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 =√(l² + b² + h²)

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 × (3l²) = 6l²
LSA= 2(l × l + l × l) = 4l²
Vol.of a cube = base area × height                                     vol. of a cube = l³

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 + 2πr² =>2πr(h + r)
Vol. of a cylinder = Base area × height = (πr²) × h = πr²h

Surface area and volume of the right circular cone-

CSA  = πrl
TSA = CSA + area of base = πrl + πr² = πr(l + r)
Vol. of a Right circular cone =(1/3)πr²h

Surface area and volume of a sphere-

CSA= TSA = 4πr²
vol. of a sphere = (4/3)πr³

surface area and volume of a hemisphere-

CSA  = 2πr²
Total Surface Area = curved surface area + area of the base circle
TSA = 3πr²
Vol. of the hemisphere= (2/3)πr³

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(r₁+r₂)l,  where l= √[h²+(r₂ – r₁)²]
TSA = CSA + the areas of the two circular faces = π(r₁ + r₂)l + π(r₁² + r₂²)
Vol. of a frustum of a cone =(1/3)πh(r₁² + r₂² + r₁r₂)

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-