Question

what is the surface area of this triangular pyramid?
3.4 cm
6.4 cm
3.4 cm
3.4 cm
2.9 cm
square centimeters
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Explanation:

Step1: Identify the base and lateral faces

The base is an equilateral triangle with side length \( 3.4 \, \text{cm} \), and the height of the base triangle is \( 2.9 \, \text{cm} \). The lateral faces are three congruent isosceles triangles with base \( 3.4 \, \text{cm} \) and slant height \( 6.4 \, \text{cm} \).

Step2: Calculate the area of the base

The area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). For the base:

$$ \text{Area of base} = \frac{1}{2} \times 3.4 \times 2.9 = 4.93 \, \text{cm}^2 $$

Step3: Calculate the area of one lateral face

For each lateral face (isosceles triangle):

$$ \text{Area of one lateral face} = \frac{1}{2} \times 3.4 \times 6.4 = 10.88 \, \text{cm}^2 $$

Step4: Calculate the total area of lateral faces

There are 3 lateral faces:

$$ \text{Total lateral area} = 3 \times 10.88 = 32.64 \, \text{cm}^2 $$

Step5: Calculate the total surface area

Add the base area and the total lateral area:

$$ \text{Total surface area} = 4.93 + 32.64 = 37.57 \, \text{cm}^2 $$

Answer:

\( 37.57 \)