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# examples of octagonal prism

From Wikipedia

Prism (optics)

In optics, a prism is a transparent optical element with flat, polished surfaces that refractlight. The exact angles between the surfaces depend on the application. The traditional geometrical shape is that of a triangular prism with a triangular base and rectangular sides, and in colloquial use "prism" usually refers to this type. Some types of optical prism are not in fact in the shape of geometric prisms. Prisms are typically made out of glass, but can be made from any material that is transparent to the wavelengths for which they are designed.

A prism can be used to break light up into its constituent spectralcolors (the colors of the rainbow). Prisms can also be used to reflect light, or to split light into components with different polarizations.

## How prisms work

Light changes speed as it moves from one medium to another (for example, from air into the glass of the prism). This speed change causes the light to be refracted and to enter the new medium at a different angle (Huygens principle). The degree of bending of the light's path depends on the angle that the incident beam of light makes with the surface, and on the ratio between the refractive indices of the two media (Snell's law). The refractive index of many materials (such as glass) varies with the wavelength or color of the light used, a phenomenon known as dispersion. This causes light of different colors to be refracted differently and to leave the prism at different angles, creating an effect similar to arainbow. This can be used to separate a beam of white light into its constituent spectrum of colors. Prisms will generally disperse light over a much larger frequency bandwidth than diffraction gratings, making them useful for broad-spectrum spectroscopy. Furthermore, prisms do not suffer from complications arising from overlapping spectral orders, which all gratings have.

Prisms are sometimes used for the internal reflection at the surfaces rather than for dispersion. If light inside the prism hits one of the surfaces at a sufficiently steep angle, total internal reflection occurs and all of the light is reflected. This makes a prism a useful substitute for a mirror in some situations.

### Deviation angle and dispersion

Ray angle deviation and dispersion through a prism can be determined by tracing a sample ray through the element and using Snell's law at each interface. The exact expressions for prism deviation and dispersion are complex, but for small angle of incidence \theta_0 and small angle \alpha they can be approximated to give a simple formula. For the prism shown at right, the indicated angles are given by

\begin{align}

\theta'_0 &\approx \frac{n_0}{n_1} \theta_0 \\ \theta_1 &= \alpha - \theta'_0 \\ \theta'_1 &\approx \frac{n_1}{n_2} \theta_1 \\ \theta_2 &= \theta'_1 - \alpha \end{align}. For a prism in air n_0=n_2 \simeq 1. Defining n=n_1, the deviation angle \delta is given by

{\delta = \theta_2 + \theta_0 \approx n \theta_1 - \alpha + \theta_0 = n \alpha - n \theta'_0 - \alpha + \theta_0 \approx (n - 1) \alpha}

The dispersion \delta (\lambda) is the wavelength-dependent deviation angle of the prism, so that for a thin prism the dispersion is given by

\delta (\lambda) \approx [ n (\lambda) - 1 ] \alpha

## Prisms and the nature of light

In Isaac Newton's time, it was believed that white light was colorless, and that the prism itself produced the color. Newton's experiments convinced him that all the colors already existed in the light in a heterogeneous fashion, and that "corpuscles" (particles) of light were fanned out because particles with different colors traveled with different speeds through the prism. It was only later that Young and Fresnel combined Newton's particle theory with Huygen's wave theory to show that color is the visible manifestation of light's wavelength.

Newton arrived at his conclusion by passing the red color from one prism through a second prism and found the color unchanged. From this, he concluded that the colors must already be present in the incoming light â€” thus, the prism did not create colors, but merely separated colors that are already there. He also used a lens and a second prism to recompose the spectrum back into white light. This experiment has become a classic example of the methodology introduced during the scientific revolution. The results of this experiment dramatically transformed the field of metaphysics, leading to John Locke's primary vs secondary quality distinction.

Newton discussed prism dispersion in great detail in his book Opticks. He also introduced the use of more than one prism to control dispersion. Newton's description of his experiments on prism dispersion was qualitative, and is quite readable. A quantitative description ofmultiple-prism dispersion was not needed until multiple prism laser beam expanders were introduced in the 1980s.

Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing (but incompatible) definitions of a cuboid in the mathematical literature. In the more general definition of a cuboid, the only additional requirement is that these six faces each be a quadrilateral, and that the undirected graph formed by the vertices and edges of the polyhedron should be isomorphic to the graph of a cube. Alternatively, the word â€œcuboidâ€� is sometimes used to refer to a shape of this type in which each of the faces is a rectangle (and so each pair of adjacent faces meets in a right angle); this more restrictive type of cuboid is also known as a right cuboid, rectangularbox, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped.

## General cuboids

By Euler's formula the number of faces (F), vertices (V), and edges (E) of any convex polyhedron are related by the formula "F + V - E" = 2 . In the case of a cuboid this gives 6 + 8 - 12 = 2; that is, like a cube, a cuboid has 6 faces, 8 vertices, and 12 edges.

Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, as is a square frustum (the shape formed by truncation of the apex of a square pyramid).

## Rectangular cuboid

In a rectangular cuboid, all angles are right angles, and opposite faces of a cuboid are equal. It is also a right rectangularprism. The term "rectangular or oblong prism" is ambiguous. Also the term rectangularparallelepipedor orthogonal parallelepiped is used.

The square cuboid, square box, or right square prism (also ambiguously called square prism) is a special case of the cuboid in which at least two faces are squares. The cube is a special case of the square cuboid in which all six faces are squares.

If the dimensions of a cuboid are a, b and c, then its volume is abc and its surface area is 2ab + 2bc + 2ac.

The length of the space diagonal is

d = \sqrt{a^2+b^2+c^2}.\

Cuboid shapes are often used for boxes, cupboards, rooms, buildings, etc. Cuboids are among those solids that can tessellate 3-dimensional space. The shape is fairly versatile in being able to contain multiple smaller cuboids, e.g. sugar cubes in a box, small boxes in a large box, a cupboard in a room, and rooms in a building.

A cuboid with integer edges as well as integer face diagonals is called an Euler brick, for example with sides 44, 117 and 240. A perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists.

Question:Here's the example I'm given: http://learn.flvs.net/webdav/educator_math2_v5/module10/imagmod10/10_03a_03.gif To find the surface area, visualize the net for this figure. The net consists of a large rectangle (the lateral faces) with sides of 12 cm and the perimeter of the base (5cm +5 cm +8 cm +8 cm = 26 cm). The area of the lateral faces would be 12 cm X 26 cm = 312 square cm The bases of the prism are rectangles with length and height of 8 cm and 5 cm. The area of this would be 8 cm X 5 cm = 40 square cm Adding the lateral area and 2 bases together will reveal the surface area. 312 sq cm + 2(40 sq cm) = 392 sq cm I dont really get it. Can somebody help me find a way to remember this, and explain it? Thanks. http://i31.tinypic.com/153ty7l.png

Answers:Just so you know, your link led to a complaint page because the system with the example expects a "cookie" which, of course, was only set on your computer. But this example is fairly straightforward, though the term "net" is a little odd. What they are doing is "unfolding" the sides of the prism to calculate the surface area. The four sides ("lateral faces") unfold into one large rectangle, and that leaves the top and bottom ("bases") to be added in. So they calculate the area of the lateral sides (height times the perimeter of the base = 312) and then add the area of the two bases, each being length x width (which gives the 2 * 40). Here's another, equivalent way to look at it: take the three dimensions of the rectangular as x, y, and z. For each pair of measurements, there will be two opposite faces that consist of rectangles with that pair of dimensions. So the area is 2(xy + xz + yz). In the example, x=5, y=8, z=12 (or you can assign them in any other order and it will come out the same). So the surface area is 2 (5*8 + 5*12 + 8*12) = 2 (40 + 60 + 96) = 2 * 196 = 392 Their approach, unfolding the sides and then adding the top and bottom, just collects the areas of the sides in a different order. You're still adding up six rectangles; it's just that they've stuck four of them together in one step.

Question:Ok there is a triangular prism that has a surface area of 100cm squared... how wud i make an exact 3-D triangular prism?? the triangle has to be a right triangle, i need to know all the lengths and stuff like that i need just one way to make one out of paper

Answers:There are an infinite number of such. The larger the ends, the shorter the length. For example, if the triangle was 5 by 5 by 5* 2 then the length = 75/(5+5+5* 2) = 4.39334 cm

Question:

Answers:Think of a box or a cube. Those are examples of rectangular prisms. There are six faces (the sides of the solid). There are eight vertices (the "corners" of the solid). Answer: 6 faces 8 vertices

Question:i asked to draw a polygon.and how many sides it is.

Answers:A polygon will have as many sides as you want, at least three, for a triangle, for example. An octagon has 8 sides. Poly = many, gone = angle, octo = 8, in Greek ;-)