Area and perimeter Questions and answers

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Elementary Mathematics

61). What is the total area of three equilateral triangles inscribed in a semi-circle of radius 2cm?

A). \( \Large 12 cm^{2} \)

B). \( \Large \frac{3\sqrt{3}}{4}cm^{2} \)

C). \( \Large \frac{9\sqrt{3}}{4}cm^{2} \)

D). \( \Large 3 \sqrt{3}cm^{2} \)

View Answer

Correct Answer: \( \Large 3 \sqrt{3}cm^{2} \)

since, \( \Large \triangle's \) AOB,BOC,COD are equilateral.

Sides = 2 cm

Now, total area = \( \Large 3\times \)\( \Large \frac{\sqrt{3}}{4}(Side)^{2} \)

= \( \Large 3\times \)\( \Large \frac{\sqrt{3}}{4}\times (Side)^{2}=3\sqrt{3} cm^{2} \)

62). How many circular plates of diameter d be taken out of a square plate of side 2d with minimum loss of material?

A). 8

B). 6

C). 5

D). 2

View Answer

Correct Answer: 5

Area of square plate = \( \Large (Side)^{2} \)

= \( \Large (2d)^{2}=4d^{2} \)

Area of circular plate = \( \Large \pi \left(\frac{d}{2}\right)^{2}=\frac{\pi d^{2}}{4} \)

Number of square plates

=\( \Large \frac{4d^{2}}{\frac{\pi d^{2}}{4}}=\frac{4\times 4}{\pi} \)=5

Since, nearest integer value is 5.

63). What is the area of a circle whose area is equal to that of a triangle with sides 7 cm, 24 cm and 25 cm?

A). \( \Large 80 cm^{2} \)

B). \( \Large 84 cm^{2} \)

C). \( \Large 88 cm^{2} \)

D). \( \Large 90 cm^{2} \)

View Answer

Correct Answer: \( \Large 84 cm^{2} \)

Given that, a = 7, b = 24 and c = 25

Semi-perimeter of triangle =\( \Large \frac{a+b+c}{2}=\frac{7+24+25}{2}=\frac{56}{2} \)=28 cm

According to the question,

Area of circle = Area of triangle

=\( \Large \sqrt{s(s-a)(s-b)(s-c)} \)

=\( \Large \sqrt{28(28 - 7)(28 - 24)(28 - 25)} \)

=\( \Large \sqrt{28\times 21\times 4\times 3}=\sqrt{7056} \)=\( \Large 84 cm^{2} \)

64). A circle of radius 10 cm has an equilateral triangle inscribed in it. The length of the perpendicular drawn from the centre to any side of the triangle is

A). \( \Large 2.5\sqrt{3} \)cm

B). \( \Large 5\sqrt{3} \)cm

C). \( \Large 10\sqrt{3} \)cm

D). None of these

View Answer

65). AB and CD are two diameters of a circle of radius r and they are mutually perpendicular. What is the ratio of the area of the circle to the area of the triangle ACD?

A). \( \Large \frac{\pi}{2} \)

B). \( \Large \pi \)

C). \( \Large \frac{\pi}{4} \)

D). \( \Large 2\pi \)

View Answer

66). The area of a rectangle is 1.8 times the area of a square. The length of the rectangle is 5 times the breadth. The side of the square is 20 cm. What is the perimeter of the rectangle?

A). 145 cm

B). 144 cm

C). 133 cm

D). 135 cm

View Answer

Correct Answer: 144 cm

Area of square =\( \Large (Side)^{2}=20^{2} \) = 400 sq cm

Area of rectangle =\( \Large 1.8\times 400 \) = 720 sq cm

Let length and breadth of rectangle be 5x and x, respectively.

Then, according to the question.

\( \Large 5x\times x \)=720

=> \( \Large 5x^{2} \)=720 => \( \Large x^{2}=\frac{720}{5} \)=144

=\( \Large \sqrt{144} \)=12 cm

Perimeter of rectangle = 2 (5x + x) = 12x

=\( \Large 12\times 12 \) = 144cm

67). The side of a square is 5 cm which is 13 cm less than the diameter of a circle. What is the approximate area of the circle?

A). 245 sq cm

B). 235 sq cm

C). 265 sq cm

D). 255 sq cm

View Answer

Correct Answer: 255 sq cm

Diameter of the circle = 13 + 5 = 18cm

Radius =\( \Large \frac{Diameter}{2}=\frac{18}{2} \)=9 cm.

Area of the circle = \( \Large \pi r^{2}=\frac{22}{7}\times 9^{2} \)

= \( \Large \frac{22\times 81}{7}=\frac{1782}{7} \)=254.57 sq cm.

=255 sq cm.

68). If area of a square is 64 sq cm, then find the area of the circle formed by the same perimeter.

A). \( \Large \frac{215}{\pi} \)sq cm

B). \( \Large \frac{216}{\pi} \)sq cm

C). \( \Large \frac{256}{\pi} \)sq cm

D). \( \Large \frac{318}{\pi} \)sq cm

View Answer

Correct Answer: \( \Large \frac{256}{\pi} \)sq cm

Area of square = 64 sq cm

\( \Large (Side)^{2} \)= 64

Side = \( \Large \sqrt{64} \) = 8 cm

According to the question,

=> \( \Large 2\pi r=4\times 8 \) => r=\( \Large \frac{4\times 8}{2\pi}=\frac{16}{\pi} \)

Area of the circle

=\( \Large \pi \times 16/ \pi\times 16/ \pi=\frac{256}{\pi} \)sq cm.

69). If area of a square is 64 sq cm, then find the area of the circle formed by the same perimeter.

A). \( \Large 25 cm^{2} \)

B). \( \Large 1.25 cm^{2} \)

C). \( \Large 12 cm^{2} \)

D). None of the above

View Answer

Correct Answer: \( \Large 12 cm^{2} \)

ABCD be the rectangle inscribed in the circle of diameter 5 cm.

Diameter = Diagonal of rectangle

Now, let x and y be the lengths and breadths of rectangle, respectively.

Now, in \( \Large \triangle ABD \)

\( \Large AB^{2}+AD^{2}=(5)^{2} \)

=> \( \Large x^{2}+y^{2} \)=25

Since, they form Pythagoras theorem.

So,x=4 and y=3

Area of rectangle = \( \Large 3\times 4=12 cm^{2} \)

70). The perimeter of a square is twice the perimeter of a rectangle. If the perimeter of the square is 72 cm and the length of the rectangle is 12 cm. what is the difference between the breadth of the rectangle and the Side of the square?

A). 9 cm

B). 12 cm

C). 18 cm

D). 3 cm

View Answer

Correct Answer: 12 cm

Perimeter of square = 72 cm

Perimeter of rectangle

=\( \Large \frac{72}{2} \)=36 cm

=> \( \Large 2\times (l+b) \)=36

[l and b are dimension of rectangle]

\( \Large 2\times (12+b) \)=36

=> 12 + b = 18

b=18-12=6cm

And side of square = \( \Large \frac{72}{4} \)=18

Difference between breadth of rectangle and side of the square

= 18 - 6 = 12 cm.

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