Multiple choice

A solid sphere of radius $r$ is sliced by the planes passing through its centre and perpendicular to each other. The total surface area of each of the pieces so formed is

  1. $\displaystyle \frac{2}{3}\pi r^{2}$
  2. $\displaystyle \pi r^{2}$
  3. $\displaystyle \frac{4}{3}\pi r^{2}$
  4. $\displaystyle 2\pi r^{2}$
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D Correct answer
Explanation

Slicing a sphere with two perpendicular planes through the center divides it into four equal parts. Each part has a surface area consisting of one-fourth of the sphere's surface (pi * r^2) plus two semicircular faces (each pi * r^2 / 2), totaling 2 * pi * r^2.

AI explanation

When the sphere is cut by two perpendicular planes through its center, it forms 4 identical pieces. The total surface area of one piece includes one quarter of the original sphere surface area plus two flat semicircular surfaces. The curved part area is (1/4) * 4 * pi * r^2 = pi * r^2, and the two flat surfaces add up to pi * r^2. The total surface area is therefore pi * r^2 + pi * r^2, resulting in 2 * pi * r^2.

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