I hope I can explain this. I can't draw the shape here, but it is a triangular prism that looks like a wedge of cheese. The height is 20 cm and the three sides are 13, 37, and 40 cm. There is also a line drawn from the corner of the two sides that are 13 and 37 cm, that makes a right angle with the side of the triangle that is 40 cm. It is 12 cm. (It is just inside the side that is 13 cm long.) I need to find the surface area and round to the nearest whole number.
For surface area, you find the area of each side (Length x Width) and then you add up all the sides.
To find the surface area of the triangular prism, we need to calculate the areas of the triangular faces and the rectangular faces.
1. Start by calculating the area of the triangular face with sides 13, 37, and 40 cm. To do this, we can use Heron's formula. The semiperimeter (s) is calculated by adding half of each side length:
s = (13 + 37 + 40) / 2 = 45
Next, we can plug the values into the formula:
area = √(s * (s - 13) * (s - 37) * (s - 40))
area ≈ √(45 * (45 - 13) * (45 - 37) * (45 - 40))
area ≈ √(45 * 32 * 8 * 5)
area ≈ √(57,600) ≈ 240 cm²
2. Now, let's calculate the area of the rectangular face. The length of this rectangle is 40 cm, and the width is the height of the prism, which is 20 cm:
area = length * width = 40 cm * 20 cm = 800 cm²
3. Finally, we calculate the area of the other triangular face. One side of this triangle is the same as the side of the rectangle (40 cm), and the other two sides are the hypotenuse (37 cm) and the given line (12 cm). Since this is a right-angled triangle, we can use the formula for the area of a right-angled triangle:
area = (1/2) * base * height = (1/2) * 40 cm * 12 cm = 240 cm²
4. To find the total surface area, we sum up the areas of all the faces:
Total surface area = 2 * (area of triangular face) + (area of rectangular face)
Total surface area = 2 * 240 cm² + 800 cm²
Total surface area ≈ 480 cm² + 800 cm² ≈ 1280 cm²
Rounding to the nearest whole number, the surface area of the triangular prism is approximately 1280 cm².