NCERT Solutions for Class 10 Maths Chapter 12 Surface Areas and Volumes Exercise 12.1#
Exercise 12.1 covers the surface area of combinations of solids — cubes, cuboids, cylinders, cones, hemispheres. The total surface area excludes the joined faces.
Key Formulas:
- Cylinder: Curved SA = $2\pi rh$, Total SA = $2\pi r(r+h)$
- Cone: Curved SA = $\pi rl$, Total SA = $\pi r(r+l)$, where $l = \sqrt{r^2+h^2}$
- Sphere: SA = $4\pi r^2$
- Hemisphere: Curved SA = $2\pi r^2$, Total SA = $3\pi r^2$
- Cube: SA = $6a^2$; Cuboid: SA = $2(lb+bh+hl)$
1. 2 cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid. #
Answer
Let the side of each cube be $a$ cm. The volume of each cube is 64 cm³, so:
$$a^3 = 64$$$$\Rightarrow a = 4$$
Joined end to end, the cubes form a cuboid with:
$l = 4 + 4 = 8$ cm
$b = 4$ cm
$h = 4$ cm
Using SA of cuboid $= 2(lb + bh + hl)$, we find the surface area:
$$= 2(8 \times 4 + 4 \times 4 + 4 \times 8)$$$$= 2(32 + 16 + 32)$$
$$= 160$$
∴ The surface area of the cuboid is 160 cm².
2. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel. #
Answer
Radius of hemisphere = radius of cylinder = $\dfrac{14}{2} = 7$ cm
Height of cylinder = 13 − 7 = 6 cm
Inner surface area = Curved SA of cylinder + Curved SA of hemisphere
So we find the inner surface area:
$$= 2\pi rh + 2\pi r^2$$$$= 2\pi r(h + r)$$
$$= 2 \times \frac{22}{7} \times 7 \times (6 + 7)$$
$$= 44 \times 13$$
$$= 572$$
∴ The inner surface area of the vessel is 572 cm².
3. A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy. #
Answer
Radius $r$ = 3.5 cm
Height of cone $h$ = 15.5 − 3.5 = 12 cm
Using $l = \sqrt{r^2 + h^2}$, we find the slant height of the cone:
$$= \sqrt{3.5^2 + 12^2}$$$$= \sqrt{12.25 + 144}$$
$$= \sqrt{156.25}$$
$$= 12.5$$
Total surface area = Curved SA of cone + Curved SA of hemisphere
So we find the total surface area:
$$= \pi rl + 2\pi r^2$$$$= \pi r(l + 2r)$$
$$= \frac{22}{7} \times 3.5 \times (12.5 + 7)$$
$$= 11 \times 19.5$$
$$= 214.5$$
∴ The total surface area of the toy is 214.5 cm².
4. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid. #
Answer
The hemisphere sits on a face of side 7 cm, so its greatest diameter is 7 cm, and its radius is $r = 3.5$ cm.
Surface area of the solid = SA of cube − Area of base of hemisphere + Curved SA of hemisphere
So we find the surface area:
$$= 6 \times 7^2 - \pi r^2 + 2\pi r^2$$$$= 294 + \pi r^2$$
$$= 294 + \frac{22}{7} \times 3.5^2$$
$$= 294 + 38.5$$
$$= 332.5$$
∴ The greatest diameter is 7 cm and the surface area of the solid is 332.5 cm².
5. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid. #
Answer
Edge of cube = $l$
Radius of hemisphere = $\dfrac{l}{2}$
Surface area of the remaining solid = SA of cube − Area of base of hemisphere + Curved SA of hemisphere
So we find the surface area:
$$= 6l^2 - \pi\left(\frac{l}{2}\right)^2 + 2\pi\left(\frac{l}{2}\right)^2$$$$= 6l^2 + \frac{\pi l^2}{4}$$
$$= \frac{l^2}{4}(24 + \pi)$$
∴ The surface area of the remaining solid is $\mathbf{\dfrac{l^2}{4}(\pi + 24)}$ square units.
6. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area. #
Answer
Radius $r = \dfrac{5}{2} = 2.5$ mm
Length of cylinder $h$ = 14 − 2 × 2.5 = 9 mm
Surface area = Curved SA of cylinder + 2 × Curved SA of hemisphere
So we find the surface area:
$$= 2\pi rh + 2 \times 2\pi r^2$$$$= 2\pi r(h + 2r)$$
$$= 2 \times \frac{22}{7} \times 2.5 \times (9 + 5)$$
$$= 220$$
∴ The surface area of the capsule is 220 mm².
7. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of ₹500 per m². (Note that the base of the tent will not be covered with canvas.) #
Answer
Radius $r = \dfrac{4}{2} = 2$ m
Height of cylinder $h$ = 2.1 m
Slant height of cone $l$ = 2.8 m
Area of canvas = Curved SA of cylinder + Curved SA of cone
So we find the area of canvas:
$$= 2\pi rh + \pi rl$$$$= \pi r(2h + l)$$
$$= \frac{22}{7} \times 2 \times (4.2 + 2.8)$$
$$= 44$$
Cost of canvas = $44 \times 500$
Cost of canvas = ₹22000
∴ The area of canvas is 44 m² and its cost is ₹22000.
8. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm². #
Answer
Radius $r = \dfrac{1.4}{2} = 0.7$ cm
Height $h$ = 2.4 cm
Using $l = \sqrt{r^2 + h^2}$, we find the slant height of the cone:
$$= \sqrt{0.49 + 5.76}$$$$= \sqrt{6.25}$$
$$= 2.5$$
The cone is hollowed out from one end, so the remaining solid has the curved surface of the cylinder, the circular base at the other end, and the curved surface of the cone inside.
Total surface area = Curved SA of cylinder + Area of base + Curved SA of cone
So we find the total surface area:
$$= 2\pi rh + \pi r^2 + \pi rl$$$$= \pi r(2h + r + l)$$
$$= \frac{22}{7} \times 0.7 \times (4.8 + 0.7 + 2.5)$$
$$= 2.2 \times 8$$
$$= 17.6$$
∴ The total surface area of the remaining solid is 17.6 cm², that is 18 cm² to the nearest cm².
9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in figure. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article. #
Answer
Radius $r$ = 3.5 cm
Height of cylinder $h$ = 10 cm
Total surface area = Curved SA of cylinder + 2 × Curved SA of hemisphere
So we find the total surface area:
$$= 2\pi rh + 2 \times 2\pi r^2$$$$= 2\pi r(h + 2r)$$
$$= 2 \times \frac{22}{7} \times 3.5 \times (10 + 7)$$
$$= 22 \times 17$$
$$= 374$$
∴ The total surface area of the article is 374 cm².