https://file.aiquickdraw.com/mathbot/file/1726407339324e127apma.jpg

Let's solve the questions step by step for the triangular prism shown in the diagram.

### Given information:
- The triangular base has a height of \( 29 \, \text{cm} \) and a base of \( 34 \, \text{cm} \).
- The length of the prism is \( 38 \, \text{cm} \).

#### Step (i) **Calculate the volume of the prism:**

The volume of a prism is given by:

\[
V = \text{Base Area} \times \text{Length}
\]

The base area of the triangular face is:

\[
\text{Base Area} = \frac{1}{2} \times \text{base of triangle} \times \text{height of triangle}
\]

Substitute the known values:

\[
\text{Base Area} = \frac{1}{2} \times 34 \, \text{cm} \times 29 \, \text{cm} = 493 \, \text{cm}^2
\]

Now, calculate the volume:

\[
V = 493 \, \text{cm}^2 \times 38 \, \text{cm} = 18,734 \, \text{cm}^3
\]

So, the volume of the prism is **18,734 cm³**.

#### Step (ii) **Calculate the length of the slanted side of the triangle:**

To find the slanted side (hypotenuse) of the triangle, we can use the Pythagorean theorem:

\[
c^2 = a^2 + b^2
\]

Where:
- \( a = 29 \, \text{cm} \) (height),
- \( b = 17 \, \text{cm} \) (half of the base, i.e., \( \frac{34}{2} \)).

Substitute the values:

\[
c^2 = 29^2 + 17^2 = 841 + 289 = 1130
\]

\[
c = \sqrt{1130} \approx 33.63 \, \text{cm}
\]

So, the length of the slanted side is approximately **33.63 cm**.

#### Step (iii) **Calculate the total surface area of the prism:**

The surface area consists of:
- The area of the two triangular faces,
- The area of the three rectangular faces.

1. **Area of the triangular faces** (2 identical triangles):
   \[
   \text{Area of one triangle} = 493 \, \text{cm}^2
   \]
   For two triangles:
   \[
   2 \times 493 = 986 \, \text{cm}^2
   \]

2. **Area of the rectangular faces**:
   - Front face: \( \text{base of the triangle} \times \text{length} = 34 \times 38 = 1292 \, \text{cm}^2 \)
   - Bottom face: \( \text{slanted side} \times \text{length} = 33.63 \times 38 = 1277.94 \, \text{cm}^2 \)
   - Back face: \( \text{height of the triangle} \times \text{length} = 29 \times 38 = 1102 \, \text{cm}^2 \)

Now, sum these areas:

\[
\text{Total Surface Area} = 986 + 1292 + 1277.94 + 1102 = 4657.94 \, \text{cm}^2
\]

So, the total surface area is approximately **4657.94 cm²**.

---

Would you

The diagram shows a triangular prism. Calculate the volume, the length of the slanted side, and the total surface area.

Learn how to calculate the volume, slanted side length, and total surface area of a triangular prism. Includes the use of Pythagoras' Theorem and surface area formulas for triangular prisms. Suitable for Grades 7-10.

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