Question

A line segment connecting two antipodal (diametrically opposite) points on a sphere will pass through the center of the sphere.

Points A(2,-7,-4) and B(6,1,2) are a pair of antipodal points on a sphere. Find the exact surface area and exact volume of this sphere.

Answer 1

`\displaystyle A=116\pi`

`\displaystyle V=(116\pi √(29))/(3)`

Explanation:

Diameter, Area, and Volume of a Sphere

Given the radius of a sphere (r), we can compute the area of the sphere as

`\displaystyle A=4\pi r^2`

The volume is

`\displaystyle V=(4\pi r^3)/(3)`

If we know the diameter of the sphere (d), the radius is half of d

`\displaystyle r=(d)/(2)`

Two antipodal (or diametrically opposite) points can be used to determine both the center and the radius of the spere. We are only interested in computing the radius, we need first to calculate the diameter which is the distance between the antipodal points A(2,-7,-4) and B(6,1,2). The distance between these two points in R^3 is

`d=√((6-2)^2+(1+7)^2+(2+4)^2)=√(116)=2√(29)`

That is the diameter, now the radius is

`\displaystyle r=(2√(29))/(2)`

`r=√(29)`

Let's compute the area

`\displaystyle A=4\pi(√(29))^2`

`\displaystyle A=4\pi(29)`

`\displaystyle A=116\pi`

Finally, the volume is

`\displaystyle V=(4\pi (√(29))^3)/(3)`

`\displaystyle V=(116\pi √(29))/(3)`