Question Video: Finding the Volume of a Sphere Given the Area of Its Great Circle | Nagwa Question Video: Finding the Volume of a Sphere Given the Area of Its Great Circle | Nagwa

# Question Video: Finding the Volume of a Sphere Given the Area of Its Great Circle Mathematics • Second Year of Preparatory School

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Find, to the nearest tenth, the volume of a sphere given that the area of its great circle is 400π inΒ².

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### Video Transcript

Find, to the nearest tenth, the volume of a sphere given that the area of its great circle is 400π square inches.

Letβs begin by sketching a sphere and recalling what we mean by its great circle. When a sphere is cut by a plane that passes through its center, the intersection of the sphere and the plane is called a great circle. A great circle cuts a sphere exactly in half, and its radius is the same as that of the sphere itself.

Weβre given that the area of the great circle of a sphere is 400π square inches. And weβre asked to find the volume of the sphere. We can use the fact that the area of a circle is ππ squared, where π is the radius of the circle, to find the radius of the sphere. And we do this by equating the area of our great circle, thatβs 400π, with ππ squared. Next, dividing through by π, we have π squared equal to 400. And then taking the positive square root on both sides, positive since π is a length and lengths are always positive, we have π equal to 20 inches.

So now that we have the radius of the great circle, which equals the radius of the sphere, recalling that the volume of a sphere is given by π equals four over three ππ cubed and making a little space for our working out, we have π equals four over three π times 20 cubed. Thatβs four over three π times 8000, which evaluates to 33,510.3216 and so on. To the nearest tenth, thatβs to one decimal place, this is 33,510.3.

Hence, given that the area of the great circle of a sphere is 400π square inches, its volume is 33,510.3 cubic inches, to the nearest tenth.

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