52
The rectangular parallelepiped and the cube
Textbook: Part 2, lesson 52, pp. 60–63
GoalThe rectangular parallelepiped and the cube; vertices, edges, faces; surface area.
New words
rectangular parallelepiped · to‘g‘ri burchakli parallelepipedcube · kubdimensions · o‘lchamlarsurface area · sirt yuzi
Explanation
A rectangular parallelepiped has 8 vertices, 12 edges and 6 faces. Its opposite faces are equal rectangles. The three edges from each vertex are its dimensions — length, width and height (a, b, c). A parallelepiped with all edges equal is a cube, and its faces are equal squares. The surface area is the sum of the areas of all faces: S = 2(ab + bc + ac); for a cube S = 6a². The sum of the lengths of all edges is 4(a + b + c); for a cube it is 12a.
Worked examples
Dimensions 6 cm, 4 cm and 3 cm: S = 2(6 · 4 + 4 · 3 + 6 · 3) = 2(24 + 12 + 18) = 108 cm². Sum of edges 4(6 + 4 + 3) = 52 cm.
A cube with edge 5 cm: S = 6 · 5² = 6 · 25 = 150 cm²; all edges 12 · 5 = 60 cm.
Class activity
“Box dimensions”: measure a small cardboard box in cm (a, b, c) and calculate its surface area. Do not cut the box; if an adult allows, cover it with paper to check.
Practice
1
Surface area (cm²) of a parallelepiped with dimensions 7 cm, 2 cm and 5 cm?
118
2
Sum of all edges (cm) of a parallelepiped with dimensions 10 cm, 6 cm and 4 cm?
80
3
Surface area (cm²) of a cube with edge 8 cm?
384
4
Why is the surface area of a cube 6a²?
A cube has 6 faces, each a square of area a².