A Triangular Prism has 12cm on the bottom right and 7cm on the top right and 7cm in the middle and 6cm on the left side and under 6cm is 4cm
Determine the area of the triangular base of the prism_______
Determine the area of the smallest rectangle of the prism_______
Determine the area of the largest rectangle of the prism_________
What is the total surface area of the triangular prism_________
To determine the area of the triangular base of the prism, we first need to find the height of the triangle.
Using the Pythagorean theorem, we find:
Height = √(12^2 - 6^2) = √(144 -36) = √108 = 6√3
Area of the triangle base = (1/2) * base * height
Area = (1/2) * 12 * 6√3 = 36√3 cm^2
To determine the area of the smallest rectangle of the prism, we use the dimensions given which are 4cm by 7cm.
Area = 4 * 7 = 28 cm^2
To determine the area of the largest rectangle of the prism, we use the dimensions given which are 12cm by 7cm.
Area = 12 * 7 = 84 cm^2
The total surface area of the triangular prism is the sum of the areas of the two triangles, the two rectangles and the two parallelograms.
Total surface area = 2(36√3) + 28 + 28 + 84 + 84 = 72√3 + 224 ≈ 352.77 cm^2
How to Determine the Surface Area of the Triangular Prism
To determine the surface area of a triangular prism, you need to calculate the areas of all the individual shapes that make up the prism and add them together.
1. Start by calculating the area of the two triangular bases. The formula for the area of a triangle is (1/2) * base * height. You have already calculated the area of one triangle base as 36√3 cm^2, which is the same for the other base.
2. Next, calculate the areas of the three rectangular faces of the prism. For this, you need to multiply the length and width of each rectangular face:
- The smaller rectangle has dimensions 4cm by 7cm - Area = 4 * 7 = 28 cm^2
- The larger rectangle has dimensions 12cm by 7cm - Area = 12 * 7 = 84 cm^2
- The bottom parallelogram has base 12cm and height 6cm. Its area is 12 * 6 = 72 cm^2.
3. Finally, add together the areas of all the faces of the prism to get the total surface area:
Total surface area = 2(36√3) + 28 + 28 + 84 + 72 = 72√3 + 212 ≈ 352.77 cm^2
Therefore, the total surface area of the triangular prism is approximately 352.77 cm^2.
whats the first one?
The first part of the equation calculates the area of the triangular base of the prism. The triangular base is formed by the two right triangles on the ends of the prism. To find the area of one of these triangles, you use the formula for the area of a triangle - (1/2) * base * height.
In this case, the base of the triangle is 12cm, and you previously calculated the height to be 6√3 cm.
So, the area of one triangular base is:
(1/2) * 12 * 6√3 = 36√3 cm^2
Since there are two triangular bases on the prism, you multiply this area by 2 to get the total area of the triangular bases which is 2 * 36√3 = 72√3 cm^2.