How to Find the Equation of a Sphere Quickly

TL;DR

To find the equation of a sphere given its center (h, k, l) and radius r, use the formula (x-h)² + (y-k)² + (z-l)² = r². If given the endpoints of a diameter, first calculate the midpoint for the center and then find the radius by measuring the distance from the center to one endpoint.

Transcript

in this video we're going to work on some practice problems involving spheres like this one find an equation of a sphere with center 2 negative 4 3 and radius 5. how do we do it well there's a formula that you need to know x minus h squared plus y minus k squared now this looks like a circle but once you add z minus l squared now we have the standa... Read More

Key Insights

• 🕡 The equation of a sphere can be written using the formula (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2, where (h,k,l) are the center coordinates and r is the radius.
• ❓ To find the center of a sphere given the endpoints of a diameter, calculate the midpoint between the two endpoints.
• ❓ The radius of a sphere can be found by calculating the distance between one of the endpoints and the center using the distance formula.

Questions & Answers

Q: How do you write the equation of a sphere given its center and radius?

To write the equation of a sphere, use the formula (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2, where (h,k,l) represents the center coordinates and r represents the radius.

Q: How do you find the center and radius of a sphere given the endpoints of a diameter?

To find the center, calculate the midpoint between the two endpoints by averaging the x, y, and z values. To find the radius, determine the distance between one endpoint and the center using the distance formula.

Q: How do you convert an equation of a sphere to the standard form?

To convert an equation to standard form, group the x, y, and z terms together, complete the square by adding the appropriate constants, and then factor the resulting trinomials. The coefficients of the factored terms will represent the center coordinates, and the square of the radius will be the sum of the added constants.

Q: Why is the midpoint of the diameter the center of the sphere?

The midpoint of the diameter is the center of the sphere because it lies equidistant from both endpoints. In a sphere, all points are equidistant from the center, so the midpoint naturally represents the center.

Summary & Key Takeaways

• The first part of the video explains how to write the equation of a sphere with a given center and radius using the formula (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2.

• The second part explains how to find the center of a sphere given the endpoints of a diameter by finding the midpoint between the two points, and how to find the radius by calculating the distance between one of the endpoints and the center.

• The third part demonstrates how to convert an equation of a sphere to the standard form (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 by completing the square and then use the coefficients to determine the center and radius.