This MCQ module is based on: Perimeter and Area of a Circle
TOPIC 16 OF 25
Perimeter and Area of a Circle
🎓 Class 9
Mathematics
CBSE
Theory
Ch 6 — Measuring Space: Perimeter and Area
⏱ ~40 min
🌐 Language: [gtranslate]
🧠 AI-Powered MCQ Assessment
▲
📐 Maths Assessment
▲
This mathematics assessment will be based on: Perimeter and Area of a Circle
Targeting Class 9 level in Mensuration, with Intermediate difficulty.
Upload images, PDFs, or Word documents to include their content in assessment generation.
6.1 Why Measure Space?
Look at a relay race on a 400 m track. Athletes run in different lanes. Should they all start at the same line, or should the outer-lane runners begin slightly ahead? The lanes are not equal in length, even though each is "one lap". To answer questions like this, we need to measure curved lengths and shaded areas. This chapter sharpens those tools.
The perimeter? of a closed plane figure is the length of its boundary, while its area? is the size of the region it encloses.
Quick Review — Standard Formulas
| Shape | Perimeter | Area |
|---|---|---|
| Square (side \(a\)) | \(4a\) | \(a^2\) |
| Rectangle (\(l, b\)) | \(2(l+b)\) | \(lb\) |
| Triangle (base \(b\), height \(h\)) | sum of three sides | \(\tfrac12 b h\) |
| Parallelogram | \(2(a+b)\) | base × height |
| Trapezium | sum of four sides | \(\tfrac12 (a+b) h\) |
| Circle (radius \(r\)) | \(2\pi r\) | \(\pi r^2\) |
6.2 The Magic Number π
Walk around a circle of radius \(r\) once and you cover a distance \(2\pi r\). The factor π is the same for every circle: π = circumference ÷ diameter. It is one of the most extraordinary constants in mathematics — irrational, non-terminating, and the same in every culture's geometry.
Definition
For any circle, \(\pi = \dfrac{\text{circumference}}{\text{diameter}} \approx 3.14159...\) — an irrational, non-terminating decimal. Common approximations: \(\pi \approx \tfrac{22}{7}\) or \(3.14\).Historical Note
By around 1900 BCE, Mesopotamian and Egyptian scribes used π ≈ 3.125 and 3.16 respectively. In India, the Shulba Sutras (~800 BCE) used a clever approximation, and Aryabhata (c. 499 CE) wrote π ≈ 3.1416. Around 250 BCE, Archimedes of Syracuse trapped π between perimeters of inscribed and circumscribed 96-gons, getting \(\tfrac{223}{71} < \pi < \tfrac{22}{7}\).Archimedes' method: trap the circle between two polygons.
6.3 Circumference and Area of a Circle
Every circle of radius \(r\) has:
Circumference \(C = 2\pi r\) and Area \(A = \pi r^2\).
To see why \(A = \pi r^2\), imagine cutting a disc into many narrow sectors and rearranging them as a near-rectangle. The "rectangle" has length \(\pi r\) (half the circumference) and height \(r\), so its area is \(\pi r \cdot r = \pi r^2\). The more sectors we use, the closer we get to a true rectangle — and the limit is exact.
Cutting a disc into sectors and rearranging gives a rectangle of area πr².
6.4 Length of an Arc and Area of a Sector
If an arc of a circle of radius \(r\) makes an angle \(\theta\) (in degrees) at the centre, then the arc takes a fraction \(\tfrac{\theta}{360°}\) of the full circumference. So:
Arc length \(\ell = \dfrac{\theta}{360°}\times 2\pi r\)
The corresponding sector takes the same fraction of the area:
Sector area \(= \dfrac{\theta}{360°}\times \pi r^2\)
A semicircle is half the disc; a quadrant is one quarter.
Worked Examples
Example 1. The perimeter of a circle is 44 cm. What is its radius? (\(\pi=\tfrac{22}{7}\))
\(2\pi r = 44 \Rightarrow r = \tfrac{44}{2\pi} = \tfrac{44 \times 7}{2 \times 22} = 7\) cm.
Example 2. Calculate to 3 significant figures the circumference of a circle with (i) radius 7 cm, (ii) radius 10 cm, (iii) radius 12 cm. Use \(\pi \approx 3.14\).
(i) \(2\pi r = 6.28 \times 7 = 44.0\) cm. (ii) \(6.28 \times 10 = 62.8\) cm. (iii) \(6.28 \times 12 = 75.4\) cm.
Example 3. Calculate the length of an arc of a circle of radius 8 cm that makes an angle of 60° at the centre.
\(\ell = \tfrac{60}{360}\cdot 2\pi(8) = \tfrac{1}{6}\cdot 16\pi = \tfrac{8\pi}{3} \approx 8.38\) cm.
Example 4. Find the area of a sector of a circle of radius 14 cm in which the arc subtends 90° at the centre. Take π = 22/7.
Sector area = \(\tfrac{90}{360}\pi r^2 = \tfrac14 \cdot \tfrac{22}{7}\cdot 196 = 154\) cm².
Activity: Measure π Yourself
L3 ApplyMaterials: three circular objects (bowl, plate, coin), string, ruler.
Predict: When you divide circumference by diameter, will the result depend on the size of the circle?
- Wrap the string once around each object. Mark the length and measure it — this is the circumference.
- Measure the diameter with a ruler.
- Compute C/D for each object.
- Compare your three values.
All three ratios should be close to 3.14, regardless of the size. The constancy of this ratio across all circles is exactly what we call π.
Competency-Based Questions
Scenario: A circular athletics track has radius 35 m. A maintenance team needs to (a) replace the white boundary paint, (b) re-grass the inside region, and (c) reset start lines for a 100° sector run for short-distance training.
Q1. Total length of paint required for the boundary (\(\pi=\tfrac{22}{7}\)):
L3 Apply(b) 220 m. Circumference = 2 × (22/7) × 35 = 220 m.
Q2. The team estimates 1 kg of grass seed covers 50 m². How many kg are needed for the full circular field?
L4 AnalyseArea = (22/7) × 35² = 3850 m². Seed = 3850 / 50 = 77 kg.
Q3. Evaluate: a coach claims "doubling the radius doubles the area." Critique.
L5 EvaluateFalse. Area scales as \(r^2\). Doubling r quadruples the area. The coach is confusing perimeter (which DOES double) with area.
Q4. Design: the 100° sector for short-distance training is to be marked. Find its area and arc length.
L6 CreateArea = (100/360) × 3850 ≈ 1069.4 m². Arc = (100/360) × 220 ≈ 61.1 m.
Assertion–Reason Questions
A: The ratio of circumference to diameter is the same for every circle.
R: π is an irrational number whose decimal never terminates.
R: π is an irrational number whose decimal never terminates.
(b) Both are true facts about π but R (irrationality) does not justify A (constancy of ratio). The constancy is what defines π in the first place.
A: If the radius of a circle is doubled, its area becomes four times the original.
R: Area = πr², which depends on the square of the radius.
R: Area = πr², which depends on the square of the radius.
(a) R explains A: \((2r)^2 = 4r^2\).
A: The arc length of a 60° sector of a circle of radius 6 cm is 2π cm.
R: Arc length = (θ/360°) × 2πr.
R: Arc length = (θ/360°) × 2πr.
(a) Plug in: (60/360) × 12π = 2π cm. R explains A.
Frequently Asked Questions
What is the formula for the circumference of a circle in Class 9?
The circumference of a circle of radius r is C = 2 pi r, or equivalently C = pi d where d is the diameter. The constant pi is approximately 3.14159 and the ratio of circumference to diameter is the same for every circle.
What is the formula for the area of a circle?
The area of a circle of radius r is A = pi r^2. It can be derived by slicing the circle into many thin sectors and rearranging them into an approximate parallelogram of base pi r and height r.
How do you find the area of a sector of angle theta degrees?
The area of a sector of a circle of radius r and central angle theta degrees is (theta/360) x pi r^2. The arc length is (theta/360) x 2 pi r. Both formulas treat the sector as a fraction theta/360 of the full circle.
How do you compute the area of a circular ring (annulus)?
If the ring has outer radius R and inner radius r, its area is pi R^2 - pi r^2 = pi (R^2 - r^2) = pi (R + r)(R - r). Subtract the area of the inner circle from the area of the outer circle.
How many times does a wheel of radius 35 cm rotate to cover 11 m?
Circumference = 2 pi r = 2 x (22/7) x 35 = 220 cm = 2.2 m. Number of rotations = total distance / circumference = 11 / 2.2 = 5 rotations.
Why is pi the same for every circle?
Because all circles are similar (they differ only in size, not in shape), the ratio of circumference to diameter is invariant: it is the same constant pi for every circle. This is a defining property of pi and of Euclidean geometry.
AI Tutor
Mathematics Class 9 — Ganita Manjari
Ready