Derivation of Surface Area of a Cuboid
The lateral and total surface areas of a cuboid are the two types of surface areas of a cuboid. The lateral surface area of a cuboid is the area occupied by its lateral faces, whereas the total surface area of a cuboid is the area occupied by all its six rectangular faces.
Total Surface Area of Cuboid
The total surface area of a cuboid can be calculated by first calculating the area of all the sides and then adding all six sides. Let us consider that “l” is the length of the cuboid, “b” is the breadth of the cuboid, and “h” is the height of the cuboid.
We know that in a cuboid, opposite faces are equal, i.e., ABCD = PQRS, APSD = BQRC, and ABQP = DCRS. Now, the total surface area of the cuboid is equal to the sum of the areas of its six rectangular faces. The total surface area of the cuboid = Area of [ABCD + APSD + ABQP + PQRS + BQRS + DCRS].
Area of [ABCD +APSD + ABQP + ABCD +APSD + ABQP] = 2 Area of [ABCD +APSD + ABQP]
- The area of the rectangle ABCD = l × h
- The area of the rectangle APSD = b × h
- The area of the rectangle ABQP = l × b
Therefore,
Total surface area of the cuboid (TSA) = 2 (lb + bh + lh) square units
Lateral Surface Area of a Cuboid
The lateral surface area of a cuboid is the area occupied by its lateral faces. Hence, the lateral surface area of the cuboid = The total surface area of the cuboid – The area of the top and bottom faces of the cuboid
= 2 (lb + bh + lh) – Area of [ABPQ + CDRS}
= 2 (lb + bh + lh) – 2 Area of (ABPQ)
= 2 (lb + bh + lh) – 2lb
= 2 (bh + lh) = 2h(l + b)
Hence,
Lateral surface area of a cuboid (LSA) = 2h(l + b) square units
Surface Area of Cuboid
The surface area of a cuboid is the total space occupied by all its surfaces/sides. In geometry, a three-dimensional shape having six rectangular faces is called a cuboid. A cuboid is also known as a regular hexahedron and has six rectangular faces, eight vertices, and twelve edges with congruent, opposite faces. It is a three-dimensional form of a rectangle with four lateral faces and two faces at the top and bottom. Some examples of cuboids that we regularly see are bricks, geometric boxes, shoe boxes, packaging boxes, etc. Let’s learn in detail about the surface area of a cuboid, along with examples.
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