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Mensuration - Perimeter and Area of 2D Shapes

Grade 10IGCSE

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

🔑Concepts

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Definition of Perimeter: The total distance around the boundary of a 2D shape.

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Definition of Area: The total space occupied by a 2D surface, measured in square units.

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Properties of Quadrilaterals: Understanding the specific attributes of squares, rectangles, parallelograms, trapeziums, rhombuses, and kites.

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Circle Geometry: Relationship between radius, diameter, circumference, and area.

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Arc Length and Sector Area: Calculating portions of a circle's circumference and area using the central angle θ.

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Composite Shapes: Breaking down complex figures into simpler, standard shapes to find total area or perimeter.

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Unit Conversion: Converting between linear units (cm to m) and area units (cm² to m²).

📐Formulae

Area of Triangle=12×base×height\text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height}

Area of Triangle (Trigonometric)=12absin⁡(C)\text{Area of Triangle (Trigonometric)} = \frac{1}{2}ab\sin(C)

Area of Rectangle=length×width\text{Area of Rectangle} = \text{length} \times \text{width}

Area of Parallelogram=base×perpendicular height\text{Area of Parallelogram} = \text{base} \times \text{perpendicular height}

Area of Trapezium=12(a+b)h\text{Area of Trapezium} = \frac{1}{2}(a + b)h

Circumference of Circle=2πr or πd\text{Circumference of Circle} = 2\pi r \text{ or } \pi d

Area of Circle=πr2\text{Area of Circle} = \pi r^2

Arc Length=θ360×2πr\text{Arc Length} = \frac{\theta}{360} \times 2\pi r

Area of Sector=θ360×πr2\text{Area of Sector} = \frac{\theta}{360} \times \pi r^2

💡Examples

Problem 1:

A sector of a circle has a radius of 9 cm and a central angle of 120°. Calculate the area of the sector and the length of the arc. (Use π=3.142\pi = 3.142)

Solution:

Area of Sector = 120360×3.142×92=13×3.142×81=84.834 cm2\frac{120}{360} \times 3.142 \times 9^2 = \frac{1}{3} \times 3.142 \times 81 = 84.834 \text{ cm}^2. Arc Length = 120360×2×3.142×9=13×18×3.142=18.852 cm\frac{120}{360} \times 2 \times 3.142 \times 9 = \frac{1}{3} \times 18 \times 3.142 = 18.852 \text{ cm}.

Explanation:

To find the sector area and arc length, we use the ratio of the central angle to the full 360° circle and multiply it by the full area and circumference formulas respectively.

Problem 2:

A trapezium has parallel sides of length 10 cm and 14 cm. The perpendicular distance between them is 5 cm. Find the area.

Solution:

Area = 12(10+14)×5=12(24)×5=12×5=60 cm2\frac{1}{2}(10 + 14) \times 5 = \frac{1}{2}(24) \times 5 = 12 \times 5 = 60 \text{ cm}^2.

Explanation:

The area of a trapezium is calculated by taking the average of the two parallel sides (sum divided by 2) and multiplying by the vertical height.

Problem 3:

Calculate the area of a triangle where two sides are 8 cm and 12 cm, and the included angle is 30°.

Solution:

Area = 12×8×12×sin⁡(30∘)=12×96×0.5=24 cm2\frac{1}{2} \times 8 \times 12 \times \sin(30^\circ) = \frac{1}{2} \times 96 \times 0.5 = 24 \text{ cm}^2.

Explanation:

When the perpendicular height is not given, we use the trigonometric formula Area = 1/2absin⁡(C)1/2 ab \sin(C), where aa and bb are the sides and CC is the angle between them.