In a 3-D figure, if the base is a rectangle or square, two of the side walls side walls are triangles and other two side walls are rectangles or squares, then the 3-D figure is called triangular prism.
Any triangular prism will look like the one shown below.
The formula to find volume of the above triangular prism is
= (1/2) x Base Area x Height
Important Note :
The above formula will work only if the given figure meets the following conditions.
(i) The base must be a rectangle or square.
(ii) Two of the side walls side walls must be triangles.
(iii) Other two side walls must be rectangles or squares.
Example 1 :
Find the volume of the prism.
Solution :
Step 1 :
In the given prism, the base is a rectangle, two of the side walls are triangles and other two side walls are rectangles.
So, the given prism is a triangular prism.
Step 2 :
Volume of a triangular prism = (1/2) x base area x height
V = (1/2) x B x h
Step 3 :
Find base area
Base Area = l x w
22 x 24
= 528 sq.m
Step 4 :
Find volume of the triangular prism
V = (1/2) x b x h
= (1/2) x (528) x 7
= 1848 cubic meters
Example 2 :
Bradley’s tent is in the shape of a triangular prism shown below. How many cubic feet of space are in his tent ?
Solution :
Step 1 :
To find the number of cubic feet of space in Bradley’s tent, we have to find the volume of his tent.
Step 2 :
Volume of a triangular prism = (1/2) x base area x height
V = (1/2) x b x h
Step 3 :
Find base area.
Base Area = l x w
= 9 x 6
= 54 sq.ft
Step 4 :
Find volume of the triangular prism.
V = (1/2) x b x h
= (1/2) x (54) x 4
= 108 cubic feet
Example 3 :
The figure is composed of a rectangular prism and a triangular prism. Find the volume of the figure.
Solution :
Volume of the shape = volume of rectangular prism + volume of triangular prism
Volume of rectangular prism = length x width x height
Length = 30 inches, width = 13 inches and height = 13 inches
= 30 x 13 x 13
= 5070 cubic inches
volume of triangular prism = (1/2) x base x height x height of prism
base of triangle = 9 inches, height of triangle = 13 inches and height of prism = 30 inches
= (1/2) x 9 x 13 x 30
= 1755 cubic inches
Volume of the shape = 5070 + 1755
= 6825 cubic inches
Example 4 :
A solid glass paperweight is in the shape of a triangular prism The density of the glass is 2.4 g/cm³ Work out the mass of the paperweight.
Solution :
density = mass / volume
mass = density x volume
base of triangle = 6 cm, height of triangle = 4 cm and height of prism = 5 cm
Volume = base area x height
= (1/2) x 6 x 4 x 5
= 60 cubic cm
Density = 2.4 g/cm³
mass = 2.4 x 60
= 144 grams
Example 5 :
Alex made a sketch for a homemade soccer goal he plans to build. The goal will be in the shape of a triangular prism. The legs of the right triangles at the sides of his goal measure 4 ft and 8 ft, and the opening along the front is 24 ft. How much space is contained within this goal
Solution :
In legs, base = 4 ft and height = 8 ft, height of prism = 24 ft
volume = base area x height
= (1/2) x 4 x 8 x 24
= 384 cubic ft
So, the space contained within this goal is 384 cubic ft.
Example 6 :
The volume of a square pyramid is 60 cubic inches and the height is 15 inches. Find the side length of the square base.
Solution :
Let x be the side lenght of square base.
volume = 60 cubic inches
base area x height = 60
x2 x 15 = 60
x2 = 60/15
x2 = 4
x = 2
So, the side length of square is 2 inches.
Example 7 :
The volume of a square pyramid is 1024 cubic inches. The base has a side length of 16 inches. Find the height of the pyramid.
Solution :
Let h be the height of the pyramid
Volume of square pyramid = 1024 cubic inches
base area x height = 1024
16 x 16 x h = 1024
256 x h = 1024
h = 1024/256
h = 4 inches
So, height of the pyramid is 4 inches.
Example 8 :
Find the value of the variable in each figure. Leave answers in simplest radical form.
Solution :
By considering the base of the triangle, it is equilateral triangle.
Area of base = area of equilateral triangle = (√3/4) a2
Here a = side length = x
Volume of the shape = (√3/4) a2 x height
18√3 = (√3/4) x2 (6)
18√3 (4/√3)/6 = x2
x2 = 12
x = √12
x = √(2 ⋅ 2 ⋅ 3)
x = 2√3
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