Buzz
Readers are invited to read the types of exercises and solving methods for finding the radius of a circumscribed sphere of a tetrahedron that we have summarized below to apply to solving exercises in the process of studying Geometry.
Formula for calculating the radius of a circumscribed sphere of a tetrahedron
The general formula for calculating the circumradius of the circumscribed sphere of any tetrahedron.
Calculation method: Suppose the tetrahedron ABCD has: BC = a, CA = b, AB = c, AD = d, BD = e, CD = f
=> The formula for calculating the circumradius of tetrahedron ABCD is:
R =
For: M = a^2d^2(b^2 + e^2 + c^2 + f^2 - a^2 - d^2)
N = b^2e^2(a^2 + d^2 + c^2 + f^2 - b^2 - e^2)
P = c^2f^2(a^2 + d^2 + b^2 + e^2 - c^2 - f^2)
Q = (abc)^2 + (aef)^2 + (bdf)^2 + (cde)^2
Given the tetrahedron ABCD with AB = 3, AC = BC = AD = BD = 4, CD = 6. Find the radius R of the circumscribed sphere.
To solve, apply the formula for the radius of the circumscribed sphere of a tetrahedron as described above, substitute the given values, and compute carefully to find the correct answer.
Introduction to calculating the radius of the circumscribed sphere for a tetrahedron.
This is the most common type of exercise in finding the radius of the circumscribed sphere of a tetrahedron.
General Procedure:
- Identify the base center, construct line d perpendicular to the base.
- Construct the perpendicular bisector plane (P) of any side edge.
- The center of the sphere is the intersection of plane (P) and line d.
Specific Problem Types:
Type 1: Pyramid with a side edge perpendicular to the base
Let's assume: r is the radius of the circumscribed circle of the polygon, h is the height:
- Formula for calculating the radius:
R = √((h/2)2 + r2)
* Application exercise: Tetrahedron OABC has OA = a, OB = OC = 2a, the edges OA, OB, OC are perpendicular to each other in pairs. Calculate the radius of the circumscribed sphere of tetrahedron OABC.
( Draw the figure and label as shown )
* Hint for approach:
- Find the length of BC (using the Pythagorean theorem in right triangle OBC)
- From the length of BC, determine r = 1/2 BC
- Given r, know h = OA => Apply the formula for calculating the radius of the circumscribed sphere of the tetrahedron to find R.
Type 2: Regular pyramid
Assuming: The pyramid has a side length of a and a height of h:
- Formula for calculating the radius:
R = a2/2h
* Application exercise: The regular tetrahedral pyramid SABCD has a side length = 2a, the base side length is a. Calculate the radius of the circumscribed sphere of the given pyramid.
( Students should draw the figure as shown )
* Tips for approach:
- Find the length of AO
- Find the length of SO (using right triangle SAO)
- Use the radius formula, substitute values, compute carefully to find the result.
The theoretical knowledge and some illustrative exercises here hopefully will assist students more easily in solving problems related to finding the radius of the circumscribed sphere of tetrahedra. To master this area, students need to memorize formulas and practice more exercises. Exercises on calculating the area of the circumscribed sphere are also related to finding the radius of the circumscribed sphere of tetrahedra, so students should refer to and remember them.
Besides, students also need to review and master the method of calculating the area of a circle, this knowledge will greatly help them when encountering exercises related to circles, spheres, spherical caps, or related shapes.
