Heron's formula (possibly known to Archimedes) gives the area of a triangle in terms of its side lengths, in a beautifully factored form. Why is it true? Does it generalize to higher dimensions? What should we mean by "generalize"?

Piero della Francesca (a renaissance painter) discovered a formula for the volume of a tetrahedron in terms of its side lengths. Is there a formula for the volume in terms of the face areas? What happens in higher dimensions? I will explain the limited amount I know about these questions and invite you to discover more.