the base of a solid right prism is a triangle whose sides are 9 cm,12cm,15cm. the height of the prism is 5 cm. then find the total surface area of the prism is?
To find the total surface area of a prism, we need to calculate the area of each face and then add them together.
In this case, the base of the prism is a triangle with sides measuring 9 cm, 12 cm, and 15 cm. To calculate the area of this triangle, we can use Heron's formula:
Step 1: Calculate the semi-perimeter (s) of the triangle:
s = (9 + 12 + 15) / 2 = 36 / 2 = 18 cm
Step 2: Calculate the area (A) of the triangle using Heron's formula:
A = √(s * (s - 9) * (s - 12) * (s - 15))
Now, let's calculate the area of the base (A_base) of the prism using the values we have:
A_base = √(18 * (18 - 9) * (18 - 12) * (18 - 15))
= √(18 * 9 * 6 * 3)
= √(2916)
= 54 cm² (approx.)
Next, we calculate the area of the lateral faces. Since a right prism has rectangular lateral faces, the area of each face is equal to the product of the length and height of the prism.
A_lateral = 2 * (length * height + width * height)
= 2 * (9 * 5 + 12 * 5)
= 2 * (45 + 60)
= 2 * 105
= 210 cm²
Finally, add the areas of the base and the lateral faces together to find the total surface area (TSA) of the prism:
TSA = A_base + A_lateral
= 54 + 210
= 264 cm²
Therefore, the total surface area of the prism is 264 cm².