Learn the formula for calculating the surface area of a circumscribed sphere, with detailed examples, easy to understand.

I. Formula for the surface area of the circumscribed sphere

S = 4.π.R²

Where: S represents the surface area of the sphere
R represents the radius of the circumscribed sphere

Illustrative figure

* Determine the sphere center:

The regular pyramid S.ABC has:
- O as the base center => SO is the base axis
- In the plane (SAO), draw d as the perpendicular bisector of SA intersecting SA at point M, intersecting SO at point I
=> The center of the circumscribed sphere of the pyramid is I.

* Calculate the radius of the circumscribed sphere:

Considering two similar triangles SMI and SOA, we have:
=> R = SI = SA² : 2.SO = IB = IB = IC (R: Radius)

* Calculate the surface area of the circumscribed sphere of the pyramid S.ABC
=> After calculating the radius of the circumscribed sphere, we apply the formula: S = 4.π.R²

b) Pyramid with a lateral edge perpendicular to the base plane

Illustrative figure

To find the center of the sphere:

Given the pyramid SABC with SA perpendicular to the base (ABC), where the base ABC is inscribed with circle O at its center:

To calculate the radius of the circumscribed sphere:

Note: If the pyramid S.ABC has SA perpendicular to the base (ABC) and triangle ABC is right-angled at B, then the center of the circumscribed circle of the pyramid is the midpoint of the line SC.

Exercise IV: Calculating the surface area of a circumscribed sphere

Exercise 1: In the triangular pyramid S.ABC, SAB is an equilateral triangle, the base ABC is an equilateral triangle with side length a, (SAB) is right (ABC). Calculate the surface area of the circumscribed sphere for S.ABC.

Solution:

Exercise 2: In the pyramid S.ABCD, the base ABCD is a square with side length a, SA = a√3, SA ⊥ (ABCD). Calculate the surface area of the circumscribed sphere for the pyramid.

Formula for calculating the surface area of the circumscribed sphere for pyramids, cubes, cones, and relatively simple prisms is straightforward and easy to remember. However, in each problem type, students need to apply it flexibly to find the most accurate answer. Besides, students also need to understand and remember how to calculate the area of a circle to easily apply it to similar shapes in solid geometry.

Additionally, students can refer to other types of problems regarding the circumscribed sphere, such as the problem formula for calculating the radius of the circumscribed sphere for tetrahedrons, which is also a problem that students often encounter.