If Each Dimension Of The Rectangular Prism Is Doubled, How Will Its Total Surface Area Change?
What is a Rectangular Prism?
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A rectangular prism is a three-dimensional figure with six faces(two faces are opposite to each other at the top and lesser and four faces are lateral to each other), eight vertices, and twelve edges. The faces of the prism are all rectangular and are such that the opposite faces are e'er congruent. Thus, it has three pairs of identical faces. A rectangular prism has a volume and a expanse. The rectangular prism is likewise called a cuboid.
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Types of Rectangular Prism
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The rectangular prism can exist of two unlike types. They are every bit follows-
- Right Rectangular Prism
- Oblique Rectangular Prism
Correct Rectangular Prism
The right rectangular prism is a prism with vi faces that are all rectangles and all the angles are right angles.
Right Rectangular prism
Oblique Rectangular Prism
An oblique prism is a prism that has bases that are not perpendicular to each other. A rectangular prism that has faces that are not exactly aligned directly to each other.
Oblique Rectangular Prism
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Properties of Rectangular Prism
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A rectangular prism has some definite properties that are equally follows-
- A rectangular prism has half dozen faces, eight vertices, and twelve edges.
- It is a iii-dimensional effigy that has a length, width, and height.
- The faces are rectangular in a right rectangular prism. The faces are parallelograms in an oblique rectangular prism.
- The opposite faces in a rectangular prism are always congruent.
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Rectangular Prism Formulae
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A rectangular prism is a three-dimensional effigy which has a surface area and volume. The dimensions of a rectangular prism length(l), width(west), top(h).
Volume of Rectangular Prism
The volume of a rectangular prism is the amount of space within it. The book of the rectangular prism is obtained by multiplying its base surface area by its height. Now,
- The base of operations surface area of the rectangular prism \(= l \times west \)
- The superlative of the rectangular prism \(= h\)
Therefore, the volume of the rectangular prism, \(V = l \times westward \times h\)
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Surface Expanse of Rectangular Prism
The surface surface area of a rectangular prism is of two types-
-
The full surface area of a rectangular prism is the sum of the areas of all the faces in the rectangular prism.
The TSA of the rectangular prism is given by the formula,
\(=2 (lw + wh + hl)\)
- The lateral surface surface area of a rectangular prism is the sum of the areas of all its side faces without the base faces.
The lateral expanse of the rectangular prism is given by the formula,
\(=two (wh + hl)\)
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Nets of a Rectangular Prism
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The surface area of a rectangular prism tin be calculated by calculating the area of the nets of a prism. The surface area of a prism is calculated from the nets.
Nets of the rectangular prism
When the prism is turned in a plane, all the sides of the prism are and then visible. The surface area of the rectangle is calculated for each rectangle and and so all the areas are added together to find the total surface surface area of the rectangular prism.
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Things to Remember
- A rectangular prism is a three-dimensional figure with six faces, viii vertices, and twelve edges.
- The faces of the prism are all rectangular and are such that the opposite faces are always congruent.
- For whatever given polyhedron like rectangular prism,
\(F + 5 - E = 2\)
where F stands for the number of faces, V stands for the number of vertices and E stands for the number of edges. This relationship is chosen Euler's formula. - A rectangular prism is a iii-dimensional effigy which has a area and book. The dimensions of a rectangular prism length(fifty), width(west), height(h).
The volume of the rectangular prism, \(V = l \times due west \times h\) - A rectangular prism is a three-dimensional effigy which has a surface area and volume. The dimensions of a rectangular prism length(50), width(due west), meridian(h).
The TSA of the rectangular prism is given by the formula,
\(=ii (lw + wh + hl)\)
The lateral surface area of the rectangular prism is given by the formula,
\(=2 (wh + hl)\) - The surface area of a prism is calculated from the nets.
Sample Questions
Ques. Detect the volume of a rectangular prism whose length, width, and height are 8cm, 6cm, and 4cm, respectively. [ii marks]
Ans. The data given are,
length, \(l = 8\ cm;\) width, \(w = 6\ cm;\) peak, \(h = 4\ cm\)
The formula that is used to find the volume of a rectangular prism is
Volume, \(Five = l \times w \times h\) cubic units
\(V = 8 \times 6 \times 4\) cubic units.
\(Five = 192\) cubic units.
Therefore, the book of a rectangular prism is \(192\ cm^three\) .
Ques. Ms. Jeniffer gives prizes to the students of her class for performing acts of kindness. She displays prizes like colorful pencils, fancy erasers, and color pens in a drinking glass case, shaped more than of a rectangular prism of length = sixteen in, width = 5 in and elevation = 3 in. Detect the total surface surface area of the prize case.[2 marks]
Ans. The prize example dimensions given are,
Length, \(l = 16\ in\)
Width, \(west = 5\ in\)
Height, \(h = iii\ in\)
So, the total surface area of the prize case,
TSA \(= 2 (lw + wh + hl)\)
\(= 2 (16 \times 5 + 5 \times iii + 16 \times three)\)
\(= ii (fourscore + 15 + 48)\)
\(= 2 \times 143\)
\(= 286\ in^2\)
Ques. Find the surface area and book of a rectangular prism with the length, width, and height are 8 cm, vi cm, and 4 cm respectively. [2 marks]
Ans. The data given is,
length, \(l = 8\ cm;\)
width, \(west = 6\ cm;\)
superlative, \(h = 4\ cm\)
The formula used to find the full surface surface area of a rectangular prism is,
\(A = 2 (lw + wh + hl)\)
\(A = 2 (eight \times 4 + 6 \times iv + 8 \times six)\)
\(A = 2 (32 + 24 + 48)\)
\(A = 2 \times 104\ cm^two\)
\(A = 208\ cm^2\)
Ques. Notice the area of a rectangular prism whose length, width, and height are given, respectively. [5 marks]
-
5 cm, 8 cm, 10 cm
-
6.ii cm, 4.4 cm, nine cm
Ans. a) The data given is,
Length, \(fifty = five\ cm\)
Width, \(w = 8\ cm\)
Pinnacle, \(h = 10\ cm\)
The formula used to find the total expanse of a rectangular prism is,
Surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (5 \times ten + 8 \times 10 + 5 \times eight)\)
\(= 2 (fifty + fourscore + 40)\)
\(= 2 \times 170\ cm^2\)
\(= 340\ cm^2\)
The formula that is used to discover the volume of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= 50 \times w \times h\) cubic units
\(= 5 \times 8 \times ten\) cubic units.
\(Five = 400\ cm^3\)
b) The information given is,
Length, \(l = vi.two\ cm\)
Width, \(w = 4.4\ cm\)
Height, \(h = 9\ cm\)
The formula used to find the total surface area of a rectangular prism is,
Surface surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (6.2 \times 4.4 + four.4 \times ix + nine \times six.2)\)
\(= 2 (27.28 + 39.6 + 55.eight )\)
\(= two \times 122.68\ cm^2\)
\(= 245.36\ cm^ii\)
The formula that is used to find the book of a rectangular prism is
Volume, \(5 = length \times width \times top\) cubic units
\(= l \times due west \times h\)
\(= half-dozen.2 \times iv.4 \times ix\) cubic units.
\(V = 245.52\ cm^3\) .
Ques. Find the volume and total surface expanse of rectangular prism with the post-obit dimensions.[4 marks]
-
3 cm ten four cm x 5 cm
-
ii.5 cm ten 6 cm x ix cm
Ans. a) The data given is,
Length, \(fifty = three\ cm\)
Width, \(w = 4\ cm\)
Elevation, \(h = v\ cm\)
The formula used to observe the total surface expanse of a rectangular prism is,
Surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (3 \times 5 + four \times 5 + 3 \times 4)\)
\(= 2 (15 + xx +12)\)
\(= two \times 47\ cm^2\)
\(= 94\ cm^2\)
The formula that is used to detect the volume of a rectangular prism is
Volume, \(V = length \times width \times meridian\) cubic units
\(= 50 \times w \times h\)
\(= 3 \times iv \times 5\) cubic units.
\(V = 60\ cm^3\)
b) The information given is,
Length, \(50 = 2.5\ cm\)
Width, \(w = 6\ cm\)
Height, \( h = nine\ cm\)
The formula used to find the full surface area of a rectangular prism is,
Expanse, \(A = 2 (lw + wh + hl)\)
\(= ii (2.5 \times 9 + 6 \times 9 + 2.5 \times half dozen)\)
\(= 2 (22.5 + 54 + 15 )\)
\(= 2 \times 91.5\ cm^2\)
\(= 183\ cm^2\)
The formula that is used to find the book of a rectangular prism is
Volume, \(5 = length \times width \times tiptop\) cubic units
\(= 50 \times w \times h\)
\(= ii.5 \times 6 \times 9\) cubic units.
\(Five = 135\ cm^3\)
Ques. Rahul has a chocolate box whose shape resembles a rectangular prism. Its dimensions are equally follows, length half-dozen in, summit 2 in, and width four in. Notice the volume of the box. [three marks]
Ans. The dimensions of the chocolate box given are,
length, \(l = 6\ in\)
width, \(w = 4\ in\)
height, \(h = two\ in\)
The formula that is used to detect the volume of a rectangular prism is
Volume, \(5 = length \times width \times superlative\) cubic units
\(= 50 \times due west \times h\)
\(= 6 \times 4 \times 2\) cubic inches.
\(= 48\ in^3\)
Ques. A gift is packed in a rectangular box (rectangular prism) of dimensions 15 in, 10 in, and viii in and information technology needs to be wrapped with gift paper. How much gift paper is required to wrap the gift box? [3 marks]
Ans. The dimensions of the given rectangular gift box are,
length, \(l = xv\ in\)
width, \(w = ten\ in\)
height, \(h = viii\ in\)
The rectangular box (rectangular prism) needs to be wrapped with gift paper therefore, the total surface area needs to exist calculated considering the wrapper will exist spread across the total surface area of the prism to encompass it completely. Thus,
The formula for Total Surface Area of a rectangular prism \(= 2 (lw + wh + hl)\)
\(= 2 (15 \times 10 + x \times 8 + 15 \times 8)\)
\(= 2 (150 + fourscore + 120)\)
\(= 2 \times 350\)
\(= 700\ in^ii\)
Ques. Olivia'south mother surprised her with pasta packed in a rectangular prism-shaped lunch box. How much quantity of pasta is served to Olivia given that the dimensions of the rectangular prism are as follows, length = 5 in, width = 4 in, peak = 3 in. [three marks]
Ans. The dimensions of the rectangular prism-shaped lunch box are
length, \(l = five\ in\)
width, \(w = 4\ in\)
height, \(h = 3\ in\)
At present, the corporeality of pasta that can be contained in the lunch box can be calculated by finding out the volume of the container. Thus,
The formula that is used to detect the book of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= 50 \times west \times h\)
\(= 5 \times 4 \times iii\) cubic inches.
\(= 60\ in^iii\)
Ques. Jenny's grandmother handed over her beautiful hand-carved chest to him for storing his stuff, especially woolens. The chest is 3 ft long and 2 ft broad having a shape of that of a rectangular prism has a capacity of 12 cubic ft. Find its elevation. [2 marks]
Ans. The dimensions of the chest given are,
Length, \(l = 3\ ft.\)
Width, \(w = 2\ ft.\)
Peak, \(h =\) to be found out
Book, \(v = 12\) cubic ft.
At present the formula for calculating the volume can exist used to calculate the height,
Volume, \(Five = length \times width \times elevation\) cubic units
\(12 = fifty \times w \times h\)
\(12 = 3 \times 2 \times h\)
\(12 = half-dozen \times h\)
\(h = 2\ ft.\)
Ques. Detect the surface area volume of the rectangular prism with the following dimensions.[4 marks]
-
14 cm ten viii cm x 5 cm
-
three.4 cm x 9.4 cm x 9.7 cm
Ans. a) The data given is,
Length, \(50 = 14\ cm\)
Width, \(w = 8\ cm\)
Height, \(h = 5\ cm\)
The formula used to find the full surface area of a rectangular prism is,
Surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (fourteen \times five + 8 \times 5 + 14 \times 8)\)
\(= 2 (lxx + twoscore + 112)\)
\(= ii \times 222\ cm^ii\)
\(= 444\ cm^two\)
The formula that is used to notice the volume of a rectangular prism is
Volume, \(Five = length \times width \times pinnacle\) cubic units
\(= l \times w \times h\)
\(= 14 \times 8 \times 5\) cubic units.
\(V = 560\ cm^3\)
b) The data given is,
Length, \(50 = 3.4\ cm\)
Width, \(westward = 9.4\ cm\)
Height, \(h = 9.seven\ cm\)
The formula used to find the total surface area of a rectangular prism is,
Surface expanse, \(A = two (lw + wh + hl)\)
\(= 2 (iii.4 \times ix.7 + ix.four \times ix.vii + three.4 \times 9.4)\)
\(= 2 (32.98 + 91.18 + 31.96 )\)
\(= 2 \times 156.12\ cm^2\)
\(= 312.24\ cm^ii\)
The formula that is used to find the volume of a rectangular prism is
Book, \(V = length \times width \times elevation\) cubic units
\(= fifty \times w \times h\)
\(= 3.four \times ix.4 \times 9.vii\) cubic units.
\(V = 310.012\ cm^3\)
Ques. Detect the expanse of a rectangular prism whose length, width, and pinnacle are given, respectively.[iv marks]
-
13 cm, 15 cm, and 6 cm
-
4.5 cm, vii.ii cm, 2.iii cm
Ans. a) The data given is,
Length, \(l = 13\ cm\)
Width, \(w = 15\ cm\)
Elevation, \(h = 6\ cm\)
The formula used to observe the total surface surface area of a rectangular prism is,
Surface area, \(A = ii (lw + wh + hl)\)
\(= 2 (13 \times vi + 15 \times 6 + 13 \times 15)\)
\(= 2 (78 + ninety + 195)\)
\(= ii \times 363\ cm^two\)
\(= 726\ cm^ii\)
The formula that is used to find the volume of a rectangular prism is
Book, \(V = length \times width \times height\) cubic units
\(= 50 \times w \times h\)
\(= 13 \times fifteen \times 6\) cubic units.
\(Five = 1170\ cm^3\)
b) The data given is,
Length, \(l = 4.v\ cm\)
Width, \(w = 7.two\ cm\)
Summit, \(h = 2.3\ cm\)
The formula used to find the full surface area of a rectangular prism is,
Surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (four.5 \times 2.3 + 7.2 \times two.iii + four.5 \times 7.2)\)
\(= 2 (10.35 + sixteen.56 + 32.4)\)
\(= ii \times 59.31\ cm^two\)
\(= 118.62\ cm^2\)
The formula that is used to find the volume of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= fifty \times west \times h\)
\(= iv.5 \times vii.2 \times 2.3\) cubic units.
\(V = 74.52\ cm^3\)
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