https://file.aiquickdraw.com/mathbot/file/1727371611498y9gsg9rr.jpg

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!

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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!

Is Rain correct? Use the drop-down menus to explain your reasoning for the surface area of the triangular pyramid.

Learn how to calculate the surface area of a triangular pyramid with base dimensions of 10 inches and height of 8.7 inches. Includes step-by-step solutions for calculating the areas of the base and lateral triangular faces. Suitable for students in Grades 7-9.

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