The total surface area of a cube with edge 'a' is given by the formula 6a². We substitute the given edge length to find this area.

The decorative block shown in Fig. 12.7 is made of two solids — a cube and a hemisphere. The base of the block is a cube with edge 5 cm, and the hemisphere fixed on the top has a diameter of 4.2 cm. Find the total surface area of the block. (Take $\pi = \frac{22}{7}$ )

Answer: The total surface area of the block is $163.86 \text{ cm}^2$ .

Step-by-step solution

Step 1: Identify given dimensions

We are given the edge length of the cube and the diameter of the hemisphere. These are the fundamental dimensions required to calculate surface areas.

Step 2: Calculate hemisphere radius

The radius of the hemisphere is half of its diameter. This value will be used in the surface area formulas for the hemisphere.

Step 3: Calculate total surface area of the cube

The total surface area of a cube with edge 'a' is given by the formula $6a^2$ . We substitute the given edge length to find this area.

Step 4: Identify area covered by hemisphere base

When the hemisphere is placed on the cube, its circular base covers a portion of the cube's top surface. This covered area cannot be part of the total exposed surface area of the block.

Step 5: Calculate area covered by hemisphere base (numerical)

We calculate the numerical value of the area of the circular base using the radius found earlier and the given value of $\pi$ .

Step 6: Calculate curved surface area of the hemisphere

The curved surface area of the hemisphere is part of the total exposed surface area of the block. The formula for the curved surface area of a hemisphere is $2\pi r^2$ .

Step 7: Calculate curved surface area of the hemisphere (numerical)

We substitute the values to find the numerical curved surface area of the hemisphere. Notice that $2\pi r^2$ is simply twice the area of its base $\pi r^2$ .

Step 8: Calculate total surface area of the block

The total surface area of the block is the surface area of the cube, minus the area of the cube's top face covered by the hemisphere, plus the curved surface area of the hemisphere. This ensures we count only the exposed surfaces.

Step 9: Final calculation of total surface area

Performing the final addition and subtraction, we arrive at the total surface area of the decorative block.

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The decorative block shown in Fig. 12.7 is made of two solids — a cube and a hemisphere. The base of the block is a cube with edge 5 cm, and the hemisphere fixed on the top has a diameter of 4.2 cm. Find the total surface area of the block. (Take π=227\pi = \frac{22}{7}π=722​)