13. SURFACE AREAS AND VOLUMES

CBSE Class 10 Chapter 13 SURFACE AREAS AND VOLUMES (NCERT)

Chapter 13. SURFACE AREAS AND VOLUMES

Chapter 13. Surface area and Volumes

Exercise 13.1 complete solution

Exercise 13.2 complete solution

1. Surface area of Cube , Cuboid , Cylinder , Cone , Sphere and Hemisphere :

(i) The Surface Area of a Cube

(ii) The lateral surface area of a cube

(iii) The surface area of a cuboid

(iv) The lateral surface area of a cuboid

(v)Curved surface area of a cylinder

(vi) Total surface area of a cylinder

(vii) Curved surface area of a cone

(viii) Total surface area of a cone

(ix) Surface Area of a sphere

(x) Curved surface area of a Hemisphere

(xi) Total surface area of a hemisphere

2. The volume of Cube , Cuboid , Cylinder , Cone , Sphere and Hemisphere :

(i) The volume of cube

(ii) The volume of a cuboid

(iii) The volume of a cylinder

(iv) The volume of cone

(v) The volume of a sphere

(vi) The volume of hemisphere

Class 10 Maths Chapter 13. SURFACE AREAS AND VOLUMES Exercise 13.1

Unless stated otherwise, taken

1. 2 cubes each of volumes 64

are joined end to end . Find the surface area of the resulting cuboid .

Solution: let be the length of the cube .

A/Q,

For cuboid : Here ,

The surface area of cuboid

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 .

Solution: Here, Radius

The height of the cylinder

Area of the inner surface of the vessel

Area of cylinder + Area of hemisphere

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 .

Solution: For cone : Radius 3.5 cm cm ,

Height cm

The slant height

12.5 cm

The curve surface area of cone

For hemisphere : Radius

The curve surface area of hemisphere

The total surface area of the toy

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 .

Solution: The cubical block of side is 7 cm .

So, the greatest diameter of the hemisphere is 7 cm .

Here , Radius ,

The surface area of the solid

Area of cubical block + Area of hemisphere – Area of circular top

5. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter

of the hemisphere is equal to the edge of cube . Determine the surface area of the remaining solid .

Solution: Here, diameter and Radius

The surface area of the remaining solid

Area of cubical block + Area of hemisphere – Area of circular top

6. A medicine capsule is in the shape of a cylinder with two hemisphere stuck to each of its ends (see Fig 13.10) . The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm .Find its surface area .

Solution: Here, Diameter Radiusr

And height of cylinder

The surface area of the capsule

S.A. of cylinder + Area of 2 hemisphere

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 Rs 500 per

. (Note that the base of the tent will not be covered with canvas)

Solution: Here , Height , Diameter , Radius and the slant height

The area of the canvas of the tent

Area of the cylinder + C.S.A. of cone

The cost of the canvas of the tent .

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

Solution: Here , Diameter,

Radius , Height

The slant height

The total surface area of the remaining solid

C.S.A of cylinder + C.S.A. of cone + Area of circular top

[appro.]

9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 13.11 .

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 .

Solution: Here , Radius Height

The total surface area of the article

C.S.A. of cylinder + Area of 2 hemisphere

Class 10 Maths Chapter 13. SURFACE AREAS AND VOLUMES Exercise 13.2 Solutions

Unless stated otherwise , take

1. A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius . Find the volume of the solid in terms of

Solution: Given ,

The volume of solid The volume of cone + The volume of hemisphere

2. Rachel , an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet . The diameter of the model is 3 cm and its length is 12 cm . If each cone has a height of 2 cm , find the volume of air contained in the model that Rachel made . (Assume the outer and inner dimensions of the model to be nearly the same.)

Solution: Here, , ,

and

The volume of air contained in the model = The volume of the Cylinder the volume of the cone

3. A gulab jamun, contains sugar syrup up to about 30% of its volume . Find approximately how much syrup would be found in 45 gulab jamuns , each shaped liked a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm (see Fig. 13.15)

Solution:

Figure of gulab jamun :

Here, ,

and

The volume of a gulab jamuns = The volume of the hemisphere + The volume of the cylinder + The volume of the hemisphere

The volume of 45 gulap jamuns

The volume of gulap jamun contains in sugar syrup

4. A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens . The dimensions of the cuboid are 15cm by 10cm by 3.5 cm . The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm . Find the volume of wood in the entire stand (see Fig. 13.16) .

Solution: For cuboid :

Here , length cm , breadth cm and height cm

The volume of a cuboidal wood

For cone : Here , and

The volume of a cone

The volume of 4 cone

Therefore, the volume of wood in the entire stand

5. A vessel is in the form of an inverted cone . Its height is 8 cm and the radius of its top , which is open, is 5 cm . It is filled with water up to the brim . When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel , one-fourth of the water flows out . Find the number of lead shots dropped in the vessel .

Solution: For cone : Here, Height of the cone cm and Radius cm

The volume of the cone

For Sphere : Here, Radius cm

The volume of the sphere

The volume of the water that flows out of the cone

(the volume of the cone)

Therefore, the number of lead shots

6. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm , which is surmounted by another cylinder of height 60 cm and radius 8 cm . Find the mass of the pole , given that 1

of iron has approximately 8g mass . (Use

)

Solution : For big cylinder : Here ,

Diameter , Radius and Height

The volume of the big cylindrical pole

For small cylinder : Here ,

Radius , Height

The volume of the big cylindrical pole

The volume of solid pole

The mass of the pole

7. A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom . Find the volume of water left in the cylinder , if the radius of the cylinder is 60 cm and its height is 180 cm .

Solution: For cone and hemisphere : Here,

Radius and Height

The volume of solid

The volume of cone + The volume of hemisphere

For Cylinder :

Here , Radius and Height

The volume of Cylinder

The volume of water left in the cylinder

Volume of Cylinder – Volume of solid

8. A spherical glass vessel has a cylindrical neck 8 cm long , 2 cm in diameter ; the diameter of the spherical part is 8.5 cm . By measuring the amount of water it holds , a child find its volume to be 345

. Check whether she is correct , taking the above as the inside measurements, and

Solution: For cylindrical neck :

Here ,

The volume of cylindrical neck

For spherical part :

Here ,

The volume of spherical part

The volume of spherical glass vessel

She is not correct . The correct answer is .