01 · Explore
Perimeter of a Shape
Perimeter is the total length around the border of a shape. It can be visualized as the distance a tiny insect travels while walking once around the boundary of a figure until it returns to the start.
For regular polygons, the perimeter is calculated by multiplying the length of one side by the total number of sides. For example, a square with side 'a' has a perimeter of 4a, and an equilateral triangle with side 'a' has a perimeter of 3a. A rectangle with length 'a' and width 'b' has a perimeter of 2(a + b).
The square is a special case of a rectangle where a = b, resulting in 2(a + a) = 4a. In all squares, the ratio of perimeter to side remains fixed at 4:1, regardless of size. Similarly, for equilateral triangles, this ratio is always 3:1.
When dealing with circles, the perimeter is specifically called the circumference. Just as squares have a fixed ratio of perimeter to side, circles have a fixed ratio of circumference (C) to diameter (D), which mathematicians call π (pi).
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Question
If the side of an equilateral triangle is doubled, what happens to its perimeter?
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02 · Explore
The C/D Ratio and the History of Pi
The ratio of a circle's circumference (C) to its diameter (D) is a constant denoted by the Greek letter π (pi).
Historically, civilizations estimated π with increasing accuracy. Mesopotamians (c. 1900 BCE) used 3.125, while Archimedes (250 BCE) used polygons to 'trap' π between 3 10/71 and 3 1/7. In China, Zu Chongzhi (480 CE) discovered the 'Close Ratio' 355/113, which remained the most accurate for 800 years.
In India, Āryabhaṭa (499 CE) provided the value 3.1416, describing it as 'asanna' (approximate). Brahmagupta (628 CE) suggested √10 ≈ 3.1622 for its mathematical elegance. Later, Mādhava of Sangamagrāma discovered the first exact formula for π using an infinite series: π/4 = 1 - 1/3 + 1/5 - 1/7 + ... (or π = 4(1 - 1/3 + 1/5 - 1/7 + ...)).
π is an irrational number, meaning it cannot be expressed as a simple fraction a/b where a and b are integers. Its decimal expansion goes on forever without a repeating pattern. Common approximations include 22/7 and 3.14.
| Mathematician/Culture | Value or Approximation of π | Method Used |
|---|---|---|
| Mesopotamia | 3.125 | Hexagon comparison |
| Archimedes | 3 10/71 < π < 3 1/7 | 96-sided polygons |
| Zu Chongzhi | 355/113 (≈ 3.1415929) | 24,576-sided polygons |
| Āryabhaṭa | 3.1416 | 62832/20000 |
| Brahmagupta | √10 (≈ 3.1622) | Algebraic elegance |
| Mādhava | π = 4(1 - 1/3 + 1/5 - 1/7 + ...) | Infinite series |
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Question
Why is 22/7 used for π if π is irrational?
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03 · Explore
Length of an Arc of a Circle
An arc is a portion of the circumference of a circle. Its length depends on the radius of the circle and the angle it subtends at the centre.
The circumference of a full circle is 2πr. A semicircle represents half a circle (180°), so its arc length is (180/360) × 2πr = πr. Similarly, a quarter circle (90°) has an arc length of (90/360) × 2πr = πr/2.
For any arc that subtends an angle θ at the centre, the length is calculated as a fraction of the total circumference. This is expressed by the formula: Length = 2πr × (θ/360).
In an athletics track, staggers are used because runners in outer lanes travel along arcs with larger radii. To ensure everyone runs exactly 400 m, the starting positions are shifted forward for outer lanes.
Calculating Arc Length
Length = 2 × (22/7) × 7 × (60/360) = 44 × (1/6) ≈ 7.33 cm
For a circle with radius 7 cm and a central angle of 60°, we substitute the values into the formula. 2 × 22/7 × 7 = 44. Then 44 × (1/6) = 44/6 ≈ 7.33 cm (to 3 significant figures).
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Question
What is the perimeter of a semicircle of radius r?
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04 · Explore
Area of Parallelograms and Triangles
The area of a shape is the amount of space it occupies in a two-dimensional plane, measured in square units.
A parallelogram can be transformed into a rectangle with the same base (b) and height (h). Therefore, the area of a parallelogram is given by the formula: Area = base × height (bh).
A triangle can be viewed as half of a parallelogram. By joining two congruent triangles, we form a parallelogram with the same base and height. Thus, the area of a triangle is 1/2 × base × height.
A core theorem states: A median of a triangle divides it into two triangles with equal area. Even if the two resulting triangles are not congruent, their areas are identical because they share the same height and have equal bases.
From Triangle to Parallelogram
- 1
Single Triangle
Start with a triangle of base 'b' and height 'h'.
- 2
Congruent Copy
Create an identical copy of the triangle.
- 3
Joining
Rotate and join the copy along one side to form a parallelogram.
- 4
Area Calculation
The parallelogram area is bh; since it's made of two triangles, one triangle is (1/2)bh.
This sequence shows how the triangle area formula is derived from the parallelogram area.
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Question
If a triangle and a parallelogram have the same base and height, what is the ratio of their areas?
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05 · Explore
Heron's Formula for Triangles
Heron's formula allows us to calculate the area of a triangle when only the lengths of its three sides are known, without needing the height.
If a triangle has sides a, b, and c, we first calculate the semi-perimeter (s), which is half the perimeter: s = (a + b + c) / 2.
The area is then found using the formula: Area = √[s(s - a)(s - b)(s - c)]. This formula is particularly useful for scalene triangles where the height is difficult to measure.
Other area formulas involve the circumcircle (radius R) and incircle (radius r). Area = abc / 4R and Area = r(a + b + c) / 2 (which is also Area = rs).
Area of a 3-4-5 Triangle
s = (3+4+5)/2 = 6; Area = √[6(6-3)(6-4)(6-5)]
First, find s = 6. Then Area = √[6 × 3 × 2 × 1] = √36 = 6 sq. units. This matches the (1/2) × base × height method for a right triangle (1/2 × 3 × 4 = 6).
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Question
Can Heron's formula be used for an equilateral triangle?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 17, 18, 19, 37.