Surface Area of Cuboid (Definition, Formula & Examples)

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 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.

What is the Surface Area of 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

  • Lateral Surface Area of cuboid
  • Total Surface Area of cuboid

Total Surface Area of Cuboid

The total surface area of cuboid or TSA Of 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) = $wh + wh + lh + lh + lw + lw = 2wh + 2lh + 2lw = 2\left(lw + wh + hl \right)$.

TSA of Cuboid Formula = 2\left(lw + wh + hl \right)$.

Lateral Surface Area of Cuboid – LSA of Cuboid Formula

The lateral surface area of 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.

LSA of Cuboid Formula = $2(l + w)h$.

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

Practice Problems

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 vertices in a cuboid meets the three faces and 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

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 $10 m^{2}$).

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 of a cuboid and example?

There are two types of surface area for a cuboid.
Lateral Surface Area: It is the sum of all four lateral faces(left side, right side, front, and back). The formula for lateral surface area is $2(l + w)h$, where $l$, $w$, and $h$ are the length, width, and height of a cuboid.
Total Surface Area: It is the sum of all the faces of a cuboid. The total surface area is equal to the Curved Surface Area plus the sum of the top and bottom faces. The formula for the total surface area is $2(lw + wh + hl)$, where $l$, $w$, and $h$ are the length, width, and height of a cuboid.
For example, for a cuboid of length $12 \text{ cm}$, width $10 \text{ cm}$, and height $5 \text{ cm}$
Lateral Surface Area = $2 \times (12 + 10) \times 5 = 2 \times 22 \times 5 = 220 \text{ cm}^2$
Total Surface Area = $2 \times (12 \times 10 + 10 \times 5 + 5 \times 12) = 2 \times (120 + 50 + 60) = 2 \times 230 = 460 \text{ cm}^2$.

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 $2h \left(l +w \right)$.

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.

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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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