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4. حساب الحجم والمساحة الكلية للقطع .  ضع أيضًا كيف حسبت. 
 أ. يتكون المبنى من قاعة وجزء مكتبي ملحق مبني على الجدار الجانبي.  يبلغ طول القاعة 25 مترا وارتفاعها 5 أمتار وعرضها 6 أمتار.  الجزء المكتبي طوله 5 أمتار وارتفاعه 3 أمتار وعرضه 4 أمتار.  (المساحة الإجمالية: المبنى مدهون من الخارج، ما هي المساحة المراد دهانها؟)

Let's break down the problem into parts:

### Part A: Volume Calculation
The building consists of two rectangular parts:
1. **Main Hall** (larger section)
   - Length: 25 m
   - Height: 5 m
   - Width: 6 m

2. **Office Section** (smaller section)
   - Length: 5 m
   - Height: 3 m
   - Width: 4 m

The **volume of a rectangular prism** is given by the formula:
\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]

#### Volume of the Hall:
\[
V_{\text{hall}} = 25 \times 6 \times 5 = 750 \, \text{m}^3
\]

#### Volume of the Office Section:
\[
V_{\text{office}} = 5 \times 4 \times 3 = 60 \, \text{m}^3
\]

### Total Volume:
\[
V_{\text{total}} = V_{\text{hall}} + V_{\text{office}} = 750 + 60 = 810 \, \text{m}^3
\]

### Part B: Surface Area to be Painted
The surface area is the **total exterior surface area** of both parts. We need to find the surface area for each part separately and then sum them, making sure to subtract the area where the two sections intersect.

#### Surface Area of the Hall:
The surface area of a rectangular prism is given by:
\[
SA = 2 \times (\text{Length} \times \text{Width} + \text{Length} \times \text{Height} + \text{Width} \times \text{Height})
\]

For the hall:
\[
SA_{\text{hall}} = 2 \times (25 \times 6 + 25 \times 5 + 6 \times 5) = 2 \times (150 + 125 + 30) = 2 \times 305 = 610 \, \text{m}^2
\]

#### Surface Area of the Office Section:
\[
SA_{\text{office}} = 2 \times (5 \times 4 + 5 \times 3 + 4 \times 3) = 2 \times (20 + 15 + 12) = 2 \times 47 = 94 \, \text{m}^2
\]

### Subtract the Intersection Area:
The intersection occurs on the side where the office is attached to the hall. The area of this intersection is the side of the office that touches the hall, which is:
\[
\text{Intersection Area} = \text{Height of Office} \times \text{Width of Office} = 3 \times 4 = 12 \, \text{m}^2
\]

### Total Surface Area to be Painted:
\[
SA_{\text{total}} = SA_{\text{hall}} + SA_{\text{office}} - \text{Intersection Area}
\]
\[
SA_{\text{total}} = 610 + 94 - 12 = 692 \, \text{m}^2
\]

### Final Answer:
- **Total Volume**: 810 m³
- **Total Surface Area to be Painted**: 692 m²

Would you like further details or have any questions about the process? 

Here are five related questions to explore this concept further:
1. How does the surface area change if the height of the office section increases by 1 meter?
2. What would be the new volume if the hall's width was reduced by 2 meters?
3. How do we calculate surface area if the office is cylindrical instead of rectangular?
4. How much paint would be required if 1 liter covers 10 m²?
5. How does the intersection area affect the total surface area?

**Tip**: When dealing with complex 3D shapes, break them down into simpler parts (like cubes or prisms) for easier calculations.

4. حساب الحجم والمساحة الكلية للقطع. ضع أيضًا كيف حسبت. أ. يتكون المبنى من قاعة وجزء مكتبي ملحق مبني على الجدار الجانبي. يبلغ طول القاعة 25 مترا وارتفاعها 5 أمتار وعرضها 6 أمتار. الجزء المكتبي طوله 5 أمتار وارتفاعه 3 أمتار وعرضه 4 أمتار. (المساحة الإجمالية: المبنى مدهون من الخارج، ما هي المساحة المراد دهانها؟)

Calculate the volume and total surface area of a building consisting of two rectangular parts. The main hall measures 25m by 6m by 5m, and the office section measures 5m by 4m by 3m. Learn how to find the volume and the total area to be painted.

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