Surface Area Of A Rectangular Prism

This article is all about how to find the surface area of a rectangular prism,  surface area of a rectangular prism formula, lateral surface area of a rectangular prism, total surface area of a rectangular prism calculator, what is the surface area of a rectangular prism, formula for surface area of a rectangular prism.

What Is The Surface Area Of A Rectangular Prism

A Prism is a 3-Dimensional solid figure with 6 sides. We can define its total surface area as the entire region which is covered within its six faces. Take note that the polygon bases of a typical Prism are identical in nature. Also, flat parallelogram sides are common characteristics of  prism solids.

Prisms come in different forms. We have the square prisms, triangular prisms, pentagonal prisms, rectangular prisms, hexagonal prisms and many others. For the purpose of this very study, we will be dwelling more on how to calculate the surface area of rectangular prisms.

Surface Area of a Rectangular Prism Formula

The surface area of a prism can be rightly described as the total region covered by all the six faces of a rectangular prism. The rectangular prism is a shape of three dimensions. All the six faces of a rectangular prism bear a rectangular shape. In line with this, we can  go forward to say that the bases of any rectangular are rectangles. Examples of objects whose surface areas can be likened to the surface area of a rectangular prism includes an aquarium with a cuboid shape, a box, a text book and the likes of them.

If for instance you bought a box to be presented as a gift and you decided to wrap it, you may need to find out the ideal size of a wrapping sheet which you will use to wrap it. In that case, you will need to calculate the specification of the wrapping sheet you need to wrap the box. To start with, you need to work out the surface area of each of the sides of the box to be wrapped. Then, you proceed to get the sum of the areas of each of the six faces of the box. This includes the top surface, the four lateral side surfaces and the bottom surface. This final answer you get is the total area of the 6 surfaces of the box and this can be referred to as the surface area of the rectangular box (prism).

On the other hand, we can also classify a rectangular prism into 2 classes. First, an oblique rectangular prism and second, a right rectangular prism. Talking about the right rectangular prism, the bases of the solid are perpendicular to each other. But in the case of an oblique rectangular prism, the bases of the prism are not aligned one above the other.

Having discussed what the rectangular prism is, let us go forward to see the basic formulas we can use to determine the total surface area of a rectangular prism area as well as the lateral surface area of a rectangular prism

Formula For Surface Area of Rectangular Prism

As we highlighted above, we can calculate the total surface area of a rectangular prism and also the lateral surface area of a rectangular prism. But before you can be able to proceed with making calculations for the surface area of a rectangular prism, you first need to find the sum of the areas of the six faces of the given prism.

Total Surface Area of Rectangular Prism Calculator

The total surface area of a rectangular prism can be worked out by first knowing area of each of the six faces of the prism and adding them up together. The formula to find the surface area of a rectangular prism can be outlined as follows:

Total surface area of a rectangular prism (TSAORP) = 2(lb + bh + lh) square units

where,

b = Breadth of the rectangular prism

h = Height of the rectangular prism

l = Length of the rectangular prism

Always bear it in mind that Square Units is the standard unit of measurement when calculating the area of a rectangular prism.

Lateral Surface Area of a Rectangular Prism Calculator

How to find lateral surface area of a rectangular prism – We can easily calculated the lateral surface area of a rectangular prism by first finding the sum of all the lateral faces of the prism. This implies adding up the values of all the lateral surfaces of the rectangular prism so as to arrive at the total area no including the area of the bases. The formula to find the surface area of a rectangular prism can be written as follows:

Lateral surface area of a rectangular prism formula (LSAORP) = 2(l + b) h square units

where,

b = Breadth

h = Height of the rectangular prism

l = Length

How Do You Find The Surface Area Of A Rectangular Prism

Let us practice how to get the surface area of a rectangular prism using the following questions presented below:

Example One: What will be the total surface area of a rectangular prism that has the the following dimensions:

Height (h) = 10 inches

Breadth (b) = 4 inches

Length (l) = 6 inches

Answer: Recall that the formula for calculating the total surface area of the rectangular prism is given as:

TSAORP = 2(lb + bh + lh)

Substituting the given dimensions into the equation, we have:

TSAORP = 2(6 × 4 + 4× 10 + 6 × 10)

TSAORP = 2(24 + 40 + 60)

TSAORP = 2 × 124

Thus, the Total Surface Area Of Rectangular Prism is = 248 inch²

How to Get the Surface Area of a Rectangular Prism

Any time you want to work out the total surface area of a rectangular prism, please follow these steps outlined below carefully:

First Step: Get the dimensions of rectangular prism and be very sure that they are presented in the same units of of measurement. If they are not, then go on to convert them into the same units so as to ensure accuracy in your calculations.

Second Step: As soon as you are done converting the dimensions of the rectangular prism into the same units, confirm from the given question whether you are asked to calculate the total surface area of a rectangular prism or rather, the lateral surface area of a rectangular prism. They are two different things as we have pointed out earlier in this article.

Third Step: If you were requested to compute the lateral surface area of a rectangular prism, then you need to use this formula = 2h(l + b). But if the questions demands that you make a calculation for the total surface area of a rectangular prism, then the right formula to use is = (lb + bh + lh).

Fourth Step: Substitute the values of your dimension into your formula and work it out carefully to the end so as to arrive at an accurate answer. Do not forget to to present your final result in the required standard square unit measurement. And that is all.

Surface Area Of Rectangular Prism Example Problem

Now that have explain the meaning of the surface area of a rectangular prism as well as the formulas used in the calculation of the lateral and total surface areas of a rectangular prism, we now need to use sordid examples to put this knowledge into practice.

Example One: You bought a rectangular gift box from the market. Calculate the minimum area of the wrapping sheets needed to wrap this rectangular gift box all around if the dimensions of the box are as follows:

Height, h = 30 inches

Breadth, b = 20 inches

Length, l = 25 inches

Answer: This situations requires that you calculate the total surface area of the rectangular gift box. So we will take the following steps to get the answer.

First Step: Work for the area of the bottom face as well as the area of the top face of this rectangular box.

Remember that the area of the top and bottom bases of a rectangular prism are equal to each other. So we can simply work for either the area of the bottom or the area of the top of the box and multiply it by 2. This can be done as follows:

2lb = 2 × (25 × 20) = 2 × 500 = 1000 inch²

Second Step: We will go further to find the area of the side surfaces of this rectangular gift box as follows:

2bh = 2 × (20 × 30) = 2 × 600 = 1200 inch²

2lh = 2 × (25 × 30) = 2 × 750 = 1500 inch²

Third Step: Add together all the areas of the six faces of the rectangular gift box so as to get the total surface area of the box.

Total Surface Area Of  A Rectangular prism = 2(lb + bh + lh)

TSAORP = 2lb + 2bh + 2lh

TSAORP = 1000 + 1200 + 1500

Thus, the total surface area of the rectangular gift box = 3700 inch².

Example Two: There are two boxes which are joined to each other face to face to form a rectangular prism solid. For each of the boxes, the measurement dimensions are 3 inches wide × 7 inches long × 9 inches high. Using the following information, calculate the total surface area of the newly formed bigger box.

Answer: The formula for calculating the total surface area of the rectangular prism is given as:

TSAORP = 2(lb + bh + lh)

We will first determine the total surface area of the first box. Hence, we substitute the given dimensions into the equation to have below:

TSAORP = 2(7 × 3 + 3× 9 + 7 × 9)

TSAORP = 2(21 + 27 + 63)

TSAORP = 2 × 111

Thus, the Total Surface Area Of this first box that made up the Rectangular Prism is = 222 inch²

Since the 2 boxes that were joined together to form the rectangular prism have the same dimensions, their respective total surface areas should be equal too. Therefore, the total surface area of the newly formed bigger box is:

2 x 222 = 444 inch²

How To Solve Surface Area Of A Rectangular Prism

A rectangular prism whose entire sides are equal to each other is known as a cube.

An oblique rectangular prism is a form of rectangular prism whose faces are not perpendicular to each other.

If all the edges of any rectangular prism are equal to each other, then the total surface area of the prism shall be calculated as six times the area of each of the prism’s face.

Lastly, If all the edges of a rectangular prism are equal to ‘l’, you can compute the volume of the rectangular prism as ‘l³’.

Surface Area of Rectangular Prism Worksheet

This Worksheet On Surface Area of a rectangular prism shows you ideal Surface Area of Rectangular Prism Examples and solutions.

Example One: Henry wants to purchase a roll of cardboard sheet to construct a wooden box without a cover. He wants the length of the box to be 11 feet, the height to be 15 feet and the width to be 8 feet. Help him to calculate the quantity of cardboard sheet he should buy to deliver the job.

Solution: From the information given above, the dimensions of the box Henry wants to build are as follows:

Height (h) = 15 feet

Breadth (b) = 8 feet

Length (l) = 11 feet

In order to find the area of cardboard sheet he needs to do the work, you have to find the total surface area of the open box without a cover. This can simply be worked out by calculating area of each of the five surfaces of the box and summing them up to get the total surface area required.

Thus, Base Area = lb

Lateral surface area (that is the Area of the 4 lateral faces) = 2(l + b)h

= 2lh + 2bh

Therefore, the Total surface area of the rectangular prism (box without cover) = lb + 2lh + 2bh

= (11 × 8) + (2 × [11 × 15]) + (2 × [8 × 15])

= 88 + 2(165) + 2(120)

= 88 + 330 + 240

= 658 inch²

So Henry should purchase 658 inch² of cardboard to construct the open box.

Example Two: A rectangular prism has a base area of 20 ft². Also, the total surface area of the same rectangular prism measures 140 ft.² while the perimeter of the base is given as 28 ft². Determine the height of this solid applying the formula for finding the surface area of a prism.

Solution: Let “l” be the length of the rectangular prism, “b” the breadth of the rectangular prism and “h” the height of the rectangular prism.

The Base Area Of The Prism lb = 20 ft.²
The Base perimeter of the prism 2(l + b) = 28 ft.²
The Total Surface area of the prism = 140 ft.²

Applying the surface area of a rectangular prism formula to determine the height of the rectangular prism, we have below:

Total surface area of a rectangular prism = 2(lb + bh + lh)

Therefore 2(lb + bh + lh) = 140 ft.²

From the above, we can derive the formula below:

2lb + 2(l+b)h = 140 ft.²

Substituting dimensions in the question into the equation, we have the following:

(2 × 20 ft.²) + (28 ft.² × h) = 140 ft.²

40 ft.² + (28 ft.² × h) = 140 ft.²

28 ft.² × h = 140 ft.² – 28 ft.²

28 ft.² × h = 112 ft.²

h = 112 ft.² ÷ 28 ft.²

h = 4 ft.²

Therefore, the height of the rectangular prism is 4 ft.²

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