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Math  /  Geometry

QuestionA triangular pyramid and its net are shown. Rain uses the calculations below to conclude that the surface area of the triangular pyramid is 238.5 square inches. 3(12×10×13)+(12×10×8.7)3(65)+43.5238.5\begin{array}{c} 3\left(\frac{1}{2} \times 10 \times 13\right)+\left(\frac{1}{2} \times 10 \times 8.7\right) \\ 3(65)+43.5 \\ 238.5 \end{array}
Is Rain correct? Use the drop-down menus to explain your reasoning.

Studdy Solution

STEP 1

1. The triangular pyramid has a base and three identical triangular faces.
2. The base of the pyramid is a triangle with a base of 10 inches and a height of 8.7 inches.
3. Each of the three triangular faces has a base of 10 inches and a height of 13 inches.
4. The surface area of the pyramid is the sum of the areas of the base and the three triangular faces.

STEP 2

1. Calculate the area of the base triangle.
2. Calculate the area of one triangular face.
3. Calculate the total surface area of the pyramid.
4. Verify Rain's calculations.

STEP 3

Calculate the area of the base triangle using the formula for the area of a triangle:
Area of base=12×base×height \text{Area of base} = \frac{1}{2} \times \text{base} \times \text{height} =12×10×8.7 = \frac{1}{2} \times 10 \times 8.7 =43.5 square inches = 43.5 \text{ square inches}

STEP 4

Calculate the area of one triangular face using the formula for the area of a triangle:
Area of one face=12×base×height \text{Area of one face} = \frac{1}{2} \times \text{base} \times \text{height} =12×10×13 = \frac{1}{2} \times 10 \times 13 =65 square inches = 65 \text{ square inches}

STEP 5

Calculate the total surface area of the pyramid by summing the area of the base and the areas of the three triangular faces:
Total Surface Area=Area of base+3×Area of one face \text{Total Surface Area} = \text{Area of base} + 3 \times \text{Area of one face} =43.5+3×65 = 43.5 + 3 \times 65 =43.5+195 = 43.5 + 195 =238.5 square inches = 238.5 \text{ square inches}

STEP 6

Verify Rain's calculations:
Rain's calculations are as follows: 3(12×10×13)+(12×10×8.7) 3\left(\frac{1}{2} \times 10 \times 13\right)+\left(\frac{1}{2} \times 10 \times 8.7\right) 3(65)+43.5 3(65)+43.5 238.5 238.5
Rain's calculations are correct. The surface area of the triangular pyramid is indeed 238.5 238.5 square inches.

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