Mensuration

Q.1 (A)(iv) Radius of a sector of a circle is 35 cm and length of its arc is 22 cm. What is the area of the sector ?

Q.3 (B)(iv) In the following figure if \(A(\text{P -- ABC}) = 154\) cm², radius of the circle is 14 cm, find : (a) \(m\angle APC\), (b) \(l(\text{arc } ABC)\).

Q.4 (iii) Find the number of coins of 2.2 cm diameter and 0.2 cm thick, can be made by melting a metallic right circular cylinder of height 0.2 m and diameter 8.8 cm.

Q.1 (A)(4) If the side of the cube is 5 cm, then the volume of that cube is ..................................... .

Q.2 (A)(3) In the following figure, side of square ABCD is 7 cm. With centre D and radius DA, sector D-AXC is drawn. Fill in the following boxes properly and find out the area of shaded region :

Solution :

Area of square = (side)²
= (7)²
∴ Area of square = ____ cm²
If area of sector (D-AXC) = 38.5 cm², then
A (shaded region) = A ____ – A ____
= 49 cm² – 38.5 cm²
= ____ cm²

Q.3 (B)(4) A bucket is frustum shaped. Its height is 28 cm. Radii of circular faces are 12 cm and 15 cm. Find the capacity of the bucket in litres. \(\left(\pi = \dfrac{22}{7}\right)\)

Q.4 (2) Surface area of sphere and total surface area of cube are same. Show that the ratio of the volume of the sphere to that of the cube is \(\sqrt{6} : \sqrt{\pi}\).

Q.1 (A)(4) If radius of cone is 5 cm and its perpendicular height is 12 cm, then the slant height is ..................... .

Q.2 (A)(1) The radius of a circle with centre ‘P’ is 10 cm. If chord AB of the circle subtends a right angle at P, find area of minor sector by using the following activity. (π = 3.14)

Activity :

r = 10 cm, θ = 90º, π = 3.14.
A(P–AXB) = \(\dfrac{\theta}{360}\) × ____
= \(\dfrac{\text{____}}{360}\) × 3.14 × 10²
= \(\dfrac{1}{4}\) × ____
A(P–AXB) = ____ sq. cm.

Q.3 (B)(4) How many solid cylinders of radius 10 cm and height 6 cm can be made by melting a solid sphere of radius 30 cm ?

Q.4 (3) The radii of the circular ends of a frustum of a cone are 14 cm and 8 cm. If the height of the frustum is 8 cm, find : (π = 3.14)
(i) Curved surface area of frustum.
(ii) Total surface area of the frustum.
(iii) Volume of the frustum.

Q.1 (A)(4) The volume of a cube of side 2 cm is ....................

Q.2 (A)(3) Measure of an arc of a circle is 90º and its radius is 14 cm. Complete the following activity to find the length of an arc.

Activity :

Length of an arc \(= \dfrac{\theta}{360} \times ____\) ......... (Formula)

\(= \dfrac{90}{360} \times 2 \times \dfrac{22}{7} \times ____\)

\(= \dfrac{1}{4} \times ____\)

Length of an arc \(= ____\) cm

Q.2 (B)(2) Find surface area of sphere of radius 7 cm.

Q.2 (B)(5) Find the volume of a cone if radius of its base is 7 cm and its perpendicular height is 15 cm.

Q.5 (2) Observe the given figure and answer the following questions :

(i) How many surfaces does a solid cone have ?
(ii) What are the names of slant height and perpendicular height in the given figure ?
(iii) If slant height of solid cone is 10 cm and perpendicular height is 8 cm, then find diameter of base of solid cone ?

Q.1 (A)(4) The volume of a cube of side 10 cm is ..............................

Q.1 (B)(4) If radius of a circle is 4 cm and length of an arc is 10 cm, then find the area of the sector.

Q.2 (A)(3) Complete the following activity to find the volume of a cone if the radius of its base is 7 cm and its perpendicular height is 6 cm.

Activity :

Here, radius (r) = 7 cm, height (h) = 6 cm

\(\therefore\) Volume of cone = ____ ....... (Formula)
\(= \dfrac{1}{3}\times\dfrac{22}{7}\times\) ____ \(\times 6\)
\(= \dfrac{1}{3}\times\dfrac{22}{7}\times\) ____ \(\times 6\)
\(\therefore\) Volume of cone = ____ cm³

Q.2 (B)(3) Find the surface area of a sphere of diameter 14 cm.

Q.3 (A)(2) A cylinder is having radius 7 cm and height 21 cm. Complete the following activity to calculate its volume and circumference of its base.

Activity :

Radius (r) = 7 cm, Height (h) = 21 cm

Volume of cylinder = ____ ........... (Formula)
\(= \dfrac{22}{7} \times 7 \times 7 \times\) ____
\(= 154 \times 21\)
\(=\) ____ cm³

Circumference of base = ____ ........ (Formula)
\(= 2 \times \dfrac{22}{7} \times\) ____
\(=\) ____ cm

Q.2 (A)(3)

In the figure given above, □ ABCD is a square and a circle is inscribed in it. All sides of a square touch the circle.

If AB = 14 cm, find the area of shaded region.

Solution :

Area of square = (____)² ............ (Formula)

= 14²

= ____ cm²

Area of circle = ____ ............ (Formula)

= \(\dfrac{22}{7} \times 7 \times 7\)

= 154 cm²

(Area of shaded portion) = (Area of square) – (Area of circle)

= 196 – 154

= ____ cm²

Q.2 (B)(1) Radius of a sector of a circle is 3.5 cm and length of its arc is 2.2 cm. Find the area of the sector.

Q.2 (B)(5) Find the surface area of a sphere of radius 7 cm.

Q.4 (2) A cylinder of radius 12 cm contains water up to the height 20 cm. A spherical iron ball is dropped into the cylinder and thus water level raised by 6.75 cm. What is the radius of iron ball ?