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06
NCERT Solutions · Maths Ch 6
Measuring Space: Perimeter and Area
65 questions · 4 exercises
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Ex 6.1
Ex 6.2
Ex 6.3
Ex 6.4
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
17 QUESTIONS
Exercise 6.1
Q1
(Use π = 22/7.) The perimeter of a circle is 44 cm. What is its radius?
(2 marks)
Q2
(Use π = 22/7.) Calculate, correct to 3 significant figures, the circumference of a circle with: (i) radius 7 cm (ii) radius 10 cm (iii) radius 12 cm.
(3 marks)
Q3
(Use π = 22/7.) 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°, and (ii) the radius is 6.3 m and the angle at the centre is 120°.
(3 marks)
Q4
(Use π = 22/7.) 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°.
(3 marks)
Q5(i)
Find the perimeter of the following shape (arcs are quarter, half or three-quarter circles as appropriate). (Use π = 22/7.)
Figure: a rectangle 80 m long and 60 m wide, with a semicircle on each 60 m side (a running-track shape). The outline is the two 80 m sides and the two semicircles.
(3 marks)
Q5(ii)
Find the perimeter of the following shape (arcs are quarter, half or three-quarter circles as appropriate). (Use π = 22/7.)
Figure: half of a ring: an outer semicircle of diameter 12 cm and an inner semicircle of diameter 8 cm, joined at the base by two straight pieces.
(3 marks)
Q5(iii)
Find the perimeter of the following shape (arcs are quarter, half or three-quarter circles as appropriate). (Use π = 22/7.)
Figure: a square of side 10 cm with a semicircle on each of its four sides, drawn outward. Only the four semicircles form the outline.
(3 marks)
Q5(iv)
Find the perimeter of the following shape (arcs are quarter, half or three-quarter circles as appropriate). (Use π = 22/7.)
Figure: an equilateral triangle of side 12 cm with a semicircle drawn outward on each side; the outline is the three semicircles.
(3 marks)
Q5(v)
Find the perimeter of the following shape (arcs are quarter, half or three-quarter circles as appropriate). (Use π = 22/7.)
Figure: a plus-shaped arrangement of 5 equal squares, each of side 14 cm. On each of the 4 outer ends of the arms is a small semicircle (diameter 14 cm), and in each of the 4 corners between arms is a quarter circle of radius 14 cm. The outline is made only of these arcs.
(3 marks)
Q5(vi)
Find the perimeter of the following shape (arcs are quarter, half or three-quarter circles as appropriate). (Use π = 22/7.)
Figure: a large semicircle on a 28 cm diameter; the base is replaced by 4 small semicircles of diameter 7 cm, alternately bulging below and above the base line.
(3 marks)
Q5(vii)
Find the perimeter of the following shape (arcs are quarter, half or three-quarter circles as appropriate). (Use π = 22/7.)
Figure: a right triangle with legs 8 cm (base) and 6 cm (vertical), hypotenuse 10 cm, with a semicircle drawn outward on each side; the outline is the three semicircles.
(3 marks)
Q5(viii)
Find the perimeter of the following shape (arcs are quarter, half or three-quarter circles as appropriate). (Use π = 22/7.)
Figure: a large semicircle on a base of 12 cm; the base is replaced by three small semicircles each of diameter 4 cm, bulging upwards inside.
(3 marks)
Q5(ix)
Find the perimeter of the following shape (arcs are quarter, half or three-quarter circles as appropriate). (Use π = 22/7.)
Figure: a large semicircle on top over a 20 cm base, with the base replaced by two semicircles of diameter 10 cm: the left one bulging upward (inside) and the right one bulging downward.
(3 marks)
Q6
(Use π = 22/7.) 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? (ii) How many revolutions does the tyre make if the car travels 10 km?
(3 marks)
Q7(i)
(Use π = 22/7.) Find the total perimeter of all the petals in the given flower.
Figure (Fig. 6.15A): a square of side 14 cm. Four semicircles of radius 7 cm are drawn inside, centred at the midpoints of the sides; each passes through two corners and the centre, forming 4 petals.
(3 marks)
Q7(ii)
(Use π = 22/7.) Find the total perimeter of all the petals in the given flower.
Figure (Fig. 6.15B): a regular hexagon of side 42 cm. Arcs centred at each vertex with radius 42 cm pass through the two neighbouring vertices and the centre, forming 6 petals.
(3 marks)
Q8
The ratio of the perimeters of two circles is 5 : 4. What is the ratio of their radii?
(1 mark)
2
11 QUESTIONS
Exercise 6.2
Q1
Find the area of triangle ADE in Fig. 6.31.
Figure: rectangle ABCD with DC = 10 cm and BC = 8 cm. E is a point on side BC; A, D and E are joined.
(2 marks)
Q2
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.
(3 marks)
Q3
Find the area of a triangle, given that its sides are 8 cm and 11 cm long, and its perimeter is 32 cm.
(3 marks)
Q4
The sides of a triangular plot are in the ratio 3 : 5 : 7; its perimeter is 300 m. Find its area.
(3 marks)
Q5
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.
(2 marks)
Q6
ABCD is a parallelogram. P and Q are any two points on side AB. What can you say about the ratio area (ΔPCD) : area (ΔQCD)?
(2 marks)
Q7
O is any point on the diagonal PR of a parallelogram PQRS. Prove that the areas of triangles PSO and PQO are equal.
(3 marks)
Q8
If the mid-points of the sides of a 4-gon (quadrilateral) are joined in order, prove that the area of the parallelogram thus formed will be half of the area of the given 4-gon.
(3 marks)
Q9
In ΔABC, the midpoint of BC is D (Fig. 6.32). Median AD is drawn. P is any point on AD. Show that area (ΔABP) = area (ΔACP).
(3 marks)
Q10
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 (ΔPAB and ΔPCD) and the green region (ΔPBC and ΔPDA)?
(3 marks)
Q11
In ΔABC, 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 (ΔBPQ) = ½ Area (ΔABC).
(3 marks)
3
10 QUESTIONS
Exercise 6.3
Q1
(Use π = 22/7.) Find the area of a sector of a circle with radius 7 cm if the angle of the sector is 60°.
(2 marks)
Q2
(Use π = 22/7.) Find the area of a quadrant of a circle whose circumference is 44 cm.
(2 marks)
Q3
(Use π = 22/7.) The length of the minute hand of a clock is 7 cm. Find the area swept by the minute hand in 10 minutes.
(2 marks)
Q4
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), and (ii) major sector (that subtends 270° at the centre). (Use π ≈ 3.14.)
(2 marks)
Q5
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.)
(5 marks)
Q6
(Use π = 22/7.) 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.
(2 marks)
Q7
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)), i.e. r²(π/6 − √3/4).
(3 marks)
Q8
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.
(3 marks)
Q9
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.
(2 marks)
Q10
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?
(3 marks)
4
EXERCISE 6.4 · 27 QUESTIONS
End-of-Chapter Exercises
Q1
Draw figures corresponding to the identities (a + b)(a – b) = a² – b² and (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
(3 marks)
Q2
An isosceles triangle has perimeter 40 cm; the equal sides are 15 cm each. Find the area of the triangle.
(3 marks)
Q3
An isosceles triangle has base 10 cm, and its area is 60 cm². What are the lengths of the equal sides?
(2 marks)
Q4
The area of a right-angled triangle is 54 sq. cm. One of its legs has length 12 cm. Find its perimeter.
(3 marks)
Q5
The sides of a triangle are in the ratio 2 : 3 : 4, and its perimeter is 45 cm. Find its area.
(3 marks)
Q6
The sides of a triangle have lengths 7 cm, 24 cm, 25 cm. Find the area of the triangle in two different ways.
(3 marks)
Q7
(Use π = 22/7.) 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.
(2 marks)
Q8
(Use π = 22/7.) Find the area of a quadrant of a circle whose circumference is 66 cm.
(2 marks)
Q9
(Use π = 22/7.) 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.
(2 marks)
Q10
Two rectangles have the same area and the same perimeter. Does this mean that they are congruent to each other?
(3 marks)
Q11
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., ½(a + b)h.
Figure (Fig. 6.42): trapezium with top side a, bottom side b, height h, divided into a parallelogram of base a (left) and a triangle of base b − a (right).
(3 marks)
Q12
By dividing a trapezium into two triangles show that its area is half the sum of the parallel sides multiplied by the height.
(2 marks)
Q13
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?
(2 marks)
Q14
Show that the area of a kite is half the product of its diagonals. Show this: (i) using algebra, and (ii) using geometry.
(3 marks)
Q15
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 4 copies of ABCD fit into PQRS? (ii) ΔABC has sides a, b, c, and ΔPQR has sides 2a, 2b, 2c. Show that ΔPQR has 4 times the area of ΔABC. Do 4 copies of ΔABC fit into ΔPQR? (iii) ΔABC has sides a, b, c, and ΔPQR has sides 3a, 3b, 3c. Show that ΔPQR has 9 times the area of ΔABC. Do 9 copies fit?
(5 marks)
Q16
What fraction of the triangle is shaded (Fig. 6.43)? What fraction of the square is shaded (Fig. 6.44)?
Fig. 6.43: triangle TLR (T at top, L bottom-left, R bottom-right). M is the midpoint of TL. TR is divided into three equal parts at U (nearer T) and V. The shaded quadrilateral is L–M–U–V. Fig. 6.44: a square in which each corner is joined to the midpoint of a side (going round the square), forming a small tilted square in the middle (shaded).
(5 marks)
Q17
What fraction of the rectangle is covered by the circles (Fig. 6.45: 3 equal circles in a row touching each other and the rectangle; Fig. 6.46: 4 such circles)?
(2 marks)
Q18
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!
(3 marks)
Q19
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.
Figure (Fig. 6.47): top row of 4 rectangles lying flat (long side horizontal); bottom row of 5 rectangles standing up (long side vertical).
(3 marks)
Q20
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.
Figure (Fig. 6.48): lines from the top vertex to the two points of trisection of the base. The blue triangle is the left third and the red triangle is the right third.
(3 marks)
Q21
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.
Figure (Fig. 6.49): A is the lens where the two semicircles overlap; B is the part of the quarter circle outside both semicircles.
(3 marks)
Q22
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. Find the perimeter and the area of this flower.
(3 marks)
Q23
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 ¼πl².
(3 marks)
Q24
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).
A and B are the semicircles on the two legs a and b; C is the semicircle on the hypotenuse c.
(2 marks)
Q25
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.
Figure: centres A and B with AB = r; the circles meet at C and D.
(3 marks)
Q26
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.
Figure: rectangle with corner O at bottom-left. P is on the top side, Q on the right side, and R is the point directly below P at the same height as Q. A = △OPR, B = △ORQ, C = △PRQ (right angle at R).
(5 marks)
Q27
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.
Figure (Fig. 6.55): O is the centre; OA = OB = OC = r with ∠AOB = 90°. Arc AFB is the quarter circle with centre O. Arc AEB is a semicircle on AB as diameter (centre D, the midpoint of AB). The shaded regions are the crescent between arcs AEB and AFB, and △AOB.
(3 marks)