Understanding the Equation of a Sphere: Derivation and Calculation of Radius

Equation of a Sphere with Center (h,k,l) and radius r

The equation of a sphere with center (h, k, l) and radius r can be written as:

(x – h)^2 + (y – k)^2 + (z – l)^2 = r^2

Here, (x, y, z) represent the coordinates of any point on the sphere

The equation of a sphere with center (h, k, l) and radius r can be written as:

(x – h)^2 + (y – k)^2 + (z – l)^2 = r^2

Here, (x, y, z) represent the coordinates of any point on the sphere.

To understand how this equation is derived, let’s break it down step by step:

1. The general equation for a sphere with center (h, k, l) can be written as:

(x – h)^2 + (y – k)^2 + (z – l)^2 = r^2

This equation represents the sum of the squared distances between each coordinate (x, y, z) and the center (h, k, l) of the sphere.

2. In order to determine the radius r, you need to know the distance between the center and any point on the surface of the sphere. This can be found using the Pythagorean theorem.

3. Let’s consider a specific point (x, y, z) on the sphere’s surface. The distance between this point and the center (h, k, l) can be found using the distance formula:

Distance = sqrt((x – h)^2 + (y – k)^2 + (z – l)^2)

4. The radius of the sphere, r, is the same for all points on the surface. So, we can set the distance equal to the radius:

sqrt((x – h)^2 + (y – k)^2 + (z – l)^2) = r

5. Squaring both sides of the equation gives:

(x – h)^2 + (y – k)^2 + (z – l)^2 = r^2

This equation represents the equation of the sphere with center (h, k, l) and radius r.

By knowing this equation, you can determine whether a given point lies on the sphere or not, and you can find the radius based on the coordinates of any point on the sphere.

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