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# 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 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.

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.

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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.

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.

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 = $₹$5,400$. ## 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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