# Surface Area of Rectangular Prism

The Surface Area of Rectangular Prism is the total area of all its six faces ie; Top, Bottom & 4 Sides. To find the surface area, we need to find the area of all the six faces and add them up.

Let's say the rectangular prism has length (l), width (w) and height (h)

Surface area of a rectangular prism formula is:

SA = 2lw + 2wh + 2lh

This formula can also be written as SA = 2 ( lw + wh + lh)Where:

• SA is the surface area of the prism
• l is the length
• w is the width
• h is the height

To find the surface area of a rectangular prism, simply substitute the values for l, w, and h into the formula and calculate the result

Practice questions:

The given dimension of a rectangular prism has length 4 cm, width 3 cm, and height 2 cm, hence its surface area would be:

SA = 2(4 x 3) + 2(3 x 2) + 2(4 x 2)

SA = 24 + 12 + 16

SA = 52 cm²

Therefore, the surface area of the rectangular prism is 52 sq. cms.

There are six rectangular faces in a rectangular prism, left and right faces, top, bottom, front & back faces. So to find the surface area of rectangular prism, add the area of each face.

In the above illustration, width is replaced with breadth (b). In some countries, width is represented as breadth (b). Hence, the rectangular prism formula can also be used as

SA = 2lb + 2bh + 2lh

Remember: It’s important to maintain same square units ie; if the dimensions are in cms, the total surface area will be in sq. cms. Same with inches- sq. inch, feet – sq. feet & so on…

Rectangular prism is also known as Cuboid according to Indian Mathematics.

Surface area of a rectangular prism is the measure of how much exposed area a solid object has and expressed in square units.
All the faces of Rectangular prisms are rectangular. The areas of Rectangle have been discussed in another section.

If the Rectangular prisms is open from top and bottom then its area is called Area of 4 walls or lateral faces or Lateral surface area.

How to find the surface area of a rectangular prism – Let’s look at some solved examples for better understanding:

Example 1:

A rectangular prism has a total surface area of 72 sq. inches and a length of 6 inches. If the width is 3 inches, what is the height of the prism?

Solution: Using the formula,

SA = 2lw + 2lh + 2wh,

we can plug in the values and simplify for h:

72 = 2(6*3) + 2(6h) + 2(3*h)

72 = 36 + 12h + 6h

72 = 36 + 18h

36 = 18h

Therefore, the height of the rectangular prism is 2 inches.Example 2:

A rectangular prism has a total surface area of 210 sq. cms and a height of 7 cms. If the length is 5 cms, what is the width of the prism?

Solution: Using the formula,

SA = 2lw + 2lh + 2wh,

we can plug in the values and simplify for w:

210 = 2(5w) + 2(57) + 2(w*7)

210 = 10w + 70 + 14w

210 = 24w + 70

140 = 24w

Answer: w = 5.83 cms (rounded to two decimal places)

Therefore, the width of the rectangular prism is approximately 5.83 cms.:Example 3:

Calculate the surface area of a rectangular prism with a length of 6 cm, a width of 4 cm, and a height of 3 cm.

Solution: Using the formula

SA = 2lw + 2lh + 2wh,

we plug in the values we have:

SA = 2(6 x 4) + 2(6 x 3) + 2(4 x 3)

SA = 48 + 36 + 24

Therefore, the total surface area of the rectangular prism is 108 cm².Example 4:

Find the surface area of a rectangular prism with a length of 12 in, a width of 3 in, and a height of 6 in.

Solution: Using the formula

SA = 2lw + 2lh + 2wh,

plug in the values we have:

SA = 2(12 x 3) + 2(12 x 6) + 2(3 x 6)

SA = 72 + 144 + 36

Answer: surface area SA = 252 in²

Therefore, the surface area of the rectangular prism is 252 in²

Example 5:

A rectangular box of length 40 cm, width 25 cm and height 20 cm is to be made of tin. What is the area of tin sheet required if the box has a lid also?
Solution: l = 40 cms ; w = 25 cms and h = 20 cms
Total surface area of box = 2 ( l x w + w x h + l x h)
Area = 2 ( 40 x 25 + 25 x 20 + 40 x 20 )
= 2 ( 1000 + 500 + 800 )
= 2 x 2300
total area = 4600 cm2Example 6:

A rectangular room of dimensions 8 m x 6 m x 3 m is to be painted. If it costs $60 per square meter, find the cost of painting the walls of the room. Solution: l = 8 m ; w = 6 m ; h = 3 m Painting walls of the room that means here we have to use the formula for area of 4 walls. Area of 4 walls = 2 x h (l + w) = 2 x 3 ( 8 + 6) = 6 x 14 Area of walls = 84 m2 Cost of painting 4 walls = 84 x 60 =$ 5040

Example 7:

The sum of length, width and depth (height) of a rectangular box is 19 cm and length of its diagonal is 11 cm. Find the surface area of the box.
Solution : l + w + h = 19
Diagonal = 11 cm
⇒ √( l2 + w2 + h2) = 11
⇒ l2 + w2 + h2 = 121
l + w + h = 19
⇒ ( l + w + h)2 = 192
⇒ l2 + w2 + h2 + 2( lw + wh + lh) = 361
⇒ 121 + 2 (lw + wh + lh) = 361
⇒ 2(lw + wh + lh ) = 240
Hence, the surface area of box is 240 m2

Example 8:

The length and width of a hall are in the ratio 4:3 and its height is 5.5 m. The cost of decorating its walls (including doors and windows) at 6.60 per square metre is 5082. Find the length and width of the room.
Solution: As the cost of decorating walls is 5082 at the ratio of 6.60 per square meter.
∴ Area of walls = 5082 / 6.60 = 770 m2
Let x be the ratio.
Length = 4x and width = 3x, Height = h = 5.5 m

Area of walls = 2 h ( l + w)
770 = 2 x 5.5 ( 4x + 3x )
770 = 11 ( 7x)
∴ 7x = 770 / 11
⇒ 7x = 70
⇒ x = 10
∴ length = 4x = 4(10) = 40 m and width = 3x = 3(10) = 30 m

Example 9:

Calculate the surface area of a rectangular prism with dimensions of length 4 cm, width 5 cm, and height 6 cm.

Solution: Using the formula, SA = 2lw + 2lh + 2wh, we can plug in the values:

SA = 2(45) + 2(46) + 2(5*6)

SA = 40 + 48 + 60

Therefore, the total surface area of the rectangular prism is 148 cm^2.Practical application in real world:

This formula will be useful to calculate sq.ft area for painting / construction work, paper required (wrapping paper) to wrap box, etc

Surface area of a cube : Cube is a solid three-dimensional figure, which has 6 equal size square faces, 8 vertices and 12 edges. It is also known as a regular hexahedron.

From the above figure, we can see that there are 6 faces and each face is of square shape. Cube formula remains same.

For more details, click the below link:
Surface Area of Cube

Surface area of a Cylinder:

A cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

For more details, click the below link:
Surface Area of Cylinder

Surface area of a Cone: A cone is a three-dimensional shape that narrows from a flat circular base to a point (which forms an axis to the centre. In real life we use this shape in funnel, ice-cream cone, joker’s cap etc. Its base is circular and has a curved surface.

For more details, click the below link:

Surface area of a sphere & hemisphere:

A sphere is a three dimensional shape where every point of its surface is equidistant or the same distance from the centre of the sphere. A hemisphere is half of a sphere.

For more details, click the below link: Surface Area of Sphere and Hemisphere

Surface area of a prism: A prism has a solid shape that is bound on all its sides by plane faces. There are two types of faces in a prism. The top and bottom faces are identical and are called bases. A prism is named after the shape of these bases. For example, if a prism has a triangular base it is called a triangular prism.

For more details, click the below link: Surface Area of Prism

Surface area of a Pyramid: We get surface area of a pyramid by adding the area of all its faces. A pyramid is three-dimensional shapes whose base is a polygon and whose side faces are triangles and meet at a point which is called the apex (or) vertex. The perpendicular distance from the apex to the centre of the base is called the altitude or height of the pyramid.

For more details, click the below link: Surface Area of Pyramid

Right rectangular Prism: To calculate the prism volume, use the formula

V=l x w x h

Where V stands for volume, l-length, w-width & h-height

Besides using above formulas, there is online rectangular prism calculator available online for speedy calculation. You can use rectangular prism calculator as per your convenience