a cuboid 8 m long, 4 m broad and 2 m high. Solution.
Total surface area of a cuboid = 2 (Length × Breadth + Breadth × Height + Length × Height)
= 2 (8 × 4 + 4 × 2 + 8 × 2) = 112 m^{2}.

Lateral surface area of a cuboid = 2 (Length + Breadth) × Height
In the figure alongside of the cuboid, lateral surface area = 2 (AB + BC) × BF.

Example
Find the lateral surface area (in m^{2}) of a cuboid 10 m long, 4 m broad and 2 m high. Solution.
Lateral surface area of a cuboid = 2 (Length + Breadth) × Height = 2 (10 + 4) × 2 = 56 m^{2}.

Diagonal of a cube = Ö3 Side
In the figure alongside of the cube, side = AB = CD = EF = GH = AD = BC = EH = FG = AE = BF = CG = DH.
In the figure alongside of the cube, diagonal = √3 AB.

Example
Find the diagonal (in cm) of a cube whose side is 12√3 cm. Solution.
Diagonal of the cube = √3 Side
= √3 × 12√3 = 36 cm.

Total surface area of a cube = 6 × Side^{2}
In the figure alongside of the cube, total surface area = 6 × AB^{2}.

Example
Find the total surface area (in m^{2}) of a cube of side 10 m. Solution.
Total surface area of a cube = 6 × Side^{2}
= 6 × 10^{2} = 600 m^{2}.

Lateral surface area of a cube = 4 × Side^{2}
In the figure alongside of the cube, lateral surface area = 4 × AB^{2}.

Example
Find the lateral surface area (in m^{2}) of a cube of side 12 m. Solution.
Lateral surface area of a cube = 4 × Side^{2}
= 4 × 12^{2} = 576 m^{2}.