## Introduction: Surface areas and volume

#### Surface area and volume of a cuboid-

Cuboid with length *l*, breadth b and height h

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

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

Lateral surface area (LSA) is the area of all the sides apart from the top and bottom faces.

LSA = Area of face AEHD + Area of face BFGC + Area of face ABFE + Area of face DHGC

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

Length of diagonal of a cuboid =√(l^{2} + b^{2} + h^{2})

The volume of a cuboid is the space occupied within its six rectangular faces.

**Vol. of a cuboid =** **(base area) × height = (lb)h = lbh**

#### Surface area and volume of a cube-

For a cube, length = breadth = height

TSA =2 × (3l^{2}) = 6l^{2}

LSA= 2(l × l + l × l) = 4l^{2}

Vol.of a cube = base area × height vol. of a cube = l^{3}

#### Surface area and volume of a cylinder-

CSA of a cylinder = 2π × r × h

TSA of a cylinder= 2π × r × h + area of two circular bases =>2π × r × h + 2πr^{2} =>2πr(h + r)

Vol. of a cylinder = Base area × height = (πr^{2}) × h = πr^{2}h

#### Surface area and volume of the right circular cone-

CSA = πrl

TSA = CSA + area of base = πrl + πr^{2} = πr(l + r)

Vol. of a Right circular cone =(1/3)πr^{2}h

Surface area and volume of a sphere-

CSA= TSA = 4πr^{2}

vol. of a sphere = (4/3)πr^{3}

#### surface area and volume of a hemisphere-

CSA = 2πr^{2}

Total Surface Area = curved surface area + area of the base circle

TSA = 3πr^{2}

Vol. of the hemisphere= (2/3)πr^{3}

#### Surface area and volume of a frustrum of a cone-

When a solid is cut by a plane, then another form of solids is formed. One such form of solid is the frustum of a cone, which is formed when a plane cuts a cone parallelly to the base of the cone. If a right circular cone is sliced by a plane parallel to its base, then the part with the two circular bases is called a Frustum.

CSA =π(r_{1}+r_{2})l, where l= √[h^{2}+(r_{2 }– r_{1})^{2}]

TSA = CSA + the areas of the two circular faces = π(r_{1 }+ r_{2})l + π(r_{1}^{2} + r_{2}^{2})

Vol. of a frustum of a cone =(1/3)πh(r_{1}^{2} + r_{2}^{2 }+ r_{1}r_{2})

#### Surface area of the combination of solids-

**TSA of new solid = CSA of one hemisphere + CSA of cylinder+ CSA of other hemisphere**

#### 2D shapes important formula-