帮我解释一下这道题---**Textual Information:**
**Question Stem:**
3. 如图是一个几何体的三视图, 根据图中数据, 可得该几何体的表面积是 ( )
(Translation: As shown in the figure is the three views of a geometric solid. According to the data in the figure, the surface area of this geometric solid is ( ))
**Options:**
A. 9π
B. 10π
C. 11π
D. 12π
**Other Relevant Text:**
俯视图 (Top view)
正(主)视图 (Front view)
侧(左)视图 (Left view)
**Chart/Diagram Description:**
**Type:** Orthographic projection (Three views) of a geometric solid.
**Main Elements:**
The image shows three views:
1. **俯视图 (Top view):** A single circle in the center.
2. **正(主)视图 (Front view):** Two shapes stacked vertically. The lower shape is a rectangle. The upper shape is a circle. Vertical dimension lines are shown: a dimension of 2 for the height of the circle, and a dimension of 3 for the height of the rectangle. A horizontal dimension line below the rectangle shows a width of 2.
3. **侧(左)视图 (Left view):** Two shapes stacked vertically, identical to the Front view. The lower shape is a rectangle with height 3 and width 2. The upper shape is a circle with height (diameter) 2.
**Interpretation of the Solid:**
Based on the three views:
- The Top view being a circle indicates that the base of the lower part is circular.
- The Front and Left views show a rectangle below a circle. The rectangle suggests a cylindrical or prismatic shape. The circle suggests a sphere or part of a sphere.
- The consistency of the Front and Left views indicates rotational symmetry around a vertical axis.
- Combining the views, the solid is a combination of a cylinder and a sphere.
- From the Front/Left views, the lower part is a cylinder with height 3 and diameter 2 (radius 1).
- From the Front/Left views, the upper part is a sphere with diameter 2 (radius 1), placed on top of the cylinder. The Top view (a single circle) is consistent with a sphere on top of a cylinder base of the same diameter, where the sphere's equatorial circle matches the cylinder's top circle, and the sphere's projection in the top view is just that circle.
**Dimensions:**
- Cylinder height: 3
- Cylinder diameter: 2 (Radius R_cylinder = 1)
- Sphere diameter: 2 (Radius R_sphere = 1)
视频信息
答案文本
视频字幕
Let's analyze this three views problem step by step. We have the front view showing a rectangle below a circle, the left view identical to the front view, and the top view showing just a circle. From these views, we can identify that this is a cylinder with a sphere on top.
Now let's construct the 3D solid from the views. The front and left views show a rectangle with height 3 and width 2, topped by a circle with diameter 2. The top view shows a single circle. This tells us we have a cylinder with height 3 and radius 1, with a sphere of radius 1 placed on top.
Now let's identify the surface area components. The external surface consists of three parts: the bottom base of the cylinder with area pi, the lateral surface of the cylinder with area 6 pi, and the complete sphere surface with area 4 pi. Note that the top of the cylinder is not part of the external surface since it's covered by the sphere.
Now let's calculate the total surface area. We add all the components: the bottom base area of pi, plus the lateral cylinder area of 6 pi, plus the sphere surface area of 4 pi. This gives us pi plus 6 pi plus 4 pi equals 11 pi. Therefore, the answer is C, 11 pi.
To summarize, when solving three views problems: first analyze each view to understand the shape, then identify the 3D solid from the combined views, extract dimensions from measurements, calculate only the external surface areas, and sum all components. In this problem, we identified a cylinder with a sphere on top, calculated the surface area as 11 pi, making the correct answer option C.