# Ex.13.1 Q5 Surface Areas and Volumes - NCERT Maths Class 9

## Question

A cubical box has each edge \(10\,\rm{ cm}\) and another cuboidal box is \(12.5\,\rm{ cm}\) long, \(10\,\rm{ cm}\) wide and \(8\,\rm{ cm}\) high.

(i) Which box has the greater lateral surface area and by how much?

(ii) Which box has the smaller total surface area and by how much?

## Text Solution

**Reasoning:**

A cube is cuboid whose length, breadth and height and equal. A cuboid has six faces and the total surface area is the sum of the surface area of the \(6\) faces.

**What is the known?**

(i) The length of the cube.

(ii) The length, breadth height of the cuboid.

**What is the unknown?**

Greater lateral surface area and by how much?

**Steps:**

Lateral surface area of the cube and cuboid is the sum of the area of the four faces.

**For cube:**

Edge of the cube is \(10\,\rm{ cm}.\)

Lateral surface area of the cube

\[\begin{align}&=4\,\rm{(edge)^2}\\&= 4 \times {10^2} \\&= 400\,\rm{cm^2} \end{align}\]

**For cuboid:**

\[\begin{align}\text{length(l)} &= 12.5\,\rm{cm}\\\text{breadth(b)} &= 10\,\rm{cm} \\\text{height(h)} &= 8\,\rm{cm} \end{align}\]

Lateral surface area of the cuboid

\(\begin{align} = 2\,(l + b)\,\,h \end{align}\)

Lateral surface area of the cuboid

\[\begin{align}&= 2\,\,(12.5 + 10) \times 8\\& = 2(12.5 + 10) \times 8\\& = 2 \times 22.5 \times 8\\ &= 16 \times 22.5\\ &= 360\,\rm{cm^2} \end{align}\]

Cubical box has the greater lateral surface area than the cuboidal box by

\(\begin{align}\,40\,\rm{cm^2}. \;(400 - 360 = 40\,\rm{cm^2}). \end{align}\)

(ii) Smaller total surface area

**What is the known?**

(i) The length of the cube.

(ii) The length, breadth height of the cuboid.

**What is the unknown?**

Smaller total surface area and by how much?

**Steps:**

Total surface area of the cube

\[\begin{align}&= 6\,\rm{(edge)^2}\\&= 6 \times {10^2}\\&= 600\,\rm{cm^2} \end{align}\]

Total surface area of the cuboid

\[\begin{align}= 2(lb + bh + hl) \end{align}\]

\(\begin{align}&\text{length(l) = 12.5}\,\rm{cm}\\&\text{breadth(b) = 10}\,\rm{cm} \\&\text{height(h) = 8}\,\rm{cm} \end{align}\)

Total surface area:

\[\begin{align}&=\!2[(12.5\!\times\!10\!+\!10\!\times\!8\!+\!8\!\times\!12.5)]\\&= 2\,\,[125 + 80 + 100]\\ &= 610\,\rm{cm^2} \end{align}\]

Cubical box has the smaller total surface area than the cuboidal box by

\(\begin{align}(610 - 600) = 10\,\rm{cm^2} \end{align}\)