To determine whether Rain's calculations are correct, let's analyze the steps.

Given Information:

• Triangular Pyramid Dimensions:
  • Each triangular face has a base of 10 inches and a height of 13 inches.
  • The base of the pyramid is a triangle with a base of 10 inches and a height of 8.7 inches.

Step 1: Surface Area Calculation

The surface area of a triangular pyramid is calculated by adding the areas of the triangular base and the three triangular lateral faces.

Lateral Faces:

The area of a triangle is given by: $\text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

For each of the three lateral faces:

• Base = 10 inches
• Height = 13 inches

$\text{Area of one lateral face} = \frac{1}{2} \times 10 \times 13 = 65 \, \text{square inches}$

Since there are 3 lateral faces: $\text{Total lateral area} = 3 \times 65 = 195 \, \text{square inches}$

Base Triangle:

The area of the base triangle is calculated similarly, but with the height being 8.7 inches: $\text{Area of base triangle} = \frac{1}{2} \times 10 \times 8.7 = 43.5 \, \text{square inches}$

Step 2: Total Surface Area

Now, summing the lateral areas and the base area: $\text{Total surface area} = 195 + 43.5 = 238.5 \, \text{square inches}$

Verification of Rain's Calculation:

Rain's calculation was: $3 \left( \frac{1}{2} \times 10 \times 13 \right) + \left( \frac{1}{2} \times 10 \times 8.7 \right) = 238.5 \, \text{square inches}$

This is correct. Therefore, Rain calculated the surface area accurately.

Let me know if you'd like further clarifications!

Here are 5 questions related to this explanation:

1. How do you calculate the area of a triangular face?
2. Why do we multiply by 3 when calculating the lateral surface area?
3. What formula is used to find the area of a triangle?
4. How would the surface area change if the base of the pyramid was different?
5. Can the surface area of a pyramid ever be less than the area of its base?

Tip: Always double-check units when calculating areas, especially in geometry!