For a regular tetrahedron with edge length $1$, let's solve the two parts:

Part (a): The angles the lateral edges make with the base

In a regular tetrahedron, each face is an equilateral triangle. The lateral edges are the ones connecting the vertices of the base to the apex of the tetrahedron. To find the angle these lateral edges make with the base, we need to consider the geometric arrangement.

The tetrahedron can be inscribed inside a cube, and using vector geometry or trigonometry, it can be shown that the angle $\theta$ that each lateral edge makes with the base is the same for all edges. The tetrahedron can be centered at the origin, and the vertices are positioned symmetrically.

One key relationship that helps here is that for a regular tetrahedron, the cosine of the angle $\theta$ between a lateral edge and the base is given by:

$\cos \theta = \frac{1}{\sqrt{3}}$

Thus, the angle is:

$\theta = \arccos \left( \frac{1}{\sqrt{3}} \right) \approx 54.74^\circ$

Part (b): The angles the lateral faces make with the base

To find the angle $\phi$ between a lateral face and the base, we need to look at the dihedral angle of the tetrahedron, which is the angle between two faces of the tetrahedron. Specifically, we want the angle between a lateral face and the triangular base face.

The cosine of the dihedral angle $\phi$ between any two adjacent faces of a regular tetrahedron is given by:

$\cos \phi = \frac{1}{3}$

Thus, the angle $\phi$ is:

$\phi = \arccos \left( \frac{1}{3} \right) \approx 70.53^\circ$

Summary of Results:

• (a) The angles the lateral edges make with the base are approximately $54.74^\circ$.
• (b) The angles the lateral faces make with the base are approximately $70.53^\circ$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How can you derive the coordinates of a regular tetrahedron in 3D space?
2. What is the volume of a regular tetrahedron with edge length 1?
3. How do you calculate the surface area of a regular tetrahedron?
4. How does the height of the tetrahedron relate to its edge length?
5. What is the solid angle subtended by a face of the tetrahedron at its apex?

Tip: For any polyhedron, understanding its symmetry can simplify the process of finding angles and other properties like volume and surface area.