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Mensuration - Area of Parallelograms and Triangles

Grade 7ICSE

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

🔑Concepts

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A parallelogram is a quadrilateral where opposite sides are parallel and equal. Visually, it looks like a slanted rectangle. The area of a parallelogram represents the total space enclosed within its four boundaries.

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The base of a parallelogram is any side of the figure. The height (or altitude) is the perpendicular distance between the chosen base and the side opposite to it. In diagrams, height is often represented by a dashed line meeting the base at a 90∘90^{\circ} angle.

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The area of a parallelogram is equal to the product of its base and its corresponding height. This relationship exists because any parallelogram can be sliced and rearranged into a rectangle of the same base and height.

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A triangle is a three-sided polygon. If you draw a diagonal across a parallelogram, you split it into two congruent (identical) triangles. This visual relationship is why the area of a triangle is exactly half of a parallelogram with the same dimensions.

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In a triangle, the height is the vertical perpendicular line dropped from a vertex to the opposite side (the base). In an obtuse-angled triangle, the height may lie outside the triangle, drawn to an imaginary extension of the base.

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For right-angled triangles, the two sides forming the right angle are particularly special; one acts as the base and the other as the height, making area calculation straightforward as they are already perpendicular to each other.

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When solving mensuration problems, always ensure that the units for base and height are the same (e.g., both in cmcm or both in mm) before performing calculations. Area is always expressed in square units like cm2cm^2 or m2m^2.

📐Formulae

Area of a Parallelogram = Base×HeightBase \times Height

Area of a Triangle = 12×Base×Height\frac{1}{2} \times Base \times Height

Base of a Parallelogram = AreaHeight\frac{Area}{Height}

Height of a Parallelogram = AreaBase\frac{Area}{Base}

Base of a Triangle = 2×AreaHeight\frac{2 \times Area}{Height}

Height of a Triangle = 2×AreaBase\frac{2 \times Area}{Base}

💡Examples

Problem 1:

A parallelogram has a base of 15 cm15\text{ cm} and a corresponding height of 8 cm8\text{ cm}. Find its area. If another side of the parallelogram is 10 cm10\text{ cm}, find the height corresponding to that side.

Solution:

  1. Find the area using the first base and height: Area=Base×Height=15×8=120 cm2Area = Base \times Height = 15 \times 8 = 120\text{ cm}^2
  2. Use the area to find the second height for the side 10 cm10\text{ cm}: Area=Base2×Height2Area = Base_2 \times Height_2 120=10×Height2120 = 10 \times Height_2 Height2=12010=12 cmHeight_2 = \frac{120}{10} = 12\text{ cm}

Explanation:

First, calculate the total area using the known pair of base and height. Since the area of the parallelogram remains constant regardless of which side is used as the base, use that area value to solve for the missing height corresponding to the second side.

Problem 2:

The area of a triangle is 50 cm250\text{ cm}^2. If the base of the triangle is 12.5 cm12.5\text{ cm}, calculate its altitude (height).

Solution:

  1. Write down the area formula for a triangle: Area=12×Base×HeightArea = \frac{1}{2} \times Base \times Height
  2. Substitute the given values: 50=12×12.5×Height50 = \frac{1}{2} \times 12.5 \times Height
  3. Solve for Height: 100=12.5×Height100 = 12.5 \times Height Height=10012.5=8 cmHeight = \frac{100}{12.5} = 8\text{ cm}

Explanation:

Plug the known area and base into the triangle area formula. Multiply the area by 22 to remove the fraction, then divide by the base to find the perpendicular height.