To calculate the surface area of a triangular pyramid (or tetrahedron), we need to find the area of the base and the areas of the three triangular faces.
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Calculate the area of the triangular base.
The base is a triangle, and we can use the formula for the area of a triangle: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Here, the base is 5 inches and the height is 4.3 inches. \[ A = \frac{1}{2} \times 5 \times 4.3 = \frac{21.5}{2} = 10.75 \text{ in}^2 \] -
Calculate the area of the three triangular sides. Each side can be calculated using the formula for the area of a triangle: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] The base of each triangular side is the same as the base of the pyramid (5 in.) and the height of the triangular sides is the slant height (6 in.). However, when calculating the area of each triangular face, we should consider the triangular face forms different shapes based on the divisions of the base.
The height of the slant triangular faces can be found using the height from the apex to the midpoint of the base. The slant height will not act as the height of the triangle face but instead provide the distance from the apex to the base.
Each triangular face has a base of 5 in and the height from the apex can be calculated using the Pythagorean theorem for the right triangle formed by:
- half the base (half of 5 in = 2.5 in)
- the height is defined directly to slant as 6 in.
\[ A_f = \frac{1}{2} \times \text{base} \times \text{slant height} \] This gives us the area of one triangular face as: \[ A_f = \frac{1}{2} \times 5 \times 6 = 15 \text{ in}^2 \] Since there are 3 identical triangular sides, we multiply this by 3: \[ A_{faces} = 3 \times 15 = 45 \text{ in}^2 \]
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Calculate total surface area. Now, we can add the area of the base to the area of the triangular faces: \[ \text{Surface Area} = A_{base} + A_{faces} = 10.75 + 45 = 55.75 \text{ in}^2 \]
Therefore, the surface area of the chocolate pyramid is 55.75 in².