Which measurements are closest to the lateral and total surface area of the triangular prism in square centimeters? The lateral surface area of the triangular prism is (Select) cm²... Which measurements are closest to the lateral and total surface area of the triangular prism in square centimeters? The lateral surface area of the triangular prism is (Select) cm² and the total surface area of the triangular prism is (Select) cm².
Understand the Problem
The question is asking for the lateral and total surface area of a triangular prism using the dimensions provided in the diagram. We will calculate the lateral surface area and total surface area based on the given measurements.
Answer
The lateral surface area is approximately $112.75 \, \text{cm}^2$ and the total surface area is approximately $138.25 \, \text{cm}^2$.
Answer for screen readers
The lateral surface area of the triangular prism is approximately $112.75 , \text{cm}^2$ and the total surface area is approximately $138.25 , \text{cm}^2$.
Steps to Solve
- Calculate the Area of the Triangular Base
To find the lateral surface area, we first need to calculate the area of the triangular base. The formula for the area (A) of a triangle is given by:
$$ A = \frac{1}{2} \times \text{base} \times \text{height} $$
Here, the base is (b = 11 \frac{2}{5} , \text{cm}) (which can be converted to an improper fraction ( \frac{57}{5} , \text{cm})) and the height is (h = 7 \frac{1}{2} , \text{cm}) (converted to an improper fraction (\frac{15}{2} , \text{cm})).
Substituting these values in:
$$ A = \frac{1}{2} \times \frac{57}{5} \times \frac{15}{2} $$
- Calculate the Lateral Surface Area
The lateral surface area (LSA) of a prism can be calculated using the formula:
$$ LSA = \text{Perimeter of the base} \times \text{height of the prism} $$
The perimeter (P) of the triangular base can be calculated as:
$$ P = 11 \frac{2}{5} + 7 \frac{1}{2} + 7 \frac{1}{2} $$
Convert the fractions to improper fractions and sum them up, and then multiply by the height of the prism, which is (h = 2 , \text{cm}):
$$ LSA = P \times 2 $$
- Calculate the Total Surface Area
The total surface area (TSA) can be found using the formula:
$$ TSA = LSA + 2 \times A $$
Substitute the values for LSA calculated previously and the area of the base:
$$ TSA = LSA + 2 \times A $$
The lateral surface area of the triangular prism is approximately $112.75 , \text{cm}^2$ and the total surface area is approximately $138.25 , \text{cm}^2$.
More Information
The lateral surface area gives the area covering the sides, while the total surface area includes both the lateral area and the area of the two triangular bases.
Tips
- Not converting mixed numbers to improper fractions before calculations.
- Forgetting to multiply the area of the triangle by 2 when calculating total surface area.
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