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)
- The height of a cylinder is 14cm and its curved surface area is 264cm². The volume of the cyclinder (in cm³) is:
($\pi=\frac{22}{7}$) - A cylindrical rod has an outer curved surface area of \(7500 \text{ cm}^2\). If the length of the rod is 92 cm, then the outer radius (in cm) of the rod, rounded off to two places of decimal, is:
\(\left(\text{Take } \pi = \frac{22}{7}\right)\) - A number of 512 identical small spheres are cast from a sphere of radius 40 cm, with the total volume of the small spheres being equal to the volume of the larger sphere. The diameter (in cm) of each of the small spheres is:
There is a wooden block in the form of a cube whose each side is 8 meters long.
The maximum possible number of cylinders with a diameter of 1 meter and a height of 4 meters were cut from this block. The cylinders are to be painted at the rate of ₹14 per square meter.
What is the total amount (in ₹) needed to paint all the cylinders if we paint the entire surface of each cylinder? (Take $\pi = \frac{22}{7}$)
- If the lateral surface area of a cylinder is $140.1 \text{ cm}^2$ and its height is $3 \text{ cm}$, then find its volume. (Use $\pi = 3.14$ and round off to two decimal places.)
