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Surface Area of Cuboid (Definition, Formula & Examples)

surface area of cuboid

The surface area of a 3D shape (solid object) is a measure of the total area that the surface of the object occupies. A cuboid is a 3D solid shape with six rectangular faces. The total surface area of a cuboid can be calculated if we calculate the area of the two bases and the area of the four lateral (side) faces. The change in any of the dimensions of a cuboid changes the value of the surface area of a cuboid.

Let’s learn how to find the surface area of cuboid and its uses.

Cuboid – A 3D Solid Shape

A cuboid is a solid shape or a three-dimensional shape. A convex polyhedron that is bounded by six rectangular faces with eight vertices and twelve edges is called a cuboid. A cuboid is also called a rectangular prism. A cuboid with six square faces is called a cube. An example of a cuboid in real life is a matchbox.

surface area of cuboid

A cuboid has the following properties:

  • A cuboid has $12$ edges, $6$ faces, and $8$ vertices.
  • All the faces are shaped as a rectangle hence the length, width(or breadth), and height are different.
  • The angles between any two faces or surfaces are $90^{\circ}$.
  • The opposite planes or faces in a cuboid are parallel to each other.
  • The opposite edges in a cuboid are parallel to each other.
  • Each of the faces in a cuboid meets the other four faces.
  • Each of the vertices in a cuboid meets the three faces and three edges.

Difference Between Cube and Cuboid

Although cube and cuboid are similar $3D$ objects, there are few differences between these two. Following are the differences between a cube and a cuboid.

surface area of cuboid
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What is the Surface Area of a Cuboid?

The surface area of a cuboid is the sum of the area of the bases($2$ bases) and the area of lateral faces($4$ lateral faces) of the cuboid. Since the faces of the cuboid are made up of $6$ rectangles therefore the total surface area of the cuboid will be numerically equal to the sum of the areas of these six rectangles. 

The surface area of a cuboid is measured as the “number of square units” ($cm^{2}$, $m^{2}$, $in^{2}$, $ft^{2}$, etc.). There are two types of surface areas of a cuboid

  • Lateral Surface Area
  • Total Surface Area

Lateral Surface Area of a Cuboid

The lateral surface area of a cuboid refers to the total area covered by the side or lateral faces of a cuboid. There are four lateral faces in a cuboid, so to calculate LSA, we find the sum of the areas of these $4$ rectangular faces.

surface area of cuboid

If the length, width(breadth), and height of a cuboid are $l$, $w$, and $h$ respectively, then the areas of the $4$ lateral faces are

  • Left face: $w \times h = wh$
  • Right face: $w \times h = wh$
  • Front face: $l \times h = lh$
  • Back face: $l \times h = lh$

Lateral Surface Area (LSA) = (Area of left face) +  (Area of right face) +  (Area of front face) +  (Area of back face) = $wh + wh + +lh + lh = 2wh + 2lh = 2\left(l + w \right)h$ sq units.

Formula for Lateral Surface Area = $2(l + w)h$.

Total Surface Area of a Cuboid

The total surface area of a cuboid refers to the total area covered by all the faces of a cube. There are six rectangular faces in a cuboid, so to calculate TSA, we find the sum of the areas of these $6$ faces.

surface area of cuboid

If the length, width(breadth), and height of a cuboid are $l$, $w$, and $h$ respectively, then the areas of the $6$ faces are

  • Left face: $w \times h = wh$
  • Right face: $w \times h = wh$
  • Front face: $l \times h = lh$
  • Back face: $l \times h = lh$
  • Top face: $l \times w = lw$
  • Bottom face: $l \times w = lw$

Total Surface Area (TSA) = (Area of left face) +  (Area of right face) +  (Area of front face) +  (Area of back face) + (Area of top face) + (Area of bottom face) = $wh + wh + lh + lh + lw + lw = 2wh + 2lh + 2lw = 2\left(lw + wh + hl \right)$.

Formula for Total Surface Area = 2\left(lw + wh + hl \right)$.

Examples

Ex 1: Find the total surface area of a cuboid of length $15 cm$, width $10 cm$, and height $6 cm$.

For the given cuboid, $l = 15 cm$, $w = 10 cm$, and $h = 6 cm$

Total Surface Area (TSA) = $2\left(lw + wh + hl \right) = 2\left(15 \times 10 + 10 \times 6 + 6 \times 15 \right) = 2\left(150 + 60 + 90\right) = 2 \times 300 = 600 cm^{2}$.

Ex 2: Find the lateral surface area of a cuboid of length $1.5 m$, width $1 m$, and height $80 cm$.

For the given cuboid, $l = 1.5 m$, $w = 1 m$, and $h = 0.80 m$

Lateral Surface Area (LSA) = $2\left(l + w \right)h = 2 \times \left(1.5 + 1 \right) \times 0.80 = 2 \times 2.5 \times 0.80 = 4 m^{2}$.

Ex 3: Find the cost of painting the four walls of a room of dimension $15 ft \times 12 ft \times 8 ft$ at the rate of ₹$12.50$ per square foot.

The room is in the form of a cuboid.

For the given room (cuboid), $l = 15 ft$, $w = 12 ft$, and $h = 8 ft$

The area of four walls of a cuboidal room = Lateral Surface Area of a cuboidal room

Therefore, area to be painted = Lateral Surface Area (LSA) = $2\left(l + w \right)h = 2 \times \left(15 + 12 \right) \times 8 = 2 \times 27 \times 8 = 432 ft^{2}$ 

Rate of painting = ₹$12.50$ per square foot.

Therefore, cost of painting the four walls of a room = $432 \times 12.50 = $ ₹ $5,400$.

Types of Coordinate Systems

Conclusion

The surface area of a 3D shape (solid object) is a measure of the total area that the surface of the object occupies. There are two types of surface areas in a cuboid, viz., lateral surface area calculated using the formula  $2\left(l + w \right)h$ and total surface area calculated using the formula $2\left(lw + wh + hl \right)$, where $l$, $w$, and $h$ are the length, width, and height of a cuboid respectively.

Practice Problems

  1. A cuboidal box has the dimensions $15 cm$, $7 cm$ and $3 cm$. Find the
    • lateral surface area
    • total surface area
  2. The length, width, and height of a cuboidal box are $b mm$, $8 mm$, and $2b mm$. Find the
    • lateral surface area
    • total surface area
  3. The total surface area of a cuboidal box is $142 cm^{2}$ and its width and height are $7 cm$ and $3 cm$ respectively. Find the length of the box.
  4. Manoj wants to paint $4$ identical doors length $0.6 m$, width $5 cm$ and height $2 m$. How much paint in litres is required to paint the $4$ doors? ($1$ litre of paint covers $10 m^{2}$).

Recommended Reading

FAQs

What is the surface area of a cuboid?

The surface area of a cuboid is the sum of the area of the bases($2$ bases) and the area of lateral faces($4$ lateral faces) of the cuboid. Since the faces of the cuboid are made up of $6$ rectangles therefore the total surface area of the cuboid will be numerically equal to the sum of the areas of these six rectangles. 

What is the surface area formula of a cuboid?

There are two surface areas for a cuboid:
a) Lateral Surface Area is the sum of areas of four lateral (side) faces and is given by $2 \left(l + w \right)h$.
b) Total Surface Area is the sum of areas of all the six faces and is given by $2 \left(lw + wh + hl \right)$.

What is the difference between the total surface area and the lateral surface area of a cuboid?

The difference between the total surface area and the lateral surface area of a cuboid is given below:
a) The total surface area of a cuboid is the sum of the areas of all $6$ faces, whereas, the lateral surface area of a cuboid is the sum of the areas of faces excluding the top and the base.
b) The total surface area of a cuboid is calculated using the formula $2 \left(lw + wh + hl \right)$, whereas, the lateral surface area of a cuboid is calculated using the formula $2h \left(l +w \right)$.

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