# Surface Areas And Volumes: Class 10 Mathematics NCERT Chapter 13

**Key Features of NCERT Material for Class 10 Mathematics Chapter 13 – Surface Areas and Volumes **

In the last chapter 12, you learned about Areas Related to Circles. In this chapter, Surface Areas and Volumes, you will learn everything about it.

**Cuboid and its Surface Area **

The surface area of a cuboid is equivalent to the total of the areas of its six rectangular faces. Consider a cuboid whose measurements are l × b × h individually.

Cuboid that has length l, breadth b and height h

The absolute surface area of the cuboid (TSA) = Sum of the areas of all its six faces

TSA (cuboid) = 2(l × b) + 2(b × h) + 2(l × h) = 2(lb + bh + lh)

Horizontal surface area (LSA) is the area of the apparent multitude of sides separated from the top and base faces.

The parallel surface area of the cuboid = Area of face AEHD + Area of face BFGC + Area of face ABFE + Area of face DHGC

LSA (cuboid) = 2(b × h) + 2(l × h) = 2h(l + b)

Length of side of a cuboid =√(l2 + b2 + h2)

Right Circular Cone and its Surface Area

Think about a right angular cone with incline length l, radius r and height h.

Cone with the base radius r and height h

CSA of right round cone = πrl

TSA is equal to CSA + area of base = πrl + πr2 = πr(l + r)

Sphere and its Surface Area

For a sphere of radius r

Curved Surface Area (CSA) = complete Surface Area (TSA) = 4πr2

Sphere with radius r

SURFACE AREA AND VOLUME OF COMBINATIONS

Cone over a Cylinder.

r = radius of cylinder and cone;

h1 = height of the cone

h2 = height of cylinder

All out Surface area is equal to Curved surface area of cone + Curved surface area of cylinder + area of round base

= πrl + 2πrh2 +πr2;

Inclination height,

Total Volume is equal to Volume of cone + Volume of cylinder

**Cone over a Hemisphere: **

h = height of cone;

l = incline height of cone =

r = radius of cone and hemisphere

Total Surface area is equal to Curved surface area of cone + Curved surface area of hemisphere = πrl + 2πr2

Volume is equal to Volume of cone + Volume of hemisphere =

Funnel shaped Cavity in Cylinder

r = radius of cone and cylinder;

h = height of cylinder and funnel shaped cavity;

l = Slant height

Total Surface area is equal to Curved surface area of cylinder + Area of base face of cylinder + Curved surface area of cone = 2πrh + πr2 + πrl

Volume is equal to Volume of cylinder – Volume of cone

**Cones on Either Side of Cylinder. **

r = radius of cylinder and cone;

h1 represents height of cylinder

h2 represents height of cones

Inclination height of cone,

Surface area is equal to Curved surface area of 2 cones + Curved surface area of cylinder = 2πrl + 2πrh1

Volume = 2(Volume of cone) + Volume of cylinder

**Cylinder with Hemispherical Ends. **

r = radius of cylinder and hemispherical closures;

h = height of cylinder

Total surface area is equal to Curved surface area of cylinder + Curved surface area of 2 hemispheres = 2πrh + 4πr2

Volume is equal to Volume of cylinder + Volume of 2 hemispheres

**Hemisphere on Cube or Hemispherical Cavity on a Cube **

a = side of cube;

r = radius of hemisphere.

Surface area is equal to Surface area of a cube– Area of hemisphere face + Curved surface area of hemisphere

= 6a2 – πr2 + 2πr2 = 6a2 + πr2

Volume = Volume of a cube + Volume of hemisphere

**Hemispherical Cavity in Cylinder **

r = radius of hemisphere;

h = height of cylinder

Complete surface area = Curved surface area of cylinder + Surface area of base + Curved surface area of hemisphere

= 2πrh + πr2 + 2πr2 = 2πrh + 3πr2

Total Volume is equal to Volume of cylinder – Volume of hemisphere =