# Difficult Geometry & Mensuration Solved QuestionAptitude Discussion

 Q. A sphere of radius $r$ is cut by a plane at a distance of $h$ from its center, thereby breaking this sphere into two different pieces. The cumulative surface area of these two pieces is 25% more than that of the sphere.  What is the value of $h$?
 ✔ A. $\dfrac{r}{\sqrt{2}}$ ✖ B. $\dfrac{r}{\sqrt{5}}$ ✖ C. $\dfrac{r}{\sqrt{7}}$ ✖ D. $\dfrac{r}{\sqrt{11}}$

Solution:
Option(A) is correct

Area  $= 4 \pi r^2$

Cumulative area of the two pieces $= 25\%$ more than the by square.

Area of 2 pieces $= 1.25 × \pi r^2 = 5 \pi r^2$

Extra area $= \pi r^2$

Extra area =  area of two new circles that are now created circles.

Area each new circle $= \dfrac{\pi r^2}{2}$

Let radius of new circle be $r_1$.

Now, $\pi r_1^2 = \dfrac{\pi r^2}{2}$

$r_1 = \dfrac{r}{\sqrt{2}}$

Now, $r_1$, $h$ and $r$ form a right angled triangle.

$h^2 + r_1^2 = r^2$

$h^2 + \left(\dfrac{r}{\sqrt{2}}\right)^2 = r^2$

$h = \dfrac{r}{ \sqrt{2}}$

## (3) Comment(s)

Ankit Aggarwal
()

Replace square by sphere in third line.

Blabla
()

sphere ka circle ho gya!

Shresth
()

Since it has been cut into two halves, sphere has made way for two circles having equal areas at the cutting plane.

P.S. above statement has been made assuming you can imagine the final shape.