Area And Perimeter 1

In this article, the learning objectives are:

1. Finding the maximum number of squares that can be fitted/cut from a rectangle
2. Finding unknown dimensions given area of a rectangle/square
3. Finding unknown dimensions given perimeter of a rectangle/square

Let’s recap P3 Area and Perimeter first! 

Perimeter

The perimeter of a shape is the total distance around the shape.

Perimeter Of A Square

\(\small \begin{aligned} \textsf{Perimeter Of A Square } &\mathsf{= \text{Length + Length + Length + Length}}\\[2ex] &\mathsf{= 4 \times \text{Length}} \end{aligned}\)

Perimeter Of A Rectangle 

\(\small{ \textsf{Perimeter Of A Rectangle} = \textsf{Length + Breadth + Length + Breadth} }\)
 

Area

The area is the space occupied by the figure. 

Area Of A Square

\(\mathsf{\small{\text{Area Of A Square}= \text{Length} \times \text{Length}}}\)
 

Area Of A Rectangle

\(\mathsf{\small{\text{Area Of A Rectangle}= \text{Length} \times \text{Breadth}}}\)

Question 1: 

Find the area and perimeter of the square below.

Solution: 

Area of square
\(= 6 \text{ cm} \times 6 \text{ cm}\)
\(= 36 \text{ cm}^2 \)

Perimeter of square
\(= 4 \times 6 \text{ cm}\)
\(= 24 \text{ cm}\)

Answer: 

Area: \(36 \text{ cm}^2\)
Perimeter: \(24 \text{ cm}\)

Question 2: 

Find the area and perimeter of the rectangle below.

Solution: 

Area of rectangle
\(= 6 \text{ cm} \times 4 \text{ cm}\)
\(= 24 \text{ cm}^2\)

Perimeter of rectangle
\(= 6 \text{ cm} + 4 \text{ cm} + 6 \text{ cm} + 4 \text{ cm}\)
\(= 20 \text{ cm}\)

Answer: 

Area: \(24 \text{ cm}^2\)
Perimeter: \(20 \text{ cm}\)

1. Finding the maximum number of squares that can be fitted / cut from a rectangle

To find the maximum number of squares that can be fitted/cut from an area, we will first find out the number of squares that are able to fit along the length and the breadth of that area.

Question 1: 

What is the maximum number of \(1 \text{ cm}\) squares that can be cut from the rectangle?

Solution:

Number of \(1 \text{ cm}\) squares along the length of the rectangle
\(= 5 \text{ cm} ÷ 1 \text{ cm}\)
\(= 5\)

Number of \(\small{1 \text{ cm}}\) squares along the breadth of the rectangle
\(= 3 \text{ cm} \div 1 \text{ cm}\)
\(= 3\)

Maximum number of squares that can be cut from the rectangle
\(= 5 \times 3\)
\(= 15\)

Answer:

\(15\) squares 

Question 2: 

What is the maximum number of \(2 \text{ cm}\) squares that can be cut from the rectangle?

Solution:

Number of \(2 \text{ cm}\) squares along the length of the rectangle
\(= 8 \text{ cm} \div 2 \text{ cm}\)
\(= 4\)

Number of \(\small{2 \text{ cm}}\) squares along the breadth of the rectangle
\(= 6 \text{ cm} \div 2 \text{ cm}\)
\(= 3\)

Maximum number of squares that can be cut from the rectangle
\(= 4 \times 3\)
\(= 12\)

Answer:

\(12\) squares 

Question 3:

What is the greatest number of \(4 \text{ cm}\) squares that can be cut from the rectangle? 

Solution: 

Number of \(4 \text{ cm}\) squares along the length of the rectangle
\(= 16 \text{ cm} \div 4\text{ cm}\)
\(= 4\)

Number of \(4 \text{ cm}\) squares along the breadth of the rectangle
\(= 10 \text{ cm} \div 4 \text{ cm}\)
\(= 2 \text{ R } 2 \text{ cm}\)

We ignore the part which is represented by the remainder of \(2 \text{ cm}\) as no squares can be cut from it.

Greatest number of squares that can be cut from the rectangle
\(= 4 \times 2\)
\(= 8\)

Answer:

\(8\) squares

2. Finding unknown dimensions given area of a rectangle / square

\(\mathsf{ \small{\text{Area Of A Rectangle} = \text{Length} \times \text{Breadth}} }\)

Therefore, 

\(\small \begin{align} \textsf{Length Of A Rectangle} &= \textsf{Area} \div \textsf{Breadth} \\[2ex] \textsf{Breadth Of A Rectangle} &= \textsf{Area} \div \textsf{Length} \end{align} ​\)

Question 1:

The area of a rectangle is \(126 \text{ cm}^2\). If its breadth is \(7 \text{ cm}\), what is the length of the rectangle?

Solution: 

Length of the rectangle
\(= 126 \text{ cm}^2 ÷ 7 \text{ cm}\)
\(= 18 \text{ cm}\)

Answer:

\(18 \text{ cm}\)

Question 2: 

The area of a rectangle is \(72 \text{ cm}^2\). Given that the length of the rectangle is \(9 \text{ cm}\), find the breadth of the rectangle. 

Solution: 

Breadth of the rectangle
\(= 72 \text{ cm}^2 \div 9 \text{ cm}\)
\(= 8 \text{ cm}\)

Answer:

\(8 \text{ cm}\)

Question 3: 

The area of a square is \(64 \text{ cm}^2\). Find the length of one side of the square.

Solution: 

Since,

\(8 \text{ cm} \times 8 \text{ cm} = 64 \text{ cm}^2\),

Length of one side of each square \(= 8 \text{ cm}\)

Answer:

\(8 \text{ cm}\)

Question 4: 

The figure below is made up of \(3\) identical squares. Given that the total area of the figure is \(75 \text{ cm}^2\), find the length of one side of each square.

Solution: 

Area of 1 square
\(= 75 \text{ cm}^2 \div 3\)
\(= 25 \text{ cm}^2\)

Since,

\(5 \text{ cm} \times 5 \text{ cm} = 25 \text{ cm}^2\),

Length of one side of each square \(= 5 \text{ cm}\)

Answer:

\(5 \text{ cm}\)

3. Finding unknown dimensions given perimeter of a rectangle/square

\(\small\begin{align} \mathsf{\text{Perimeter Of Rectangle }} &\mathsf{= \text{Length + Length + Breadth + Breadth}}\\[2ex] \mathsf{\text{Length Of Rectangle }} &\mathsf{= \text{(Perimeter - Breadth - Breadth)} \div 2}\\[2ex] \mathsf{\text{Breadth Of Rectangle }} &\mathsf{= \text{(Perimeter - Length - Length)} \div 2} \end{align}\)

Question 1: 

The perimeter of a rectangle is \(36 \text{ cm}\). Given that its breadth is \(5 \text{ cm}\), find its length.

Solution:

Perimeter of a rectangle \(= \textsf{Length + Breadth + Length + Breadth}\)

Total length of 2 lengths
\(= 36 \text{ cm} - 5 \text{ cm} - 5 \text{ cm}\)
\(= 26 \text{ cm}\)

Length of rectangle
\(= 26 \text{ cm} \div 2\)
\(= 13 \text{ cm}\)

Answer:

\(13 \text{ cm}\)

Question 2: 

The perimeter of a rectangle is \(72 \text{ cm}\). Given that its length is \(24 \text{ cm}\), find its breadth.

Solution:

 Perimeter of a rectangle \(= \textsf{Length + Breadth + Length + Breadth}\)

 Total length of 2 lengths
\(= 72 \text{ cm} - 24 \text{ cm} - 24 \text{ cm}\)
\(= 24 \text{ cm}\)

 Breadth of rectangle
\(= 24 \text{ cm} \div 2\)
\(= 12 \text{ cm}\)

Answer:

\(12 \text{ cm}\)

\(\small​\begin{align} \textsf{Perimeter of a square} &= \mathsf{4 \times Length} \\[2ex] \textsf{Length of one side of a square} &= \mathsf{Perimeter \div 4} \end{align}\)

Question 3: 

The perimeter of a square is \(60 \text{ cm}\). Find the length of one side of the square. 

Solution: 

Perimeter of a square \(= 4 \times \small{\textsf{ Length }}\)

Length of one side of the square
\(= 60 \text{ cm} \div 4\)
\(= 15 \text{ cm}\)

Answer:

\(15 \text{ cm}\)

Question 4: 

The area of a rectangular garden is \(168 \text{ m}^2\). Its breadth is 8 m. 

1. Find the length of the garden. 
2. Vincent jogged round the entire rectangular garden twice. Find the distance he jogged. 

Solution:

A. Length of the rectangle\(\begin{align} &= 168 \text{ m}^2 \div 8 \text{ m} \\[2ex] &= 21 \text{ m}  \end{align}\)

B. Perimeter of garden\(\begin{align}&= 21 \text{ m} + 8 \text{ m} + 21 \text{ m} + 8 \text{ m} \\[2ex] &= 58 \text{ m} \end{align}\)

Distance he jogged \(\begin{align}​​ &= 58 \text{ m} \times 2\\[2ex] &= 116 \text{ m}  \end{align}\)

Answer:

A. \(21 \text{ m}\)
B. \(116 \text{ m}\)

Question 5: 

Elaine jogged \(36 \text{ m}\) round a square sand pit. 

1. Find the length of one side of the sand pit. 
2. Find the area of the square sand pit. 

Solution: 

A. Length of one side of the sand pit\(\begin{align} &= 36 \text{ m} \div 4\\[2ex] &= 9 \text{ m} \end{align}\)

B. Area of the sand pit\(\begin{align} &= 9 \text{ m} \times 9 \text{ m}\\[2ex] &= 81 \text{ m}^2  \end{align}\)

Answer:

 A. \(9 \text{ m}\)
B. \(81 \text{ m}^2\)