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February 4, 2010

Frustum

Set of pyramidal frusta

Examples: Pentagonal and square frusta
Faces n trapezoids,
2 n-gons
Edges 3n
Vertices 2n
Symmetry group Cnv
Dual polyhedron -
Properties convex

In geometry, a frustum [1] (plural: frusta or frustums) is the portion of a solid (normally a cone or pyramid) which lies between two parallel planes cutting it.  

The term is commonly used in computer graphics to describe the three-dimensional region which is visible on the screen (which is formed by a clipped pyramid); in particular, frustum culling is a method of hidden surface determination.  

In the aerospace industry, frustum is the common term for the fairing between two stages of a multistage rocket (such as the Saturn V), which is shaped like a truncated cone.   

Note 

  1. ^ The term comes from Latin frustum meaning “piece” or “crumb”. The English word is often misspelled as frustrum, probably because of a similarity with the common words “frustrate” and “frustration“, also of Latin origin.

Pyramidal Frustum
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A pyramidal frustum is a frustum made by chopping the top off a pyramid. It is a special case of a prismatoid.  

For a right pyramidal frustum, let s be the slant height, h the height, p_1 the bottom base perimeter, p_2 the top base perimeter, A_1 the bottom area, and A_2 the top area. Then the surface area (of the sides) and volume of a pyramidal frustum are given by

S = 1/2(p_1+p_2)s
(1)
V = 1/3h(A_1+A_2+sqrt(A_1A_2)). 

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Frustum of a Cone

                        If a cone is cut by a plane parallel to its base, the portion of a solid between this plane and the base is known as frustum of a cone.             The volume denoted by ABCD in figure is a frustum of the cone ABE.  

   

// //

Volume of Frustum of a Cone:
            Since, we know that cone is a limit of a pyramid therefore; frustum of a cone will be the limit of frustum of a pyramid. But volume of a pyramid is
                       
Where             
                       
                   
                          

  

Volume of a Frustum of a Cone

A frustum may be formed from a cone with a circular base by cutting off the tip of the cone with a cut perpendicular to the height, forming a lower base and an upper base that are circular and parallel.   

Let h be the height, R the radius of the lower base, and r the radius of the upper base. One picture of the frustrum is the following.   

   

Given R, r, and h, find the volume of the frustum.   

Cone (geometry)

From Wikipedia, the free encyclopedia

A right circular cone and an oblique circular cone 

A cone is a three-dimensional geometric shape that tapers smoothly from a flat, usually round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface (called the lateral surface) formed by the locus of all straight line segments joining the apex to the perimeter of the base. The term “cone” sometimes refers just to the surface of this solid figure, or just to the lateral surface.  

 

  

Volume of Cone

A cone is a solid with a circular base. It has a curved surface which tapers (i.e. decreases in size) to a vertex at the top. The height of the cone is the perpendicular distance from the base to the vertex.  

 
The volume of a cone is given by the formula:  

  

Volume of cone = Area of base × height  

V = where r is the radius of the base and h is the height of the prism.  

Pyramid (geometry)

Set of pyramids
Faces n triangles,
1 n-gon
Edges 2n
Vertices n + 1
Symmetry group Cnv
Dual polyhedron Self-dual
Properties convex

The 1-skeleton of pyramid is a wheel graph

This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural pyramids, see Pyramid (disambiguation).

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base. 

A pyramid with an n-sided base will have n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual. 

Square Pyramid

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SquarePyramid
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A square pyramid is a pyramid with a square base. It is a pentahedron

The lateral edge length e and slant height s of a right square pyramid of side length a and height h are

e = sqrt(h^2+1/2a^2)
(1)
s = sqrt(h^2+1/4a^2).
(2)

The corresponding surface area and volume are

S = a(a+sqrt(a^2+4h^2))
(3)
V = 1/3a^2h.
(4)
SquarePyramidCube

Prismatoid

From Wikipedia, the free encyclopedia

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In geometry, a prismatoid is a polyhedron where all vertices lie in two parallel planes. (If both planes have the same number of vertices, and the lateral faces are either parallelograms or trapezoids, it is called a prismoid.) 

If the areas of the two parallel faces are A1 and A3, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A2, and the height (the distance between the two parallel faces) is h, then the volume of the prismatoid is given by (This formula follows immediately by integrating the area parallel to the two planes of vertices by Simpson’s rule, since that rule is exact for integration of polynomials of degree up to 3, and in this case the area is at most a quadratic in the height.) 

[edit] Prismatoid families

 

Families of prismatoids include: 

The volume of such a solid is the same as for a prismatoid

 V=1/6h(A_1+4M+A_2).  

Cylinder (geometry)

A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity. 

In differential geometry, a cylinder is defined more broadly as any ruled surface spanned by a one-parameter family of parallel lines. A cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder

 

Volume of a cylinder

Volume of a cylinder
 
Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h
  

Rectangular Prism

A solid (3-dimensional) object which has six faces that are rectangles.

It is a prism because it has the same cross-section along a length.

cuboid is a box-shaped object. 

It has six flat sides and all angles are right angles

And all of its faces are rectangles. 

It is also a prism because it has the same cross-section along a length. In fact it is a rectangular prism

 

Volume and Surface Area

The volume is found using the formula: 

Volume = Height × Width × Length 

Which is usually shortened to: 

V = h × w × l 

Or more simply: 

V = hwl 

Surface Area

And the surface area is found using the formula: 

A = 2wl + 2lh + 2hw 

Cube

From Wikipedia, the free encyclopedia

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This article is about the geometric shape. For other uses, see Cube (disambiguation).

Regular Hexahedron
Cube
(Click here for rotating model)
Type Platonic solid
Elements F = 6, E = 12
V = 8 (χ = 2)
Faces by sides 6{4}
Schläfli symbol {4,3}
Wythoff symbol 3 | 2 4
Coxeter-Dynkin CDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.png
Symmetry Oh or (*432)
References U06, C18, W3
Properties Regular convex zonohedron
Dihedral angle 90°
Cube
4.4.4
(Vertex figure)
Octahedron.png
Octahedron
(dual polyhedron)
Cube
Net

In geometry, a cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and of trigonal trapezohedron. The cube is dual to the octahedron. It has cubical symmetry (also called octahedral symmetry). 

A cube is the three-dimensional case of the more general concept of a hypercube

It has 11 nets.[2] If one were to colour the cube so that no two adjacent faces had the same colour, one would need 3 colours. 

If the original cube has edge length 1, its dual octahedron has edge length \sqrt{2}

Volume of a Cube

A cube is a three-dimensional figure with six matching square sides.  

cube  

The figure above shows a cube. The dotted lines indicate edges hidden from your view.  

If s is the length of one of its sides, the  

Volume of the cube = s3  

Since the cube has six square-shape sides, the  

Surface area of a cube = 6s2  


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