NCERT Solutions for Class 10 Maths Exercise 12.1 Chapter 12 Surface Areas and Volumes
1. 2 cubes each of volume 64 cm3 are joined end to end. Find the surface area of the resulting cuboid.
Step 1: Calculate the side of each cube.
Volume of cube = 64 cm³
Side = ∛64 = 4 cm
Step 2: Determine the dimensions of the resulting cuboid.
Length = 4 cm + 4 cm = 8 cm
Breadth = 4 cm
Height = 4 cm
Step 3: Calculate the surface area of the cuboid.
Surface area = 2(lb + bh + hl)
= 2(8×4 + 4×4 + 4×8)
= 2(32 + 16 + 32)
= 2 × 80
= 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.
Step 1: Find the radius of the hemisphere.
Radius = Diameter / 2 = 14 cm / 2 = 7 cm
Step 2: Calculate the height of the cylinder.
Total height = Height of cylinder + Radius of hemisphere
13 cm = Height of cylinder + 7 cm
Height of cylinder = 6 cm
Step 3: Calculate the inner surface area of the vessel.
Inner surface area = Surface area of hemisphere + Curved surface area of cylinder
= 2πr² + 2πrh
= 2π(7)² + 2π(7)(6)
= 2π(49 + 42)
= 182π
= 182 × (22/7)
= 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.
Step 1: Calculate the height of the cone.
Total height = Height of cone + Radius of hemisphere
15.5 cm = Height of cone + 3.5 cm
Height of cone = 12 cm
Step 2: Calculate the slant height of the cone.
l = √(radius² + height²)
= √(3.5² + 12²)
= √(12.25 + 144)
= √156.25
= 12.5 cm
Step 3: Calculate the total surface area of the toy.
Total surface area = Curved surface area of cone + Curved surface area of hemisphere
= πrl + 2πr²
= π(3.5)(12.5) + 2π(3.5)²
= 43.75π + 24.5π
= 68.25π
= 68.25 × (22/7)
≈ 213.5 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.
Step 1: Determine the greatest diameter of the hemisphere.
Greatest diameter = Side of cube = 7 cm
Radius = 3.5 cm
Step 2: Calculate the surface area of the solid.
Surface area = Surface area of cube + Curved surface area of hemisphere – Area of base of hemisphere
= 6 × side² + 2πr² – πr²
= 6(7)² + 2π(3.5)² – π(3.5)²
= 294 + 24.5π – 12.25π
= 294 + 12.25π
= 294 + 12.25 × (22/7)
≈ 431.5 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.
Step 1: Let the side of the cube (also the diameter of the hemisphere) be l cm.
Step 2: Calculate the surface area of the remaining solid.
Surface area = Surface area of cube – Area of face with depression + Curved surface area of hemisphere
= 6l² – l² + 2π(r)²
= 5l² + 2π(l/2)²
= 5l² + π(l²/4)
= 5l² + πl²/4
Step 3: Substitute the value of π.
Surface area = 5l² + (22/7) × l²/4
= 5l² + 22l²/28
= 5l² + 11l²/14
= (70l² + 11l²) / 14
= 81l² / 14
The surface area of the remaining solid is 81l² / 14 cm².
6. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig. 12.10). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.
Step 1: Calculate the radius of the hemisphere and the cylinder.
Radius (r) = Diameter / 2 = 5 mm / 2 = 2.5 mm
Step 2: Determine the height of the cylindrical part.
Total length of capsule = Length of cylinder + Diameter of hemisphere + Diameter of hemisphere
14 mm = Height of cylinder + 2.5 mm + 2.5 mm
Height of cylinder (h) = 14 mm – 5 mm = 9 mm
Step 3: Calculate the curved surface area of the cylinder.
Curved surface area of cylinder = 2πrh
= 2 × (22/7) × 2.5 mm × 9 mm
= 22 × 2.5 × 9 / 7
= 22 × 2.5 × 1.2857
≈ 70.7143 mm²
Step 4: Calculate the total surface area of the two hemispheres.
Surface area of one hemisphere = 2πr²
Surface area of two hemispheres = 2 × 2πr²
= 4πr²
= 4 × (22/7) × (2.5 mm)²
= 4 × (22/7) × 6.25 mm²
= 4 × 22 × 0.8929 mm²
≈ 78.5714 mm²
Step 5: Calculate the total surface area of the capsule.
Total surface area = Curved surface area of cylinder + Surface area of two hemispheres
≈ 70.7143 mm² + 78.5714 mm²
≈ 149.2857 mm²
The surface area of the medicine capsule is approximately 149.29 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.)
Step 1: Calculate the radius of the cylinder and cone (which are the same).
Radius (r) = Diameter / 2 = 4 m / 2 = 2 m
Step 2: Calculate the curved surface area of the cylindrical part.
Curved surface area of cylinder = 2πrh
= 2 × (22/7) × 2 m × 2.1 m
= 4 × 22 × 0.3 m²
= 26.4 m²
Step 3: Calculate the surface area of the conical part.
Surface area of cone = πrl (l is the slant height)
= (22/7) × 2 m × 2.8 m
= 22 × 0.4 m²
= 17.6 m²
Step 4: Calculate the total area of the canvas used.
Total area of canvas = Curved surface area of cylinder + Surface area of cone
= 26.4 m² + 17.6 m²
= 44 m²
Step 5: Find the cost of the canvas.
Cost of canvas = Area of canvas × Cost per m²
= 44 m² × ₹ 500/m²
= ₹ 22000
The area of the canvas used for making the tent is 44 m² and the cost of the canvas 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.
Step 1: Calculate the radius of the cylinder and cone.
Radius (r) = Diameter / 2 = 1.4 cm / 2 = 0.7 cm
Step 2: Calculate the curved surface area of the cylinder.
Curved surface area of cylinder = 2πrh
= 2 × (22/7) × 0.7 cm × 2.4 cm
= 22 × 0.2 cm²
= 4.4 cm²
Step 3: Calculate the surface area of the cone.
Surface area of cone = πrl (l is the slant height)
Since the cone is hollowed out, we don’t include its area.
Step 4: Calculate the surface area of the remaining solid.
Total surface area = Curved surface area of cylinder + Area of cylinder top – Area of cone base
= 4.4 cm² + πr² – πr²
= 4.4 cm² + (22/7) × (0.7 cm)² – (22/7) × (0.7 cm)²
= 4.4 cm² (since the areas of the cylinder top and cone base cancel each other out)
Rounded to the nearest cm², the total surface area of the remaining solid is approximately 4 cm².
9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 12.11. 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.
Step 1: Calculate the surface area of the cylindrical part.
The radius (r) of the base is 3.5 cm, and the height (h) is 10 cm.
Curved surface area of the cylinder = 2πrh
= 2 × (22/7) × 3.5 cm × 10 cm
= 220 cm²
Step 2: Calculate the total surface area of the two hemispheres that were scooped out.
Since the hemispheres are scooped out, only their outer surface contributes to the total surface area of the article.
Surface area of one hemisphere = 2πr²
Surface area of two hemispheres = 2 × 2πr²
= 4πr²
= 4 × (22/7) × (3.5 cm)²
= 4 × 22 × 0.5 cm²
= 44 cm²
Step 3: Find the total surface area of the article.
Total surface area of the article = Curved surface area of the cylinder + Surface area of two hemispheres
= 220 cm² + 44 cm²
= 264 cm²
The total surface area of the wooden article is 264 cm².