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Mensuration - Area and Perimeter of Plane Figures (Triangle, Quadrilaterals)

Grade 9ICSE

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

🔑Concepts

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Perimeter is the total length of the boundary of a closed figure, while Area is the measure of the region or surface enclosed by it. Visualize perimeter as a fence around a garden and area as the grass covering the ground inside the fence.

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For a scalene triangle with side lengths aa, bb, and cc, where the height is unknown, Heron's Formula is used. First, calculate the semi-perimeter ss, which is half the sum of all sides, then apply the square root formula to find the space occupied.

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An equilateral triangle is a visual representation of perfect symmetry where all three sides are equal and all internal angles are 60∘60^{\circ}. Its area can be calculated using only the side length aa, making it more efficient than the standard base-height formula.

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A parallelogram can be visualized as a rectangle that has been tilted. Despite the tilt, its area remains the product of its base and its vertical (perpendicular) height, not the length of its slanted side. The opposite sides are parallel and equal in length.

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A rhombus is a special quadrilateral where all four sides are equal. Visually, its diagonals bisect each other at right angles (90∘90^{\circ}), effectively splitting the rhombus into four identical right-angled triangles. The area is half the product of these two diagonals.

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A trapezium is a quadrilateral with at least one pair of parallel sides (often called bases aa and bb). The area is the product of the average of the parallel sides and the perpendicular distance (height hh) between them.

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Rectangles and squares are fundamental plane figures. A rectangle has opposite sides equal and all angles at 90∘90^{\circ}, while a square is a special rectangle with all four sides equal. The diagonal of these figures creates two congruent right-angled triangles.

📐Formulae

Perimeter of a Triangle: P=a+b+cP = a + b + c

Area of a Triangle (General): A=12×base×heightA = \frac{1}{2} \times \text{base} \times \text{height}

Semi-perimeter (ss): s=a+b+c2s = \frac{a + b + c}{2}

Heron's Formula: Area=s(s−a)(s−b)(s−c)\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}

Area of an Equilateral Triangle: A=34a2A = \frac{\sqrt{3}}{4} a^2

Area of a Rectangle: A=l×bA = l \times b

Perimeter of a Rectangle: P=2(l+b)P = 2(l + b)

Area of a Square: A=a2A = a^2 or A=12d2A = \frac{1}{2} d^2 (where dd is the diagonal)

Area of a Parallelogram: A=base×heightA = \text{base} \times \text{height}

Area of a Rhombus: A=12×d1×d2A = \frac{1}{2} \times d_1 \times d_2

Area of a Trapezium: A=12×(a+b)×hA = \frac{1}{2} \times (a + b) \times h

💡Examples

Problem 1:

Find the area of a triangle whose sides are 13 cm13 \text{ cm}, 14 cm14 \text{ cm}, and 15 cm15 \text{ cm}.

Solution:

  1. Find the semi-perimeter ss: s=13+14+152=422=21 cms = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \text{ cm}
  2. Apply Heron's Formula: Area=21(21−13)(21−14)(21−15)\text{Area} = \sqrt{21(21-13)(21-14)(21-15)} Area=21×8×7×6\text{Area} = \sqrt{21 \times 8 \times 7 \times 6} Area=7056=84 cm2\text{Area} = \sqrt{7056} = 84 \text{ cm}^2

Explanation:

Since all three sides of the triangle are given and it is a scalene triangle, Heron's formula is the most direct method to find the area.

Problem 2:

The area of a rhombus is 96 cm296 \text{ cm}^2 and one of its diagonals is 12 cm12 \text{ cm}. Find the length of the other diagonal and the length of its side.

Solution:

  1. Find the second diagonal (d2d_2): Area=12×d1×d2\text{Area} = \frac{1}{2} \times d_1 \times d_2 96=12×12×d296 = \frac{1}{2} \times 12 \times d_2 96=6×d2  ⟹  d2=16 cm96 = 6 \times d_2 \implies d_2 = 16 \text{ cm}
  2. Find the side (aa) using the property that diagonals bisect at right angles: side=(d12)2+(d22)2\text{side} = \sqrt{(\frac{d_1}{2})^2 + (\frac{d_2}{2})^2} a=62+82=36+64=100=10 cma = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm}

Explanation:

We use the area formula for a rhombus to find the missing diagonal. Then, we use the Pythagorean theorem on one of the four internal right-angled triangles formed by the diagonals to find the side length.