An illustration shows a triangular prism. The front triangular face is visible. The right rectangular face is visible. The sides and edges that are not visible are indicated by dashed lines. Parts are labeled with measurements. The left side and right side of the triangle measure 17 meters. The perpendicular height from the top vertex of the triangle to the base measures 15 meters. The base of the triangle measures 16 meters. The width of the rectangular face measures 11 meters.

Question

what is the surface area of the triangular prism

Answer 1

To find the surface area of the triangular prism, we need to calculate the area of its triangular bases and the area of the three rectangular faces.

1. Area of the triangular base: The formula for the area of a triangle is:

   \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

   In this case, the base of the triangle is 16 meters and the height is 15 meters.

   \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times 16 , \text{m} \times 15 , \text{m} = \frac{1}{2} \times 240 , \text{m}^2 = 120 , \text{m}^2 \]

   Since there are two triangular bases, the total area for both triangular bases is:

   \[ \text{Total Area}_{\text{triangles}} = 2 \times 120 , \text{m}^2 = 240 , \text{m}^2 \]

2. Area of the rectangular faces: There are three rectangular faces:

   • The first rectangular face (height of the triangular prism) that corresponds to the base of the triangle:

     • Width (base of triangle) = 16 m
     • Height = 11 m (width of the rectangular face)

     \[ \text{Area}_{\text{rect1}} = \text{base length} \times \text{height} = 16 , \text{m} \times 11 , \text{m} = 176 , \text{m}^2 \]

   • The second rectangular face corresponding to the height of one side of the triangle:

     • Width = 17 m (left side of triangle)
     • Height = 11 m

     \[ \text{Area}_{\text{rect2}} = 17 , \text{m} \times 11 , \text{m} = 187 , \text{m}^2 \]

   • The third rectangular face corresponding to the height of the other side of the triangle:

     • Width = 17 m (right side of triangle)
     • Height = 11 m

     \[ \text{Area}_{\text{rect3}} = 17 , \text{m} \times 11 , \text{m} = 187 , \text{m}^2 \]

   Now we can find the total area of the three rectangular faces:

   \[ \text{Total Area}_{\text{rectangles}} = 176 , \text{m}^2 + 187 , \text{m}^2 + 187 , \text{m}^2 = 550 , \text{m}^2 \]

3. Total Surface Area of the prism: Finally, we add the areas of the triangular bases and the rectangular faces:

   \[ \text{Surface Area} = \text{Total Area}{\text{triangles}} + \text{Total Area}{\text{rectangles}} = 240 , \text{m}^2 + 550 , \text{m}^2 = 790 , \text{m}^2 \]

Therefore, the surface area of the triangular prism is 790 square meters.