What is the Formula for the Surface Area of a Triangular Prism?
A triangular prism is a three-dimensional shape with two triangular faces and three rectangular faces. The rectangular faces are known as the lateral faces and the triangular faces are the bases of the prism. The surface area of a triangular prism is calculated by adding up the area of the lateral faces and the triangular bases. Here is the formula for the surface area of a triangular prism.
Surface area of a triangular prism = bh + (a + b + c)H
Let’s see what each term refers to:
‘a’, ‘b’, and ‘c’ are the side lengths of the triangular bases.
‘b’ and ‘h’ are the base and height of the triangular faces.
H is the height of the triangular prism.
Rapid Recall
Surface area of a triangular prism = bh + (a + b + c)H
Solved Examples
Example 1:
Find the surface area of the triangular prism with the measurements seen in the image.
Solution:
From the image, we can observe that the side lengths of the triangle are a = 5 cm, b = 6 cm and c = 5 cm.
The base and height of the triangular faces are b = 6 cm and h = 4 cm.
The height of the triangular prism is H = 15 cm
We can find the surface area of the triangular prism by applying the formula,
Surface area of a triangular prism = bh + (a + b + c)H
= 6 \(\times\) 4 + (5 + 6 + 5) \(\times\) 15 [Substitute the value]
= 24 + (16) \(\times\) 15 [Apply PEMDAS rule]
= 24 + 240 [Multiply]
= 264 \(cm^2\)
Hence, the surface area of a triangular prism is 264 square centimeters.
Example 2:
Cathy recently purchased a new triangular kaleidoscope in which the sides are 9 cm long. The surface area of the kaleidoscope is 576 \(cm^2\), and its base height is 7.8 cm. Find the height of the kaleidoscope.
Solution:
As stated, the length of each side of the kaleidoscope is 7.8 cm.
That is, a = b = c = 9 cm.
It is mentioned that the surface area of the kaleidoscope is 576 \(cm^2\) and the base height is 7.8 cm.
Since the kaleidoscope is in the shape of a triangular prism, we can use the formula for the surface area to find its height.
So,
Surface area of a triangular prism = bh + (a + b + c)H
576 = 9 \(\times\) 7.8 + (9 + 9 + 9)H [Substitute the value]
576 = 70.2 + (27)H [Apply PEMDAS rule]
576 – 70.2 = (27)H [Subtract 36 from both the sides]
505.8 = (27)H
\(\frac{505.8}{27}\) = H [Divide both sides by 27]
18.73 = H
Hence, the height of Cathy’s kaleidoscope is 18.73 cm.
Example 3:
Adam and Mac have set up a tent while they are on vacation. The image gives you the dimensions of the tent. How much cloth is used for a tent?
Solution:
In the image, all the sides of the triangular faces of the tents are a = b = 5 ft and c = 8 ft
The height of a triangular face is 3 ft and the total length of the tent is 6 ft. So, to find the quantity of cloth used for a tent we add the area of two triangular faces and two rectangular faces.
Area of cloth required = \(2\times \frac{1}{2}\) bh + (a + b)H
= \(2\times \frac{1}{2}\times 8 \times 3+(5+5)\times 6\) [Substitute the value]
= 24 + 10 \(\times\) 6 [Apply PEMDAS rule]
= 24 + 60 [Multiply]
= 84 \(ft^2\) [Add]
Therefore, 84 square feet of cloth is required for a tent.
There are many examples of triangular prisms around us. Some of them are: triangular glass prisms, camping tents, triangular roof tops, kaleidoscopes and even triangular pieces of cheese.
Three rectangular faces and two triangle faces make up the five faces of a triangular prism.
The surface area of a triangular prism is measured in square units, such as square meters,square centimeters, square inches and square feet among many other squared units of length.
A net is a two-dimensional representation of a three-dimensional object and is used to calculate the surface area of the three dimensional object.