The circumscribed sphere of a polyhedron is a fundamental concept in solid geometry. It is defined as a sphere that passes through all vertices of the polyhedron. Here we see a cube with its circumscribed sphere. Notice how every vertex of the cube lies exactly on the surface of the red sphere.
To find the circumscribed sphere, we follow two main steps. First, we locate the center of the sphere, which must be equidistant from all vertices. For symmetric shapes like a cube, the center is at the geometric center. Second, we calculate the radius as the distance from this center to any vertex. The green lines show these equal distances from center to each vertex.
For a cube with side length a, the radius of the circumscribed sphere can be calculated using the formula R equals a times square root of 3, divided by 2. This comes from the fact that the space diagonal of a cube has length a times square root of 3, and the radius is half of this diagonal. In our example with side length 2, the radius is square root of 3, approximately 1.73 units.
For a regular tetrahedron with edge length a, the circumscribed sphere has radius R equals a times square root of 6, divided by 4. The center of this sphere is located at the centroid of the tetrahedron. Unlike the cube, the tetrahedron's center is not at the geometric middle but at a point that balances all four vertices equally. With edge length 2, the radius is approximately 1.22 units.
In summary, the circumscribed sphere is a fundamental concept in solid geometry. It passes through all vertices of a polyhedron, with its center equidistant from every vertex. For regular shapes like cubes and tetrahedra, we have specific formulas to calculate the radius. This concept has important applications in 3D modeling, computer graphics, crystallography, molecular geometry, and engineering design, making it essential for understanding spatial relationships in three dimensions.