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Mensuration - Area of Rectangles and Squares

Grade 6ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The area of a rectangle represents the total surface covered by it in a two-dimensional plane. It is calculated by multiplying its dimensions: Area=length×breadthArea = length \times breadth.

Diagram showing a rectangle with length and breadth labeled.
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A square is a special rectangle where all sides are equal. Therefore, its area is the product of its sides: Area=side×side=(side)2Area = side \times side = (side)^2.

Diagram showing a square with all sides equal.
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Area is always measured in square units, such as cm2cm^2, m2m^2, or km2km^2. When solving problems, ensure all dimensions are in the same unit before calculation.

A square divided into unit squares to illustrate square units.
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To find a missing dimension (length or breadth) of a rectangle when the area and one side are known, use division: length=Area÷breadthlength = Area \div breadth or breadth=Area÷lengthbreadth = Area \div length.

Visualizing the derivation of length from area.

📐Formulae

Area of a Rectangle=length×breadth\text{Area of a Rectangle} = length \times breadth

Area of a Square=side×side=(side)2\text{Area of a Square} = side \times side = (side)^2

length=Area of Rectanglebreadthlength = \frac{\text{Area of Rectangle}}{breadth}

breadth=Area of Rectanglelengthbreadth = \frac{\text{Area of Rectangle}}{length}

side=Area of Squareside = \sqrt{\text{Area of Square}}

💡Examples

Problem 1:

A rectangular floor is 15 m15\text{ m} long and 10 m10\text{ m} wide. Calculate the cost of carpeting the floor if the rate of carpeting is ₹50₹ 50 per square meter.

Solution:

Step 1: Identify the given values. Length (ll) = 15 m15\text{ m} Breadth (bb) = 10 m10\text{ m}

Step 2: Calculate the area of the rectangular floor. Area=l×b\text{Area} = l \times b Area=15 m×10 m=150 m2\text{Area} = 15\text{ m} \times 10\text{ m} = 150\text{ m}^2

Step 3: Calculate the total cost. Total Cost=Area×Rate\text{Total Cost} = \text{Area} \times \text{Rate} Total Cost=150×50=7500\text{Total Cost} = 150 \times 50 = 7500

Final Answer: The cost of carpeting is ₹7500₹ 7500.

Explanation:

First, we find the total surface space (area) of the floor using the rectangle formula. Then, we multiply this area by the cost per unit area to find the total expense.

Problem 2:

The area of a square plot is 144 m2144\text{ m}^2. Find the length of its side and also calculate its perimeter.

Solution:

Step 1: Find the side of the square. Area=side×side=144 m2\text{Area} = side \times side = 144\text{ m}^2 side=144=12 mside = \sqrt{144} = 12\text{ m}

Step 2: Calculate the perimeter of the square. Perimeter=4×side\text{Perimeter} = 4 \times side Perimeter=4×12=48 m\text{Perimeter} = 4 \times 12 = 48\text{ m}

Final Answer: The side of the plot is 12 m12\text{ m} and the perimeter is 48 m48\text{ m}.

Explanation:

We use the inverse of the area formula (square root) to find the length of one side. Once the side is known, we multiply it by 4 to find the total boundary length (perimeter).

Problem 3:

A square tile has a side of 25 cm25 \text{ cm}. How many such tiles are required to cover a rectangular floor of length 5 m5 \text{ m} and breadth 4 m4 \text{ m}?

Diagram showing a large rectangular floor and a small square tile inside it.

Solution:

Area of the floor=length×breadthArea \text{ of the floor} = length \times breadth Area of the floor=5 m×4 m=20 m2Area \text{ of the floor} = 5 \text{ m} \times 4 \text{ m} = 20 \text{ m}^2 Convert to cm2:20×100×100=2,00,000 cm2Convert \text{ to } cm^2: 20 \times 100 \times 100 = 2,00,000 \text{ cm}^2 Area of one square tile=side×side=25 cm×25 cm=625 cm2Area \text{ of one square tile} = side \times side = 25 \text{ cm} \times 25 \text{ cm} = 625 \text{ cm}^2 Number of tiles=Area of floorArea of one tileNumber \text{ of tiles} = \frac{Area \text{ of floor}}{Area \text{ of one tile}} Number of tiles=2,00,000625=320Number \text{ of tiles} = \frac{2,00,000}{625} = 320 Total tiles required=320\text{Total tiles required} = 320

Explanation:

First, find the area of the rectangular floor and the square tile. Ensure units match (convert meters to centimeters). Then, divide the total floor area by the area of a single tile to find the quantity.

Problem 4:

Find the area of a rectangular garden if its breadth is 12 m12 \text{ m} and its length is three times its breadth.

Diagram of a garden with length labeled as 3 times the breadth.

Solution:

Breadth(b)=12 mBreadth (b) = 12 \text{ m} Length(l)=3×12 m=36 mLength (l) = 3 \times 12 \text{ m} = 36 \text{ m} Area=length×breadthArea = length \times breadth Area=36 m×12 mArea = 36 \text{ m} \times 12 \text{ m} 36×1272360432\begin{array}{r} 36 \\ \times 12 \\ \hline 72 \\ 360 \\ \hline 432 \end{array} Area=432 m2\text{Area} = 432 \text{ m}^2

Explanation:

First, calculate the length based on the given ratio (3 times the breadth). Then, multiply the calculated length by the given breadth to find the area.