Lessons 4–5 · 2 lessons · B. Xaydarov, E. Sariqov, A. Qo‘chqorov. Geometry, Grade 9 (textbook for general secondary schools), revised 4th edition. Huquq va Jamiyat Publishing, Tashkent, 2019
4
3D geometry: plane-geometry problems in solids
Textbook: pp. 18–25
GoalRecall the polyhedron, prism and pyramid, and find the diagonal of a cuboid using plane geometry (the Pythagorean theorem).
New words
polyhedron · ko‘pyoqprism · prizmapyramid · piramidaspace diagonal of a cuboid · parallelepiped diagonali
Explanation
A bounded part of space is a solid, and its boundary is its surface. A solid bounded by flat polygons is a polyhedron, with faces, edges and vertices. A prism has two equal polygons as bases and parallelograms as lateral faces; a pyramid has one polygon as base and triangles with a common vertex as lateral faces. For the space diagonal of a cuboid, d² = a² + b² + c² (a, b, c are its three dimensions). It follows from applying the Pythagorean theorem twice: first to the diagonal of the base, then to the solid. A solid problem is often reduced to a plane problem by picking out a suitable cross-section.
Worked examples
A cuboid has edges 2 cm, 3 cm and 6 cm. Its diagonal: d² = 4 + 9 + 36 = 49, so d = 7 cm.
A cube with edge 5 cm has diagonal d² = 25 + 25 + 25 = 75, so d = 5√3 cm.
Class activity
“A stick in a box”: how long a stick fits inside a cardboard box 30 × 40 × 120 cm? Try with a box and a stick and compare with the calculation.
Practice
1
A cuboid has edges 1, 2 and 2 cm. How long is its diagonal (cm)?
3
2
A cube has diagonal 6√3 cm. What is its edge (cm)?
6
3
How many faces, edges and vertices does a pentagonal prism have?
7 faces, 15 edges, 10 vertices
4
Why is the square of a cuboid's diagonal the sum of the squares of its three dimensions?
The square of the base diagonal is a² + b²; adding the square of the height c² (a second use of Pythagoras) gives a² + b² + c².