Question

How do you find the area of the cross-section of a sphere formed by a plane intersecting the sphere at an equator, if the radius of the sphere is 11 inches?

Answer 1

Hint: Here the cross-section of the sphere which is formed by a plane that intersects the sphere at an equator will be a circle of the same radius as that of the sphere. So to find the area of the cross-section, we will find the area of the circle of the same radius as that of the sphere by using its formula.

Formula used:
Area of the circle \[ = \pi {r^2}\], where \[r\] is the radius of the circle.

Complete step by step solution:
Here we need to find the area of the cross-section of the sphere of a given radius.
Here the cross-section of the sphere which is formed by a plane intersecting the sphere at an equator will be a circle of the same radius as that of the sphere and the given radius of the sphere is 11 inches.
So we will draw the diagram for the same.

So to find the area of the cross-section, we will find the area of the circle of the same radius as that of the sphere.
We know that the formula of the area of a circle is equal to \[\pi {r^2}\].
Therefore, area of cross section \[ = \pi {\left( {11} \right)^2}\]
Now, we will substitute the value of pi. So, we get

\[ \Rightarrow \] Area of cross section \[ = \dfrac{{22}}{7}{\left( {11} \right)^2}\]

On multiplying the numbers, we get

\[ \Rightarrow \] Area of cross section \[ = 380.286\] square inches.

Note: A sphere is defined as a three-dimensional figure or a solid figure in which the distance of every point on its surface from its center is equidistant. However, a circle is defined as a two-dimensional figure or a plane figure in which the distance of every point on its circumference from its center is equidistant. Half of the sphere is called a hemisphere whereas half of the circle is called a semicircle.