A triangular prism measures 81 cm long. Its cross section is an equilateral traingle with a base of 16 centimeters and a height of 14 cm. Find the surface area of this triangular prism.
Surface area of a prism:
SA = (area of the base) + (lateral area)
Imagine the shape. Triangular prism is the shape of a toblerone box. So,
SA = 2*(area of triangle) + 3*(area of rectangle)
Recall that area of triangle is just
A = bh/2
And area of rectangle is
A = lw
Therefore,
SA = 2(16)(14)/2 + 3(81)(16)
SA = ?
Units in cm^2. Hope this helps~ `u`
To find the surface area of the triangular prism, we need to calculate the area of each individual side and then sum them up.
There are three types of surfaces for a triangular prism:
1. The two triangular bases
2. The three rectangular faces
1. Let's start with the triangular bases. Since the base is an equilateral triangle, we can use the formula for the area of an equilateral triangle, which is A = (sqrt(3) / 4) * (side length)^2.
In this case, the side length is 16 cm. Substituting it into the formula:
A = (sqrt(3) / 4) * (16 cm)^2
A = (1.732 / 4) * 256 cm^2
A = 1.732 * 64 cm^2
A = 110.848 cm^2
Since there are two triangular bases, the total area for them is 2 * 110.848 cm^2 = 221.696 cm^2.
2. Now let's calculate the area of the three rectangular faces. The length of each rectangular face is the same as the length of the prism, which is 81 cm. The width of each rectangular face is the same as the base of the triangular cross-section, which is 16 cm. Thus, the area of each rectangular face is length * width = 81 cm * 16 cm = 1296 cm^2.
Since there are three rectangular faces, the total area for them is 3 * 1296 cm^2 = 3888 cm^2.
To find the total surface area, we need to sum the area of the triangular bases and the three rectangular faces:
Total surface area = 221.696 cm^2 + 3888 cm^2 = 4109.696 cm^2.
Therefore, the surface area of the given triangular prism is 4109.696 cm^2.