CBSE Class 7 Maths: Perimeter and Area - Exercise 11.2

Welcome to a deep dive into Perimeter and Area Exercise 11.2 for CBSE Class 7 Maths! In previous classes, you've explored the perimeter and area of basic shapes like rectangles and squares. Now, we're going to expand our knowledge to discover how to measure the space covered by two very important geometric figures: parallelograms and triangles.

Understanding the area of these shapes isn't just about formulas; it's about developing your problem-solving skills and applying mathematics to real-world scenarios, from designing a garden to understanding architectural plans. By the end of this page, you'll not only master the formulas for parallelograms and triangles but also confidently solve problems involving unknown dimensions and composite figures. Let's begin this exciting journey with YoLearn.ai!

Understanding Area of Parallelograms and Triangles

To truly grasp Exercise 11.2, we must first clearly understand the concepts behind the area of parallelograms and triangles. The area of a closed figure is the measure of the surface enclosed by its boundary. It's always measured in square units, like square centimetres (cm²) or square meters (m²).

Area of a Parallelogram

A parallelogram is a quadrilateral where opposite sides are parallel and equal in length. Think of a rectangle that has been pushed over, making it 'lean'. To find its area, we need two key measurements: its base and its height.

The base is any one of its sides. The height (or altitude) is the perpendicular distance from that chosen base to the opposite side. It's crucial that the height is perpendicular to the base, forming a 90-degree angle. Imagine dropping a straight line from one vertex to the base, making sure it hits at a right angle. The formula is beautifully simple:

Area of a Parallelogram = base × height

Area of a Triangle

A triangle is a three-sided polygon. You might already know that a diagonal divides a parallelogram into two congruent triangles. This observation is key to understanding the area of a triangle. If we take a parallelogram and cut it exactly in half along its diagonal, we get two identical triangles. Therefore, the area of one such triangle will be half the area of the parallelogram.

Just like with a parallelogram, we need a base and a corresponding height. The base can be any side of the triangle. The height is the perpendicular distance from the vertex opposite to the base, down to that base (or its extension). The formula reflects its relation to the parallelogram:

Area of a Triangle = ½ × base × height

Remember, the height must be perpendicular to the base. This is a common point of confusion, so always look for that right angle!

Step-by-Step Calculation of Area

  1. For a Parallelogram — 1. Identify the Base (b): Choose any side of the parallelogram as its base. It's usually the side given in the problem or indicated with a length. 2. Identify the Corresponding Height (h): Find the perpendicular distance from the chosen base to the opposite parallel side. This will often be drawn as a dotted line inside or outside the parallelogram, forming a right angle with the base. 3. Apply the Formula: Use the formula: Area = base × height. 4. State the Units: Always write your final answer with appropriate square units (e.g., cm², m²).
  2. For a Triangle — 1. Identify the Base (b): Select any side of the triangle as its base. 2. Identify the Corresponding Height (h): Locate the perpendicular distance from the vertex opposite the chosen base, down to that base (or an extension of the base if it's an obtuse triangle). Look for the right angle symbol. 3. Apply the Formula: Use the formula: Area = ½ × base × height. 4. State the Units: Express your answer in square units (e.g., cm², m²).

Worked Examples: Putting Formulas into Practice

  • Example 1: Finding the Area of a Parallelogram A parallelogram has a base of 12 cm and a corresponding height of 7 cm. Find its area. Given: Base (b) = 12 cm, Height (h) = 7 cm Formula: Area of parallelogram = b × h Calculation: Area = 12 cm × 7 cm = 84 cm² Answer: The area of the parallelogram is 84 square centimetres.
  • Example 2: Finding the Area of a Triangle Calculate the area of a triangle with a base of 10 meters and a height of 8 meters. Given: Base (b) = 10 m, Height (h) = 8 m Formula: Area of triangle = ½ × b × h Calculation: Area = ½ × 10 m × 8 m = 5 m × 8 m = 40 m² Answer: The area of the triangle is 40 square meters.
  • Example 3: Finding the Height of a Parallelogram (Inverse Problem) The area of a parallelogram is 60 cm². If its base is 15 cm, find its height. Given: Area = 60 cm², Base (b) = 15 cm Formula: Area = b × h => 60 = 15 × h Calculation: To find h, divide the area by the base: h = 60 cm² / 15 cm = 4 cm Answer: The height of the parallelogram is 4 centimetres.

Exam Tip: Avoiding Common Mistakes

When solving problems from Exercise 11.2, students often make a few common errors. Be mindful of these to score full marks!

  1. Confusing Height with Side Length: Remember, the height must always be the perpendicular distance to the base. Sometimes a problem might give you the slanted side length of a parallelogram or triangle, but this is not the height unless it's a right-angled side.
  2. Missing the ½ for Triangles: It's very easy to forget the '½' factor when calculating the area of a triangle. Always double-check your formula: Area = ½ × base × height.
  3. Incorrect Units: Area is always measured in square units (e.g., cm², m²). Perimeter is in linear units (cm, m). Ensure you use the correct unit in your final answer.
  4. Identifying Corresponding Height: For a triangle, the height corresponds to a specific base. If the base changes, the corresponding height also changes. Always draw or visualize the perpendicular from the opposite vertex to the chosen base.

Practice Questions with Solutions

  • Q: Find the area of a parallelogram with a base of 7 cm and a height of 4.5 cm. A: Step 1: Identify the given values. Base (b) = 7 cm, Height (h) = 4.5 cm. Step 2: Recall the formula for the area of a parallelogram: Area = b × h. Step 3: Substitute the values into the formula: Area = 7 cm × 4.5 cm. Step 4: Perform the multiplication: Area = 31.5 cm². Final answer: The area of the parallelogram is 31.5 cm².
  • Q: A triangle has a base of 14 cm and its height is 9 cm. What is its area? A: Step 1: Identify the given values. Base (b) = 14 cm, Height (h) = 9 cm. Step 2: Recall the formula for the area of a triangle: Area = ½ × b × h. Step 3: Substitute the values into the formula: Area = ½ × 14 cm × 9 cm. Step 4: Perform the multiplication: Area = 7 cm × 9 cm = 63 cm². Final answer: The area of the triangle is 63 cm².
  • Q: The area of a parallelogram is 88 m². If its height is 11 m, find the length of its base. A: Step 1: Identify the given values. Area = 88 m², Height (h) = 11 m. We need to find the base (b). Step 2: Recall the formula for the area of a parallelogram: Area = b × h. Step 3: Substitute the known values: 88 m² = b × 11 m. Step 4: Solve for b: b = 88 m² / 11 m = 8 m. Final answer: The length of the base of the parallelogram is 8 m.
  • Q: The area of a triangle is 72 cm². If its base is 16 cm, what is its height? A: Step 1: Identify the given values. Area = 72 cm², Base (b) = 16 cm. We need to find the height (h). Step 2: Recall the formula for the area of a triangle: Area = ½ × b × h. Step 3: Substitute the known values: 72 cm² = ½ × 16 cm × h. Step 4: Simplify the equation: 72 cm² = 8 cm × h. Step 5: Solve for h: h = 72 cm² / 8 cm = 9 cm. Final answer: The height of the triangle is 9 cm.

Frequently Asked Questions

What is the main difference between perimeter and area?

Perimeter is the total length of the boundary of a closed figure, measured in linear units like cm or m. Area, on the other hand, is the measure of the surface enclosed within that boundary, measured in square units like cm² or m².

Why is the area of a triangle half the area of a parallelogram?

This is because any parallelogram can be divided into two identical (congruent) triangles by drawing one of its diagonals. Since the two triangles are equal, each triangle's area is exactly half of the parallelogram's area from which it was formed, assuming they share the same base and height.

Can the height of a triangle be outside the triangle?

Yes, for an obtuse-angled triangle (a triangle with one angle greater than 90 degrees), the perpendicular height from a vertex to an extended base can sometimes fall outside the triangle. You might need to extend the base line to draw the altitude (height).