Class 9 › Mathematics › Ganita Manjari Measuring Space: Perimeter and Area Class 9 Mathematics · Ganita Manjari · Chapter 6
Summary Questions & answers Practise this chapter → This chapter measures the boundary and the region of plane shapes. The perimeter is the total distance around a shape: \(4a\) for a square of side \(a\), and \(2(l+w)\) for a rectangle. For a circle the perimeter is the circumference \(C = 2\pi r = \pi d\), where \(\pi\) is the constant ratio of circumference to diameter; historical estimates include Archimedes' bound \(3\tfrac{10}{71} < \pi < 3\tfrac{1}{7}\), Aryabhata's \(\pi \approx 3.1416\) and Madhava's exact infinite series. Area measures the surface enclosed: a rectangle has area \(l \times w\) and a triangle \(\dfrac{1}{2} \times \text{base} \times \text{height}\). When the height is not known, Heron's formula gives the area of a triangle from its sides: \(\sqrt{s(s-a)(s-b)(s-c)}\), where \(s = \dfrac{a+b+c}{2}\) is the semi-perimeter. The area of a circle is \(\pi r^2\), explained by Nilakantha as equal to a triangle with base \(C\) and height \(r\). The chapter also treats the area of a sector as a fraction of the whole circle and applies these ideas to real problems such as relay-race lane staggers.
Perimeter
The total length of the boundary of a closed plane figure.
Circumference
The perimeter of a circle, \(C = 2\pi r = \pi d\).
\(\pi\) (pi)
The constant ratio of a circle's circumference to its diameter, approximately \(3.1416\).
Heron's formula
The area of a triangle from its sides: \(\sqrt{s(s-a)(s-b)(s-c)}\) with \(s=\dfrac{a+b+c}{2}\).
Semi-perimeter
Half the perimeter of a triangle, \(s = \dfrac{a+b+c}{2}\).
Sector
A part of a circle bounded by two radii and the arc between them.
Q1 What is the perimeter of a square of side \(a\)? Answer Q2 Write the formula for the circumference of a circle of radius \(r\). Answer Q3 What is the area of a triangle with base \(b\) and height \(h\)? Answer Q4 State Heron's formula for the area of a triangle. Answer Q5 Find the area of a triangle with sides \(3, 4, 5\) using Heron's formula. Answer Q6 What is the area of a circle of radius \(r\)? Answer Q7 How is the area of a circle related to a triangle? Answer Q8 What value did Aryabhata give for \(\pi\)? Answer Q9 What is the perimeter of a rectangle of length \(l\) and width \(w\)? Answer Q10 How is the area of a sector found? Answer Q1 In my school, the playground is too small to have a 400 m track, so the school constructed a 200 m track instead. Does this mean that we need a smaller stagger for the race tracks in my school (i.e., smaller than the stagger used in the Olympics), for the same 4 × 100 m relay race? Answer Q1 What is the connection between this question and the one about the 400 m athletics track? Answer Q1 You can do a simple measurement at home to estimate the C/D ratio. Take a cotton reel with thin thread around it. Measure the diameter D of the reel as accurately as possible. Unwrap and then tightly wrap the thread around the reel 20 times. Unwrap it again; measure its length L, and calculate L/20D. This is the ratio we want. For accuracy, the thread should be very thin. Please do the experiment! Do you get a ratio between 3 and 4? Between 3.1 and 3.2? Answer Q2 It is also possible to estimate the C/D ratio using pure geometry, i.e., without any measurements at all! Can you imagine how? Answer Q1 What is the difference in radius between the first and second lanes? Use the Fig. 6.11 to find the stagger needed by the runner in the second lane. Will an equal stagger be needed between the third and second lanes? Answer Q1 What happens if the parallelogram is ‘thin’ (Fig. 6.18) and the foot of the perpendicular from C to AD does not lie on side AD? The construction then does not seem to work. How do we fix this ‘gap’? Answer Q1 The area of a rectangle can be found when we know the lengths of its sides. Is the same true for a parallelogram? That is, can we find the area of a parallelogram when we know the lengths of its sides? Why or why not? Answer Q1 Since ΔABD and ΔACD have equal area, you may wonder — Can we divide ΔABD using straight cuts into two or more pieces that we can then rearrange to exactly cover ΔACD? What do you think? Is it possible? Answer Q1 Suppose we are given two polygons P and Q with equal area. Will it always be possible to divide one of them using straight cuts into two or more pieces and then rearrange the pieces to exactly cover the other polygon? Try this out for familiar shapes, e.g., 1. A square and non-square rectangle with equal area, 2. Two triangles with different shapes but equal area, 3. A triangle and a square with equal area. Formulate a conjecture of your own about this. Answer Q2 Think of various rectangles with perimeter 40 units (the sides do not have to be integers). 1. How many such rectangles are there? Answer Q3 Think of various rectangles with perimeter 40 units (the sides do not have to be integers). 2. Among them, is there one whose area is the largest? What are its dimensions? Answer Q4 Think of various rectangles with perimeter 40 units (the sides do not have to be integers). 3. Among all these rectangles, is there one whose area is the smallest? What are its dimensions? Do either of these answers come as a surprise to you? Answer Q1 What procedure would you use to square a given triangle? Here, the task is to construct a square whose area is equal to the area of some given triangle. Think carefully. How would you proceed? Answer Q1 1. The perimeter of a circle is 44 cm. What is its radius? Answer Q2 2. Calculate, correct to 3 significant figures, the circumference of a circle with: (i) radius 7 cm Answer Q3 2. Calculate, correct to 3 significant figures, the circumference of a circle with: (ii) radius 10 cm Answer Q4 2. Calculate, correct to 3 significant figures, the circumference of a circle with: (iii) radius 12 cm Answer Q5 3. Calculate the length of the arc of a circle if: (i) the radius is 3.5 cm and the angle at the centre is 60° Answer Q6 3. Calculate the length of the arc of a circle if: (ii) the radius is 6.3 m and the angle at the centre is 120° Answer Q7 4. Find the perimeter of a sector (i.e., the curved portion as well as the two straight portions) of a circle of radius 14 cm and sector angle 75°. Answer Q8 5. Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14): (i) 60 m by 80 m shape Answer Q9 5. Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14): (ii) 8 cm by 12 cm shape Answer Q10 5. Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14): (iii) 10 cm shape Answer Q11 5. Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14): (iv) 12 cm shape Answer Q12 5. Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14): (v) 14 cm shape Answer Q13 5. Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14): (vi) 28 cm shape Answer Q14 5. Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14): (vii) 8 cm by 6 cm shape Answer Q15 5. Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14): (viii) 4 cm 4 cm 4 cm shape Answer Q16 5. Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14): (ix) 10 cm 10 cm shape Answer Q17 6. If the diameter of a car tyre is 56 cm, then: (i) How far does the car need to travel for the tyre to complete one revolution? Answer Q18 6. If the diameter of a car tyre is 56 cm, then: (ii) How many revolutions does the tyre make if the car travels 10 km? Answer Q19 7. Find the total perimeter of all the petals in each of the given flowers. (i) Fig. 6.15A: The centres of the arcs are the midpoints of the sides of the square 14 cm Answer Q20 7. Find the total perimeter of all the petals in each of the given flowers. (ii) Fig. 6.15B: The centres of the arcs are the vertices of the hexagon 42 cm Answer Q21 8. The ratio of the perimeters of two circles is 5:4. What is the ratio of their radii? Answer Q1 Why were human beings so fond of using circular shapes? Was this only for practical reasons, or could there have been other reasons too? What kinds of uses have human beings found for the circular shape? Answer Q1 1. Find the area of triangle ADE in Fig. 6.31. Answer Q2 2. The parallel sides of a trapezium are 40 cm and 20 cm. If its non-parallel sides are both equal, each being 26 cm, find the area of the trapezium. Answer Q3 3. Find the area of a triangle, given that its sides are 8 cm and 11 cm long, and its perimeter is 32 cm. Answer Q4 4. The sides of a triangular plot are in the ratio 3:5:7; its perimeter is 300 m. Find its area. Answer Q5 5. One diagonal of a rhombus is twice as long as the other diagonal. If the rhombus has area 128 cm2, find the length of the shorter diagonal. Answer Q6 6. ABCD is a parallelogram. P and Q are any two points on side AB. What can you say about the ratio area (∆PCD): area (∆QCD)? Answer Q7 7. O is any point on the diagonal PR of a parallelogram PQRS. Prove that the areas of triangles PSO and PQO are equal. Answer Q8 8. If the mid-points of the sides of a 4-gon (also known as a quadrilateral, but we prefer to call it a ‘4-gon’) are joined in order, prove that the area of the parallelogram thus formed will be half of the area of the given 4-gon. (You may wonder whether the 4-gon thus formed is always a parallelogram, and if so, why? These questions will be tackled and answered in the chapter on quadrilaterals.) Answer Q9 9. In ∆ABC, the midpoint of BC is D (Fig. 6.32). Median AD is drawn. P is any point on AD. Show that area (∆ABP) = area (∆ACP). Answer Q10 10. Given a square ABCD, let P be a point within it. Join PA, PB, PC, PD (Fig. 6.33). What is the ratio of the areas of the red region (∆PAB and ∆PCD) and the green region (∆PBC and ∆PDA)? Answer Q11 11. In ∆ABC, D is the midpoint of AB. P is any point on BC, and Q is a point on AB such that CQ || PD. PQ is joined (Fig. 6.34). Prove that Area (∆BPQ) = 1/2 Area (∆ABC). Answer Q1 1. Find the area of a sector of a circle with radius 7 cm if the angle of the sector is 60°. Answer Q2 2. Find the area of a quadrant of a circle whose circumference is 44 cm. Answer Q3 3. The length of the minute hand of a clock is 7 cm. Find the area swept by the minute hand in 10 minutes. Answer Q4 4. A chord of a circle of radius 10 cm subtends 90° at the centre. Find the area of the corresponding: (i) minor sector (that subtends 90° at the centre) Answer Q5 4. A chord of a circle of radius 10 cm subtends 90° at the centre. Find the area of the corresponding: (ii) major sector (that subtends 270° at the centre). (Use π ≈ 3.14.) Answer Q6 5. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre of the circle. Find the areas of the corresponding minor and major segments of the circle. (Use π ≈ 3.14 and √3 ≈ 1.73.) Answer Q7 6. A car has two wipers which do not overlap. Each wiper has a blade of length 28 cm and sweeps through an angle of 120°. Find the total area cleaned at each sweep of the blades. Answer Q8 7. A chord of a circle of radius r subtends an angle of 60° at the centre of the circle. Show that the area of the corresponding minor segment of the circle is equal to πr²(1/6 - √3/4π?) . Answer Q9 8. An equilateral triangle is inscribed in a circle of radius r. Show that the ratio of the area of the triangle to the area of the circle is equal to 3√3/4π ≈ 0.413. Answer Q10 9. A square is inscribed in a circle of radius r. Show that the ratio of the area of the square to the area of the circle is equal to 2/π ≈ 0.637. Answer Q11 10. A hexagon is inscribed in a circle of radius r. Show that the ratio of the area of the hexagon to the area of the circle is equal to 3√3/2π ≈ 0.827. Can you see why the answer is exactly twice the answer to Question 8? Answer Q1 1. Identities in algebra can sometimes be shown as area relationships. For example: The figure shown corresponds to the identity (a + b)² = a² + 2ab + b². Do you see how? Draw figures corresponding to the identities (a + b)(a – b) = a² – b² and (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. Answer Q2 2. An isosceles triangle has perimeter 40 cm; the equal sides are 15 cm each. Find the area of the triangle. Answer Q3 3. An isosceles triangle has base 10 cm, and its area is 60 cm². What are the lengths of the equal sides? Answer Q4 4. The area of a right-angled triangle is 54 sq. cm. One of its legs has length 12 cm. Find its perimeter. Answer Q5 5. The sides of a triangle are in the ratio 2:3:4, and its perimeter is 45 cm. Find its area. Answer Q6 6. The sides of a triangle have lengths 7 cm, 24 cm, 25 cm. Find the area of the triangle in two different ways. Answer Q7 7. If the wheel of a bicycle has a diameter of 60 cm, find how far a cyclist will have travelled after the wheel has rotated 100 times. Answer Q8 8. Find the area of a quadrant of a circle whose circumference is 66 cm. Answer Q9 9. The wheel of a car has an outer radius of 28 cm. Calculate how far the car travels after one complete turn of the wheel, and how many times the wheel turns during a journey of 1 km. Answer Q10 10. Two rectangles have the same area and the same perimeter. Does this mean that they are congruent to each other? Answer Q11 11. You know that the area of a parallelogram is base × height. Using this and the figure, show that the area of a trapezium is half the sum of the parallel sides × height, i.e., 1/2(a + b)h. Answer Q12 12. By dividing a trapezium into two triangles show that its area is, half the sum of the parallel sides multiplied by the height (the same formula as the one given above). Answer Q13 13. Show how we can use two identical copies of a trapezium to make a parallelogram. How will this give us the formula for the area of a trapezium? Answer Q14 14. Show that the area of a kite is half the product of its diagonals. Show this: (i) using algebra Answer Q15 14. Show that the area of a kite is half the product of its diagonals. Show this: (ii) using geometry Answer Q16 15. Three problems about fitting congruent shapes together: (i) Rectangle ABCD has sides a, b, and rectangle PQRS has sides 2a, 2b. Show that PQRS has 4 times the area of ABCD. Does this mean that 4 copies of rectangle ABCD will fit into rectangle PQRS? Check and see! Answer Q17 15. Three problems about fitting congruent shapes together: (ii) ∆ABC has sides a, b, c, and ∆PQR has sides 2a, 2b, 2c. Show that ∆PQR has 4 times the area of ∆ABC. Does this mean that 4 copies of ∆ABC will fit into ∆PQR? Check and see! Answer Q18 15. Three problems about fitting congruent shapes together: (iii) ∆ABC has sides a, b, c, and ∆PQR has sides 3a, 3b, 3c. Show that ∆PQR has 9 times the area of ∆ABC. Does this mean that 9 copies of ∆ABC will fit into ∆PQR? Check and see! Answer Q19 16. What fraction of the triangle is shaded? Fig. 6.43 Answer Q20 16. What fraction of the square is shaded? Fig. 6.44 Answer Q21 17. What fraction of the rectangle is covered by the circles? Fig. 6.45 Answer Q22 17. What fraction of the rectangle is covered by the circles? Fig. 6.46 Answer Q23 18. Use the above to make a conjecture about the area occupied by circles fitted into a rectangle in the manner shown. Test your conjecture for particular cases: 10 circles; 20 circles; 50 circles. Then prove your conjecture! Answer Q24 19. The figure shows nine identical rectangles fitted together to make a large rectangle whose area is 72 cm². Find the perimeter of each small rectangle. Answer Q25 20. Show that the areas of the shaded blue triangle and the shaded red triangle are equal. Find a way of cutting up the blue triangle into some number of pieces and rearranging the pieces to cover the red triangle. Answer Q26 21. The figure shows a quarter circle in a square. Its centre is at one vertex, and it passes through two adjacent vertices. There are two semicircles on two adjacent sides as diameters. They create the shaded regions A and B. Show that A and B have equal area. Answer Q27 22. In Fig. 6.50, four semicircles have been drawn within the given square whose side is 2 units. The centres of these semicircles are the midpoints of the sides. They create a 4-petalled flower (shown in blue). Find the perimeter and the area of this flower. Answer Q28 23. In Fig. 6.51 we see two concentric circles with a common centre O. A chord BC of the larger circle is drawn, touching the smaller circle at A. The length of BC is l. Show that the area of the green region enclosed between the two circles is 1/4 πl². Answer Q29 24. In Fig. 6.52, semicircles have been drawn on all the sides of a right-angled triangle as shown. Show that Area (A) + Area (B) = Area (C). Answer Q30 25. Fig. 6.53 shows two circles passing through each other’s centres. Find the area of the region enclosed by the two circles in terms of the common radius r. Answer Q31 26. In Fig. 6.54, we see three triangles within a rectangle. The areas of the triangles are A, B, C, as marked. Show that the area of the rectangle is 2(A + C)(B + C)/C. Answer Q32 27. In the figure we see two shaded regions formed by a quarter circle, a semicircle, and a triangle. Show that the areas of the two shaded regions are equal. Answer
The perimeter of a square of side \(a\) is:
A \(4a\) B \(a^2\) C \(a^3\) D \(2a\)
Flashcards & interactive practice 
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