For a certain regular solid: number of faces + number of vertices = number of edges+2. For three such distinct (not touching each other) objects, what is the total value of faces + vertices – edges?
Regular Solid Properties Using Euler's Formula
The question relates the number of faces (F), vertices (V), and edges (E) of a regular solid using the formula:
$ F + V = E + 2 $
This formula is a form of Euler's formula for polyhedra ($V - E + F = 2$).
Calculating Value for One Solid
We need to determine the value of the expression $ F + V - E $. Let's rearrange the given formula:
- Given: $ F + V = E + 2 $
- Subtract $ E $ from both sides: $ F + V - E = 2 $
So, for any single regular solid that satisfies the condition, the value of $ F + V - E $ is 2.
Total Value for Three Objects
The question asks for the total value of $ F + V - E $ for three distinct objects.
Calculate the total value by multiplying the value for one object by three:
$ \text{Total Value} = 3 \times (F + V - E)_{\text{one solid}} $
$ \text{Total Value} = 3 \times 2 $
$ \text{Total Value} = 6 $
Final Answer
The total value of faces + vertices – edges for three distinct regular solids is 6.
Important Questions from Mensuration 3D (Notes)
- On a spherical balloon of 10 cm radius, a circular colour patch has an area of 25 cm². If the balloon is uniformly expanded to a sphere of 50 cm radius, the area of the colour patch in cm² would be
- A block of marble 5 m x 4 m x 2 m in size is cut into rectangular tiles of 1 m x 0.5 m size having thickness of 10 cm. Assuming 10% wastage in cutting, how many tiles will be made?
- The height of a cylinder is 14cm and its curved surface area is 264cm². The volume of the cyclinder (in cm³) is:
($\pi=\frac{22}{7}$) - What is the volume of a 6 m deep tank having rectangular shaped top 6m X 4 m and bottom 4 m X 2 m? (use mean-area method).
- The surface area of the solid generated by revolving the curve $x = e^t \cos t, y = e^t \sin t$ about y-axis $0 \leq t \leq \pi/2$ is
