The number of diagonals in each face a cube is
2
Understanding the Structure of a Cube
A cube is a three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex. All edges are of equal length, and all angles are right angles.
The question asks about the number of diagonals in each face of a cube. This means we need to focus on the shape of a single face of the cube.
Analyzing the Shape of a Cube's Face
As mentioned, each face of a cube is a square. Let's consider a single square face. A square is a four-sided polygon with four vertices and four sides of equal length, where opposite sides are parallel and all four internal angles are right angles.
Finding Diagonals in a Square Face
A diagonal in a polygon is a line segment connecting two non-adjacent vertices. In a square, there are four vertices. Let's label them A, B, C, and D in order around the square.
- Vertices adjacent to A are B and D. The side connecting A and B is a side, and the side connecting A and D is also a side.
- Vertices non-adjacent to A is C. The line segment connecting A and C is a diagonal.
- Vertices adjacent to B are A and C.
- Vertices non-adjacent to B is D. The line segment connecting B and D is a diagonal.
- Vertices adjacent to C are B and D.
- Vertices non-adjacent to C is A. The line segment connecting C and A is the same diagonal as AC.
- Vertices adjacent to D are A and C.
- Vertices non-adjacent to D is B. The line segment connecting D and B is the same diagonal as BD.
So, there are exactly two distinct line segments connecting non-adjacent vertices in a square:
- The diagonal connecting vertices A and C (AC).
- The diagonal connecting vertices B and D (BD).
Let's illustrate with a simple table:
| Vertex | Adjacent Vertices | Non-Adjacent Vertex | Diagonal from this Vertex |
|---|---|---|---|
| A | B, D | C | AC |
| B | A, C | D | BD |
| C | B, D | A | CA (Same as AC) |
| D | A, C | B | DB (Same as BD) |
From this analysis, it is clear that there are only two diagonals in a square. Since each face of a cube is a square, the number of diagonals in each face of a cube is 2.
This applies to all six faces of the cube.
Conclusion
A cube has square faces. A square has 2 diagonals. Therefore, the number of diagonals in each face of a cube is 2.
Important Questions from Geometry
If the angles of a triangle are in the ratio of 2:3:4, then the difference of the measure of greatest angle and smallest angle is
In ΔABC, ∠A = 90°, AD ┴ BC and AD = BD = 2 cm. The length of CD is
How many lines of symmetry does a rectangle have?
Which of the following is/are the geometric figures with the line of symmetry?
I. Rectangle
II. Isosceles triangle
Euclid’s Postulate III is:
