Example problem calculating surface area and volume of a triangular prism with given dimensions.

Problem Analysis:

The image shows a composite 3D shape consisting of a rectangular prism and a triangular pyramid (or tetrahedron) on top. The task is to calculate the Surface Area (SA) and Volume (V) of the entire shape.

#### Given Dimensions:
- Rectangular Prism:
- Length (\( l \)) = 30 cm
- Width (\( w \)) = 24 cm
- Height (\( h \)) = 18 cm

- Triangular Pyramid:
- Base triangle sides: 17 cm, 17 cm, 30 cm (isosceles triangle)
- Height of the triangular pyramid (\( H \)) = 8 cm
- Slant height of the triangular faces = 17 cm (as shown in the diagram)

---

Step-by-Step Solution:

#### 1. Surface Area (SA) Calculation

The total surface area of the composite shape is the sum of:
- The surface area of the rectangular prism (excluding the top face, as it is covered by the triangular pyramid).
- The surface area of the triangular pyramid (including its three triangular faces).

##### a. Surface Area of the Rectangular Prism:
The formula for the surface area of a rectangular prism is:
\[
\text{SA}_{\text{rect}} = 2(lw + lh + wh)
\]
However, since the top face is covered by the triangular pyramid, we exclude the top face (\( lw \)) from the calculation.

\[
\text{SA}_{\text{rect}} = 2(lh + wh) + lw_{\text{(bottom only)}}
\]

Substitute the given dimensions:
\[
\text{SA}_{\text{rect}} = 2(30 \times 18 + 24 \times 18) + 30 \times 24
\]
\[
\text{SA}_{\text{rect}} = 2(540 + 432) + 720
\]
\[
\text{SA}_{\text{rect}} = 2(972) + 720
\]
\[
\text{SA}_{\text{rect}} = 1944 + 720 = 2664 \, \text{cm}^2
\]

##### b. Surface Area of the Triangular Pyramid:
The triangular pyramid has four triangular faces:
1. One base triangle.
2. Three lateral triangular faces.

###### i. Base Triangle Area:
The base triangle is an isosceles triangle with sides 17 cm, 17 cm, and 30 cm. We use Heron's formula to find its area.

First, calculate the semi-perimeter (\( s \)):
\[
s = \frac{17 + 17 + 30}{2} = 32 \, \text{cm}
\]

Now, apply Heron's formula:
\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \( a = 17 \), \( b = 17 \), and \( c = 30 \):
\[
\text{Area} = \sqrt{32(32-17)(32-17)(32-30)}
\]
\[
\text{Area} = \sqrt{32 \times 15 \times 15 \times 2}
\]
\[
\text{Area} = \sqrt{32 \times 450} = \sqrt{14400} = 120 \, \text{cm}^2
\]

###### ii. Lateral Triangle Areas:
Each lateral face is a triangle with a base of 30 cm and a slant height of 17 cm. The area of one lateral triangle is:
\[
\text{Area}_{\text{lateral}} = \frac{1}{2} \times \text{base} \times \text{slant height}
\]
\[
\text{Area}_{\text{lateral}} = \frac{1}{2} \times 30 \times 17 = 255 \, \text{cm}^2
\]

Since there are three lateral faces:
\[
\text{Total lateral area} = 3 \times 255 = 765 \, \text{cm}^2
\]

###### iii. Total Surface Area of the Triangular Pyramid:
\[
\text{SA}_{\text{pyramid}} = \text{Base area} + \text{Lateral area}
\]
\[
\text{SA}_{\text{pyramid}} = 120 + 765 = 885 \, \text{cm}^2
\]

##### c. Total Surface Area of the Composite Shape:
The total surface area is the sum of the surface area of the rectangular prism (excluding the top face) and the surface area of the triangular pyramid:
\[
\text{Total SA} = \text{SA}_{\text{rect}} + \text{SA}_{\text{pyramid}}
\]
\[
\text{Total SA} = 2664 + 885 = 3549 \, \text{cm}^2
\]

---

#### 2. Volume (V) Calculation

The total volume of the composite shape is the sum of:
- The volume of the rectangular prism.
- The volume of the triangular pyramid.

##### a. Volume of the Rectangular Prism:
The formula for the volume of a rectangular prism is:
\[
V_{\text{rect}} = l \times w \times h
\]
Substitute the given dimensions:
\[
V_{\text{rect}} = 30 \times 24 \times 18
\]
\[
V_{\text{rect}} = 12960 \, \text{cm}^3
\]

##### b. Volume of the Triangular Pyramid:
The formula for the volume of a pyramid is:
\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
We already calculated the base area of the triangular pyramid as 120 cm², and the height (\( H \)) is 8 cm:
\[
V_{\text{pyramid}} = \frac{1}{3} \times 120 \times 8
\]
\[
V_{\text{pyramid}} = \frac{1}{3} \times 960 = 320 \, \text{cm}^3
\]

##### c. Total Volume of the Composite Shape:
\[
\text{Total V} = V_{\text{rect}} + V_{\text{pyramid}}
\]
\[
\text{Total V} = 12960 + 320 = 13280 \, \text{cm}^3
\]

---

Final Answers:

\[
\boxed{3549 \, \text{cm}^2, 13280 \, \text{cm}^3}
\]