Find the surface area of the triangular prism shown below.
Understand the Problem
The question is asking to calculate the surface area of a triangular prism based on the dimensions provided in the image. To solve this, we'll use the formula for the surface area of a prism, which includes calculating the areas of the triangular bases and the rectangular sides.
Answer
The surface area of the triangular prism is $544 \, \text{units}^2$.
Answer for screen readers
The surface area of the triangular prism is $544 , \text{units}^2$.
Steps to Solve
- Identify the dimensions
From the image, we have:
- Length of the prism (height) = 14 units
- Base of the triangular face = 12 units
- Height of the triangular face = 8 units
- The other two sides of the triangle = 10 units each (isosceles triangle).
- Calculate the area of the triangular base
To find the area of a triangle, we use the formula: $$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$
Substituting the values: $$ \text{Area} = \frac{1}{2} \times 12 \times 8 = 48 , \text{units}^2 $$
- Calculate the perimeter of the triangular base
The perimeter of the triangular base can be calculated as: $$ \text{Perimeter} = \text{side}_1 + \text{side}_2 + \text{base} = 10 + 10 + 12 = 32 , \text{units} $$
- Calculate the lateral surface area
The lateral surface area is calculated as the product of the perimeter of the base and the length of the prism: $$ \text{Lateral Surface Area} = \text{Perimeter} \times \text{Length} $$ $$ \text{Lateral Surface Area} = 32 \times 14 = 448 , \text{units}^2 $$
- Calculate the total surface area
Finally, the total surface area of the prism is the sum of the lateral surface area and twice the area of the triangular base: $$ \text{Total Surface Area} = \text{Lateral Surface Area} + 2 \times \text{Area of the Base} $$ $$ \text{Total Surface Area} = 448 + 2 \times 48 = 448 + 96 = 544 , \text{units}^2 $$
The surface area of the triangular prism is $544 , \text{units}^2$.
More Information
The calculated surface area combines the areas of the triangular bases with the lateral surfaces, providing a complete measurement of the outside of the prism. Understanding how to break down the calculations into individual components is key to successfully solving these types of problems.
Tips
- Forgetting to multiply the area of the triangular base by 2 when calculating total surface area.
- Miscalculating the perimeter of the triangle by not including all three sides.
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