A cubical block of side 14 cm is surmounted by a hemisphere of radius 7 cm. What is the total surface area of the solid thus formed ? (take π = 22/7)
1330 cm 2
Calculating Total Surface Area of the Combined Solid
The problem asks us to find the total surface area of a solid formed by placing a hemisphere on top of a cubical block. We are given the side length of the cube and the radius of the hemisphere.
Understanding the Solid and its Surface Area
The solid consists of a cube with a hemisphere on its top face. To find the total surface area of this combined solid, we need to consider the areas that are exposed to the outside.
- The cube has 6 faces. However, one face (the top one) is partially covered by the base of the hemisphere.
- The surface area includes the area of 5 faces of the cube.
- It also includes the area of the part of the top face of the cube that is *not* covered by the hemisphere.
- Finally, it includes the curved surface area of the hemisphere.
Alternatively, we can think of it as:
- Total surface area of the cube initially (\(6 \times \text{side}^2\)).
- Minus the area of the base of the hemisphere (which is on the cube's face and is not exposed).
- Plus the curved surface area of the hemisphere.
So, Total Surface Area of Solid = (Total Surface Area of Cube) - (Area of base of Hemisphere) + (Curved Surface Area of Hemisphere)
We know the formula for the Curved Surface Area (CSA) of a hemisphere is \(2\pi r^2\) and the area of its base (a circle) is \(\pi r^2\). The total surface area of a cube is \(6 \times \text{side}^2\).
Substituting these into the formula:
Total Surface Area of Solid = \(6 \times \text{side}^2 - \pi r^2 + 2\pi r^2\)
Total Surface Area of Solid = \(6 \times \text{side}^2 + \pi r^2\)
Given Information
- Side of the cubical block, \(s = 14\) cm
- Radius of the hemisphere, \(r = 7\) cm
- Value of \(\pi = 22/7\)
Note that the diameter of the hemisphere (\(2 \times 7\) cm = 14 cm) is equal to the side length of the cube. This means the base of the hemisphere fits exactly onto the top face of the cube.
Step-by-Step Calculation
Let's calculate the areas required:
- Calculate the total surface area of the cube:
- Area of cube = \(6 \times s^2\)
- Area of cube = \(6 \times (14 \text{ cm})^2\)
- Area of cube = \(6 \times 196 \text{ cm}^2\)
- Area of cube = \(1176 \text{ cm}^2\)
- Calculate the area of the base of the hemisphere:
- Area of base = \(\pi r^2\)
- Area of base = \((22/7) \times (7 \text{ cm})^2\)
- Area of base = \((22/7) \times 49 \text{ cm}^2\)
- Area of base = \(22 \times 7 \text{ cm}^2\)
- Area of base = \(154 \text{ cm}^2\)
- Calculate the curved surface area of the hemisphere:
- CSA of hemisphere = \(2\pi r^2\)
- CSA of hemisphere = \(2 \times (22/7) \times (7 \text{ cm})^2\)
- CSA of hemisphere = \(2 \times 154 \text{ cm}^2\)
- CSA of hemisphere = \(308 \text{ cm}^2\)
Now, use the formula for the total surface area of the solid:
Total Surface Area of Solid = (TSA of Cube) - (Area of base of Hemisphere) + (CSA of Hemisphere)
Total Surface Area of Solid = \(1176 \text{ cm}^2 - 154 \text{ cm}^2 + 308 \text{ cm}^2\)
Total Surface Area of Solid = \(1022 \text{ cm}^2 + 308 \text{ cm}^2\)
Total Surface Area of Solid = \(1330 \text{ cm}^2\)
Alternatively, using the simplified formula: Total Surface Area of Solid = \(6 \times s^2 + \pi r^2\)
Total Surface Area of Solid = \(6 \times (14 \text{ cm})^2 + (22/7) \times (7 \text{ cm})^2\)
Total Surface Area of Solid = \(6 \times 196 \text{ cm}^2 + (22/7) \times 49 \text{ cm}^2\)
Total Surface Area of Solid = \(1176 \text{ cm}^2 + 22 \times 7 \text{ cm}^2\)
Total Surface Area of Solid = \(1176 \text{ cm}^2 + 154 \text{ cm}^2\)
Total Surface Area of Solid = \(1330 \text{ cm}^2\)
Both methods yield the same result.
Final Answer
The total surface area of the solid thus formed is \(1330 \text{ cm}^2\).
Revision Table: Surface Area Formulas
| Shape | Formula | Notes |
|---|---|---|
| Cube | \(6 \times \text{side}^2\) | Total Surface Area |
| Hemisphere | \(2\pi r^2\) | Curved Surface Area (CSA) |
| Hemisphere | \(3\pi r^2\) | Total Surface Area (CSA + Base Area) |
| Circle | \(\pi r^2\) | Area of the base of the hemisphere |
Additional Information: Combining Solids
When different solid shapes are combined, the total surface area of the resulting solid is the sum of the exposed areas of the individual shapes. Areas that are hidden or covered when the shapes are joined are subtracted from the total surface area.
In this specific problem, the base of the hemisphere is placed on the top face of the cube. The area where they join (\(\pi r^2\)) is no longer part of the exposed surface. Thus, we subtract the area of the hemisphere's base from the total surface area of the cube and the total surface area of the hemisphere (which includes its base if considering TSA of hemisphere itself). A simpler way is to take the surface area of the cube (including the top face), subtract the area covered by the hemisphere's base, and add the curved surface area of the hemisphere.
This can be visualized as:
- Area of 5 faces of the cube: \(5 \times s^2\)
- Area of the exposed part of the top face: \(s^2 - \pi r^2\)
- Curved surface area of the hemisphere: \(2\pi r^2\)
Total Area = \(5s^2 + (s^2 - \pi r^2) + 2\pi r^2 = 6s^2 + \pi r^2\), which is the formula we used.
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