Question 1

Two circles of equal radius are located such that each circle passes through the centre of the other circle (Fig. 6.12).

Given that the radius of each circle is r units, find the perimeter of the shape formed by the two circles in terms of r units. (Ignore the dotted portions that lie within the circles.)

Accepted Answer

Let the two circles have centers A and B, each with radius r units. Each circle passes through the center of the other, so AB = r. The circles intersect at points C and D. Triangle ABC is equilateral with sides r, so angle CAB = 60°. The arcs forming the perimeter are each 120° arcs (since the angle subtended by the chord CD at the center is 120°). Each red arc is therefore 1/3 of the circumference of a circle with radius r.

The perimeter consists of two such arcs, so total perimeter = 2 × (1/3 × 2πr) = (4/3) × 2πr = (8/3)πr units. — Since AB = r and AC = BC = r, triangle ABC is equilateral with each angle 60°. The arcs CD on each circle correspond to 120° (360° - 2×60°) arcs. Each arc length = (120/360) × 2πr = (1/3) × 2πr. Two such arcs make the perimeter, so total perimeter = 2 × (1/3 × 2πr) = (8/3)πr.

Question 2

In Fig. 6.13, we see points P and Q and two paths connecting them. The first path is made up of the semicircle a. The other path is made up of three semicircles (b, c and d). Which path is longer? Choose one: (i) Path a is longer. (ii) Path b + c + d is longer. (iii) The two paths have equal length. (Try to answer this before reading on.)

Accepted Answer

The two paths have equal length.

Explanation: Let the radii of semicircles a, b, c, d be a', b', c', d' respectively. Length of semicircle a = πa'. Length of semicircles b, c, d are πb', πc', πd' respectively. Total length of second path = π(b' + c' + d'). Since the length of PQ = 2a' = 2b' + 2c' + 2d', it follows that a' = b' + c' + d'. Hence, lengths of both paths are equal. — Since PQ = 2a' and also PQ = 2b' + 2c' + 2d', we get a' = b' + c' + d'. Therefore, length of path a = πa' = π(b' + c' + d') = length of path b + c + d.

Question 3

The perimeter of a circle is 44 cm. What is its radius?

Accepted Answer

Given perimeter (circumference) C = 44 cm.
We know circumference C = 2πr.
Using π = 22/7,
44 = 2 × (22/7) × r
=> 44 = (44/7) × r
=> r = 44 × (7/44) = 7 cm.
So, the radius of the circle is 7 cm. — Using the formula for circumference C = 2πr, substitute given values and solve for r.

Question 4

Calculate, correct to 3 significant figures, the circumference of a circle with: (i) radius 7 cm (ii) radius 10 cm (iii) radius 12 cm.

Accepted Answer

Formula: Circumference C = 2πr.
(i) r = 7 cm
C = 2 × (22/7) × 7 = 44 cm
(ii) r = 10 cm
C = 2 × (22/7) × 10 = (44/7) × 10 = 440/7 ≈ 62.857 cm
Rounded to 3 significant figures: 62.9 cm
(iii) r = 12 cm
C = 2 × (22/7) × 12 = (44/7) × 12 = 528/7 ≈ 75.429 cm
Rounded to 3 significant figures: 75.4 cm — Apply circumference formula for each radius and round the results to 3 significant figures.

Question 5

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°.

Accepted Answer

Formula for arc length L = (θ/360) × 2πr
(i) r = 3.5 cm, θ = 60°
L = (60/360) × 2 × (22/7) × 3.5 = (1/6) × 2 × (22/7) × 3.5
= (1/6) × 2 × 22 × 0.5 = (1/6) × 22 = 3.666... cm ≈ 3.67 cm
(ii) r = 6.3 m, θ = 120°
L = (120/360) × 2 × (22/7) × 6.3 = (1/3) × 2 × (22/7) × 6.3
= (1/3) × 2 × 22 × 0.9 = (1/3) × 39.6 = 13.2 m — Use arc length formula for each case substituting radius and central angle values.

Question 6

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°.

Accepted Answer

Perimeter of sector = length of arc + 2 × radius
Arc length L = (θ/360) × 2πr = (75/360) × 2 × (22/7) × 14
= (75/360) × 2 × 22 × 2 = (75/360) × 88 = (75 × 88)/360 = 6600/360 = 18.33 cm
Perimeter = 18.33 + 2 × 14 = 18.33 + 28 = 46.33 cm — Calculate arc length using sector angle and radius, then add twice the radius for the two straight sides.

Question 7

Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14i to 6.14ix):

Accepted Answer

Since the question refers to multiple figures (Fig. 6.14i to 6.14ix), the perimeter of each shape is calculated by adding the lengths of straight sides and the lengths of arcs (quarter, half, or three-quarters of circles) as per the figure.

General approach:
- Calculate arc length = fraction × circumference = fraction × 2πr
- Add straight sides as given

Specific calculations depend on the dimensions given in each figure (not provided here). — Use the fraction of the circle indicated (quarter, half, three-quarters) to find arc length, then add straight sides to find total perimeter for each shape.

Question 8

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?

Accepted Answer

(i) Diameter d = 56 cm, radius r = 28 cm
Distance travelled in one revolution = circumference = 2πr = 2 × (22/7) × 28 = 2 × 22 × 4 = 176 cm = 1.76 m

(ii) Distance travelled = 10 km = 10,000 m
Number of revolutions = total distance / distance per revolution = 10,000 / 1.76 ≈ 5681.82 revolutions — Calculate circumference for one revolution distance, then divide total distance by this to find number of revolutions.

Question 9

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
(ii) Fig. 6.15B: The centres of the arcs are the vertices of the hexagon

Accepted Answer

The total perimeter of all petals is the sum of the perimeters of each petal.

(i) For Fig. 6.15A:
- Each petal is formed by arcs centered at midpoints of square sides.
- Calculate length of each arc and multiply by number of petals.

(ii) For Fig. 6.15B:
- Each petal is formed by arcs centered at vertices of hexagon.
- Calculate length of each arc and multiply by number of petals.

Exact calculations require dimensions from the figures (not provided here). The method involves finding arc lengths and summing over petals. — Identify arcs forming petals, calculate arc lengths using radius and central angles, then sum for all petals.

Question 10

The ratio of the perimeters of two circles is 5:4. What is the ratio of their radii?

Accepted Answer

Let the radii be r1 and r2.
Perimeters (circumferences) are C1 = 2πr1 and C2 = 2πr2.
Given C1 : C2 = 5 : 4
=> 2πr1 : 2πr2 = 5 : 4
=> r1 : r2 = 5 : 4
So, the ratio of their radii is 5:4. — Since circumference is directly proportional to radius, ratio of perimeters equals ratio of radii.

Question 11

Find the area of triangle ADE in Fig. 6.31.

Accepted Answer

To find the area of triangle ADE, use the given figure (Fig. 6.31). Identify the base and height of triangle ADE from the figure. Calculate the area using the formula: Area = (1/2) × base × height. Substitute the values from the figure and compute the area. — Step 1: Identify base and height of triangle ADE from Fig. 6.31.
Step 2: Use area formula for triangle: (1/2) × base × height.
Step 3: Substitute the values and calculate the area.

Question 12

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.

Accepted Answer

Given: Parallel sides a = 40 cm, b = 20 cm; non-parallel sides each = 26 cm.

Step 1: Let the trapezium be ABCD with AB and DC as parallel sides.
Step 2: Drop perpendiculars from vertices of the shorter base to the longer base to find height.
Step 3: Use Pythagoras theorem to find height h.
Step 4: Area of trapezium = (1/2) × (sum of parallel sides) × height = (1/2) × (40 + 20) × h = 30 × h.

Calculations:
Let the trapezium be ABCD with AB = 40 cm, DC = 20 cm, and AD = BC = 26 cm.
Drop perpendiculars from C and D to AB, meeting at points E and F respectively.
Let AE = x, then EB = 40 - x.
Since AD = BC = 26 cm, triangles AED and BFC are right angled.
Using Pythagoras theorem:
AE^2 + h^2 = 26^2
EB^2 + h^2 = 26^2
So, x^2 + h^2 = 676 and (40 - x)^2 + h^2 = 676
Subtracting:
x^2 - (40 - x)^2 = 0
Expanding:
x^2 - (1600 - 80x + x^2) = 0
x^2 - 1600 + 80x - x^2 = 0
80x = 1600
x = 20

Now, h^2 = 676 - x^2 = 676 - 400 = 276
h = sqrt(276) = 2 * sqrt(69) cm

Area = 30 × 2 × sqrt(69) = 60 × sqrt(69) cm² ≈ 499.2 cm². — Step 1: Assign variables to sides and drop perpendiculars to find height.
Step 2: Use Pythagoras theorem to find height h.
Step 3: Calculate area using trapezium area formula.
Step 4: Substitute values and compute final area.

Question 13

Find the area of a triangle, given that its sides are 8 cm and 11 cm long, and its perimeter is 32 cm.

Accepted Answer

Given sides: a = 8 cm, b = 11 cm, perimeter = 32 cm.

Step 1: Find the third side c.
Perimeter = a + b + c = 32
=> c = 32 - (8 + 11) = 13 cm.

Step 2: Use Heron's formula to find area.
Semi-perimeter, s = (a + b + c)/2 = 32/2 = 16 cm.

Area = \sqrt{s(s - a)(s - b)(s - c)}
= \sqrt{16(16 - 8)(16 - 11)(16 - 13)}
= \sqrt{16 × 8 × 5 × 3}
= \sqrt{1920}
= 43.82 cm² (approx).

Therefore, the area of the triangle is approximately 43.82 cm². — Step 1: Calculate the third side using perimeter.
Step 2: Calculate semi-perimeter.
Step 3: Apply Heron's formula.
Step 4: Compute the square root to find area.

Question 14

The sides of a triangular plot are in the ratio 3: 5: 7; its perimeter is 300 m. Find its area.

Accepted Answer

Given ratio of sides = 3:5:7, perimeter = 300 m.

Step 1: Let the sides be 3x, 5x, 7x.
Perimeter = 3x + 5x + 7x = 15x = 300
=> x = 300 / 15 = 20 m.

So, sides are 60 m, 100 m, and 140 m.

Step 2: Use Heron's formula to find area.
Semi-perimeter, s = (60 + 100 + 140)/2 = 150 m.

Area = \sqrt{s(s - a)(s - b)(s - c)}
= \sqrt{150(150 - 60)(150 - 100)(150 - 140)}
= \sqrt{150 × 90 × 50 × 10}
= \sqrt{6,750,000}
= 2598.08 m² (approx).

Therefore, the area of the triangular plot is approximately 2598.08 m². — Step 1: Find actual side lengths using ratio and perimeter.
Step 2: Calculate semi-perimeter.
Step 3: Apply Heron's formula.
Step 4: Compute the square root to find area.

Question 15

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.

Accepted Answer

Let the shorter diagonal be d cm.
Then the longer diagonal = 2d cm.

Area of rhombus = (1/2) × product of diagonals = 128 cm².

So, (1/2) × d × 2d = 128
=> d² = 128
=> d = \sqrt{128} = 8\sqrt{2} cm.

Therefore, the length of the shorter diagonal is 8√2 cm. — Step 1: Let shorter diagonal be d.
Step 2: Express longer diagonal as 2d.
Step 3: Use area formula of rhombus.
Step 4: Solve for d.
Step 5: Calculate numerical value.

Measuring Space

Measuring Space — Study Notes

Measuring Space: Perimeter and Area

6.1 PERIMETER OF A SHAPE

6.2 PERIMETER OF A CIRCLE—THE C/D RATIO

Practice Questions — Measuring Space

All 8 Chapters in Ganita Manjari (English)