Question

Two identical solid hemispheres of equal base radius r cm are stuck together along their bases. The total surface area of the combination is $6{\mathrm{\pi r}}^2$. Write ‘True’ or ‘False’ and justify your answer.

• A. True
• B. False

Answer 1

The correct option is B False

Sphere:

A three-dimensional rounded, solid figure with equal distance from its center to any point on its surface is called a sphere.

Hemisphere:

A three-dimensional solid figure of exactly half of the sphere is called a hemisphere.

Calculating the total surface area of the combination:

The radius of two identical hemispheres is given as r cm.

Curved surface area of the hemisphere$=2{\mathrm{\pi r}}^2$ square units

The base area of the hemisphere is in the shape of a circle.

Base area of the hemisphere$={\mathrm{\pi r}}^2$ square units

Total surface area of hemisphere$=2{\mathrm{\pi r}}^2+{\mathrm{\pi r}}^2$

$\Rightarrow =3{\mathrm{\pi r}}^2$ square units

Total surface area of two identical hemispheres$=2\left(3{\mathrm{\pi r}}^2\right)$

$\Rightarrow =6{\mathrm{\pi r}}^2$square units

The two identical hemispheres are stuck together.

If the two identical hemispheres combine, it forms a sphere.

Surface area of the sphere$=2$(Surface area of two identical hemispheres)$-2$(Base area of the hemisphere)

$$\begin{gathered} \Rightarrow =6{\mathrm{\pi r}}^2-2\left({\mathrm{\pi r}}^2\right) \\ \Rightarrow =4{\mathrm{\pi r}}^2 \end{gathered}$$

Thus, the total surface area of the combination is $4{\mathrm{\pi r}}^2$ square units.

Hence, the given statement is false.