Practice Set 3 – Solutions പരപ്പളവ് – ചോദ്യങ്ങൾ
Seven questions on areas of circles. Each one has a hint, a step-by-step animation with the textbook figure, and a hidden final answer. Try each in your notebook first. Answers are given in terms of π and, where asked, as decimals.
📖 Tools you need
- Area of a circle = πr²; a sector of angle θ° is θ/360 of the circle.
- The perpendicular from the centre bisects a chord: r² = (distance)² + (half chord)².
- A square or rectangle inside a circle has its diagonal as a diameter; a regular hexagon's side equals the radius; its area is 6 × (√3/4)s².
Question 1 · A chord and the area
Q1A chord of length 4 cm is 3 cm away from the centre of a circle. Find the area of the circle.
ഒരു വൃത്തത്തിന്റെ കേന്ദ്രത്തിൽ നിന്നും 3 സെന്റിമീറ്റർ അകലെയുള്ള ഞാണിന്റെ നീളം 4 സെന്റിമീറ്ററാണ്. വൃത്തത്തിന്റെ പരപ്പളവ് കണക്കാക്കുക.
ഒരു വൃത്തത്തിന്റെ കേന്ദ്രത്തിൽ നിന്നും 3 സെന്റിമീറ്റർ അകലെയുള്ള ഞാണിന്റെ നീളം 4 സെന്റിമീറ്ററാണ്. വൃത്തത്തിന്റെ പരപ്പളവ് കണക്കാക്കുക.
The perpendicular from the centre cuts the chord in half: a right triangle with sides 3 and 2, hypotenuse r.
🎬 Solution animation
Answer: r² = 3² + 2² = 13, so the area is 13π sq cm ≈ 40.84 sq cm. (We never need r itself – πr² uses r² directly.)
Question 2 · Circle minus polygon
Q2In the figures, find the difference between the areas of the circle and the regular polygon, correct to two decimal places: (a) a square whose diagonal is 4 cm drawn in a circle (b) a regular hexagon whose longest diagonal is 4 cm drawn in a circle.
In both figures the 4 cm line is a diameter, so the radius is 2. A square's area is half the product of its diagonals; the hexagon is six equilateral triangles of side 2.
🎬 Solution animation
Answer: circle area 4π ≈ 12.566 in both. (a) square = ½ × 4 × 4 = 8 ⇒ difference 4π − 8 ≈ 4.57 sq cm. (b) hexagon = 6 × (√3/4) × 2² = 6√3 ≈ 10.392 ⇒ difference 4π − 6√3 ≈ 2.17 sq cm. The hexagon, with more sides, leaves less over.
Question 3 · Circles through four corners
Q3Circles are drawn through the four corners of a square of side 3 cm and of a rectangle 4 cm by 2 cm. Find the area of each circle.
The diagonal of each figure is a diameter. Find (diagonal)², then r² = (diagonal)²/4.
🎬 Solution animation
Answer: square: diagonal² = 9 + 9 = 18, r² = 4.5 ⇒ 4.5π sq cm ≈ 14.14. Rectangle: diagonal² = 16 + 4 = 20, r² = 5 ⇒ 5π sq cm ≈ 15.71.
Question 4 · Four small circles, one big circle
Q4Draw a square; with its four corners as centres and half the side as radius, draw circles. Then draw a square made of four squares of the same size and the circle fitting exactly inside it. Prove that the area of the big circle equals the sum of the areas of the four small circles.
Call the side of the small square s. The small circles have radius s/2; the big square has side 2s, so the big circle has radius s.
🎬 Solution animation
Answer: four small circles: 4 × π(s/2)² = 4 × πs²/4 = πs². Big circle (radius s): πs². Equal. ∎ Doubling the radius makes the area four times.
Question 5 · Two green regions
Q5The two squares are the same size. In the first, quarter circles are drawn at the four corners with radius half the side; in the second, the circle touching all four sides is drawn. Prove that the green parts have equal areas.
Four quarter circles of radius a/2 together make one circle of radius a/2 – the same as the inscribed circle.
🎬 Solution animation
Answer: square side a. Left: yellow = 4 × ¼ × π(a/2)² = πa²/4, green = a² − πa²/4. Right: yellow circle π(a/2)² = πa²/4, green = a² − πa²/4. Equal. ∎
Question 6 · A star from a hexagon
Q6With the corners of a regular hexagon of side 2 cm as centres, parts of circles are drawn (radius 1 cm) and the star-shaped figure left is cut out. Find its area.
Each interior angle of a regular hexagon is 120°, so each cut-off piece is 120/360 = ⅓ of a circle of radius 1. Hexagon area = 6 × (√3/4) × 2².
🎬 Solution animation
Answer: hexagon = 6√3 ≈ 10.392 sq cm. Six sectors, each ⅓ × π × 1² ⇒ total 2π ≈ 6.283. Star = 6√3 − 2π ≈ 4.11 sq cm.
Question 7 · Half the square
Q7Inside a square, parts of circles are drawn as in the figure: a semicircle on the upper half and two quarter circles at the bottom corners. Prove that the blue part has half the area of the square.
Side a. The semicircle has radius a/2 (its centre is the centre of the square). The two quarter circles also have radius a/2. Compare what is added above with what is removed below.
🎬 Solution animation
Answer: blue = semicircle + (bottom half − two quarter circles) = πa²/8 + (a²/2 − 2 × πa²/16) = πa²/8 + a²/2 − πa²/8 = a²/2, half the square. ∎ The semicircle exactly makes up for the two quarter circles.
🌍 Why this matters
Tile patterns, rangoli and jewellery designs are built from squares, hexagons and circle pieces. Splitting a fancy shape into "polygon minus sectors" is how designers work out how much material or paint is needed.