Lesson 2 — Rectangular Solids
Big idea: A rectangular solid is a six-faced polyhedron, and its interior diagonal is found by using the Pythagorean Theorem twice.
Words to know
| Word | Meaning |
|---|---|
| Polyhedron | A solid bounded by parts of intersecting planes. |
| Face | One of the polygonal regions formed by the intersecting planes. |
| Edge | A side of a face of a polyhedron. |
| Vertex | A corner point where edges of a polyhedron meet. |
| Rectangular solid | A polyhedron that has six rectangular faces. |
| Diagonal of a rectangular solid | A segment connecting two opposite vertices (vertices not on the same face). |
| Dimensions | The lengths of the three edges that meet at one vertex: length l, width w, height h. |
| Cube | A rectangular solid whose three dimensions are equal. |
Concepts to know
- A rectangular solid has 6 faces, 12 edges, and 8 vertices.
- Opposite faces lie in parallel planes; intersecting faces lie in perpendicular planes.
- Every rectangular solid has four interior diagonals, and they all have the same length.
- The diagonal formula comes from applying the Pythagorean Theorem to two right triangles: one in a face, then one through the solid.
- Theorem 79: The length of a diagonal of a rectangular solid with dimensions l, w, and h is √(l² + w² + h²).
- Corollary to Theorem 79: The length of a diagonal of a cube with edges of length e is e√3.
- The surface area comes from adding the areas of all six rectangles.
Formulas / rules
- Interior diagonal: d = √(l² + w² + h²)
- Cube diagonal: d = e√3
- Face diagonal: use the Pythagorean Theorem on that rectangle, d = √(l² + w²)
- Surface area: A = 2(lw + lh + wh)
- Volume is studied in Lesson 4: V = lwh.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find the interior diagonal | Substitute the three dimensions into d = √(l² + w² + h²). |
| Find the diagonal of a cube | Use d = e√3. |
| Find a diagonal of one face | Use the Pythagorean Theorem on that rectangle only. |
| Find the surface area | Add the areas of the six faces: 2(lw + lh + wh). |
| Compare a cube and a square | Square diagonal is e√2; cube diagonal is e√3. |
Watch out
- Do not confuse an interior diagonal with a diagonal of a face.
- A cube’s diagonal is e√3, not e√2.
- All dimensions must be in the same units before substituting.
- Diagonal length is in linear units; surface area is in square units.
- The four interior diagonals are equal, but they are not edges of the solid.