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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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Area is defined as the total region or surface enclosed by a closed plane figure. If you place a flat object on a piece of paper, the amount of space it covers is its area.

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The standard unit of area is the square unit. For a small shape like a stamp, we use square centimeters (cm2cm^2), while for larger areas like a room, we use square meters (m2m^2). Visually, 1 cm21\text{ cm}^2 is the space occupied by a small square with each side measuring 1 cm1\text{ cm}.

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A rectangle is a quadrilateral where opposite sides are equal and each angle is 90∘90^\circ. Visually, it has a horizontal dimension called 'length' (ll) and a vertical dimension called 'breadth' (bb). The area is found by multiplying these two dimensions.

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A square is a special rectangle where all four sides are equal. Imagine a grid where the number of rows is exactly equal to the number of columns; this represents the area of a square as side×sideside \times side.

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To visualize the area of a rectangle, imagine dividing it into unit squares. For instance, a rectangle of 4 cm4\text{ cm} by 3 cm3\text{ cm} can be viewed as 3 rows containing 4 squares of 1 cm21\text{ cm}^2 each, totaling 12 squares.

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The relationship between units is crucial: since 1 m=100 cm1\text{ m} = 100\text{ cm}, a square meter (1 m×1 m1\text{ m} \times 1\text{ m}) is equivalent to 100 cm×100 cm=10,000 cm2100\text{ cm} \times 100\text{ cm} = 10,000\text{ cm}^2.

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If the area and one dimension of a rectangle are known, the missing dimension can be calculated by dividing the area by the known dimension. For a square, the side can be found by taking the square root of the 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).