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Class 11 NCERT Math Notes
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/ Preparation Maths Mensuration
Area and perimeter are important concepts for any geometrical shape and figure. We can calculate the space occupied by that shape and the boundaries. In real life, these concepts are used widely in terms of construction or measuring the fence boundary. Different geometrical shapes have different formulas to calculate the area and perimeter.
This topic from Mensuration chapter covers the basic characteristics of any geometrical shape and their uses in our day-to-day life. For example, before constructing any building, the architect needs to figure out the building's area and perimeter to build walls and flooring purposes.
Area and perimeter calculation require the fundamental knowledge of the geometric shapes that are being used. CBSE board covers this chapter in earlier classes to ensure that students can easily grasp advanced concepts later on.
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11th Math NCERT Solutions
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12th Math NCERT Solutions
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- What are Area and Perimeter?
- What is the Area of Different Shapes?
- What is the Perimeter of Different Shapes?
- Difference Between Area and Perimeter
- Illustrative Examples on Area and Perimeter
What are Area and Perimeter?
We will now learn about area and perimeter, one by one. Let us begin with understanding what the area is.
What is Area?
In mathematics, area measures the amount of “flat” space within a two-dimensional shape. Think of it as the number of square units required for covering a shape without any gaps or overlaps.
An area of a square will be equal to the number of squares that will fit in if you break a region into grids of 1×1 squares. Area is measured as “units²” such as cm², m², in², etc.
Let us now understand what the perimeter is.
What is Perimeter?
The perimeter of any shape is the total length of the boundary. For composite figures, one should split the figure into known pieces. Then sum up their perimeters. In such cases, the perimeters of shared edges must be ignored. For an irregular curve, approximate the perimeter by many small straight segments.
What is the Area of Different Shapes?
The following table represents the area of basic shapes:
Area of Basic Shapes
| Shape | Area Formula |
|---|---|
| Rectangle | A = width × height |
| Square | A = side² |
| Triangle | A = ½ × base × height |
| Parallelogram | A = base × height |
| Trapezoid | A = (b₁ + b₂)/2 × height |
| Circle | A = π r² |
| Ellipse | A = π a b |
Let us take a look at the area of other shapes as well, beyond the basic shapes.
Area of Other Shapes
| Shape | Area Formula | Notes |
|---|---|---|
| Rhombus (or Kite) | A = (d₁ × d₂) / 2 | d₁, d₂ are diagonal lengths |
| Equilateral Triangle | A = (√3 / 4) × a² | all sides = a |
| Regular n-gon | A = (1/4) · n · s² · cot(π/n) | s = side length |
| Circle Sector | A = ½ · r² · θ | θ in radians |
| Annulus (ring) | A = π · (R² − r²) | R = outer radius, r = inner radius |
| Circular Segment | A = (R² / 2) · (θ − sin θ) | θ = central angle in radians |
What is the Perimeter of Different Shapes?
Difference Between Area and Perimeter
The following table explains the difference between area and perimeter:
| Aspect | Area | Perimeter |
|---|---|---|
| Application | Amount of surface covere (2-D space). | Total distance around the boundary of shape. |
| Units | Square units (m², cm², ft², etc.). | Linear units (m, cm, ft, etc.). |
| Basic formula (rectangle) | . | . |
| Basic formula (circle) | . | (also called circumference). |
| Dimensionality | 2d measurement (length × width). | 1d measurement (length only). |
| Effect of scaling | When each side is doubled, the area will become 4 × larger (since it scales with square of the factor). | When each side is doubled, the perimeter will become 2 times large (scales linearly). |
| Typical uses | Flooring, painting walls, crop fields, land zoning. | Fencing a yard, framing pictures, outlining tracks. |
| Zero-value case | A line segment has zero area. | A line segment’s “perimeter” is its length (non-zero). |
Illustrative Examples on Area and Perimeter
Let us solve some questions that can be asked in exams like CUET, and students must be well prepared to answer these questions
1. Find the area of the given quadrilateral PQRS.
(Source: NCERT)
Solution.
In the given case, d = 5.5 cm,
h1 = 2.5cm, h2 = 1.5 cm,
Area = 1/2 * d ( h1 + h2 )
= 1 2 × 5.5 × (2.5 + 1.5) cm^2
= 1 2 × 5.5 × 4 cm^2 = 11 cm^2
2. Find the rhombus area whose diagonals are of lengths 10 cm and 8.2 cm.
Solution.
Area of the rhombus = 1/2/* d1* d2
Where d1 and d2 are the lengths of the diagonals of the rhombus.
= 1/2 × 10 × 8.2 cm^2 = 41 cm^2
3. The rhombus area is 240 cm^2, and one of the diagonals is 16 cm. Find the other diagonal.
Solution.
One diagonal length (d1) = 16 cm & other diagonal length = (d2)
Hence, rhombus area = 1/2 *d1* d2 = 240
So, 1/2 *16* d = 240 Therefore, d2 = 30 cm
Maths Mensuration Exam
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