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Surface Areas and Volumes

The surface areas and volumes are the parameters that are present in 3D geometric figures only. The surface area of any shape is the area or region covered by its surface (flat or curved) in the 3D space. The volume is the value of space present inside the solid. In geometry, several distinct figures exist; some are regularly shaped while others have irregular symmetry and design. We are familiar with the cube, cuboid, cone, cylinder, sphere, hemisphere, and trapezoid shapes. These are well- defined shapes that have their own formulas for the surface area as well as volume. These shapes are present in day-to-day life like cylindrical boxes, cuboidal cupboards, ice cream cones, Rubik’s cubes, hemispheric bowls, spherical football, and more. Let us learn about each parameter one at a time.

Surface Area

As defined above, the area is the space swept or occupied by a two-dimensional flat surface, and the surface area of a 3D figure refers to the area filled by a three-dimensional object's exterior surface. Both the 2D and 3D surface areas are always expressed in square units. Generally, the 3D surface is of two types: the total surface area (TSA) and the curved surface area (CSA) (in the case of round solids) or the lateral surface area (LSA) (in the case of flat solids). Let us now define the TSA and CSA/LSA.

Total Surface Area

Total surface area refers to the area of the entire shape. This includes the area of all the faces, including the roof and base in case of flat solids and the curved surface area, plus the area of all the bases present in the curved solids. It is the sum of all the areas occupied by the surface of the solid. If the shape has a round surface and a flat surface, then the total area is the sum of both areas individually.

Curved Surface Area/Lateral Surface Area

Curved surface area or CSA infers only the curved part (usually concave) of a curved solid like the cylinder and sphere and does not include the flat part. The lateral surface area or the LSA is the area of all the faces of a regular solid like cuboid and cube, excluding the bottom and top faces.

Volume

Volume is expressed in cubic units (cm or m) and is the value of space within the solid. Experimentally, we can say that volume is none other than the liters of fluids (water or air) an object can hold inside it. Only 3D objects have volume, and 2D figures are volume-less shapes. A circle cannot hold any amount of water in it, while a sphere can store liters of water in it.

Some common surface area and volume are listed in the table below:

 Shape TSA CSA/LSA Volume Cuboid 2(lw + wd + ld) 2d (l + w) l x w x d Cube 6s2 4s2 s3 Cylinder 2 π k (k + d) 2π k d π k2 d Cone π k (k +c) π k c 1/3π k2 d Sphere 4 π k2 4 π k2 4/3π k3 Hemisphere 3 π k2 2 π k2 2/3π k3

In the table:

l = length of the cuboid

w = width of the cuboid

d = height of each 3d figure

s = side of a cube

k = radius of each round solid

c = slant height of a cone

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