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From Wikipedia

Octagon

In geometry, an octagon (from the Greekokto, eight) is a polygon that has eight sides. A regular octagon is represented by the Schläfli symbol {8}.

Regular octagons

A regular octagon is a closed figure with sides of the same length and internal angles of the same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles is 1080° (as for any octagon). The area of a regular octagon of side length a is given by

A = 2 \cot \frac{\pi}{8} a^2 = 2(1+\sqrt{2})a^2 \simeq 4.828427125\,a^2.

In terms of R (circumradius), the area is

A = 4 \sin \frac{\pi}{4} R^2 = 2\sqrt{2}R^2 \simeq 2.828427\,R^2.

In terms of r (inradius), the area is

A = 8 \tan \frac{\pi}{8} r^2 = 8(\sqrt{2}-1)r^2 \simeq 3.3137085\,r^2.

These last two coefficients bracket the value of pi, the area of the unit circle.

The area can also be derived as follows:

\,\!A=S^2-a^2,

where S is the span of the octagon, or the second shortest diagonal; and a is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside (making sure that four of the eight sides touch the four sides of the square) and then taking the corner triangles (these are 45-45-90 triangles) and placing them with right angles pointed inward, forming a square. The edges of this square are each the length of the base.

Given the span S, the length of a side a is:

S=\frac{a}{\sqrt{2}}+a+\frac{a}{\sqrt{2}}=(1+\sqrt{2})a
S=2.414a\, (approximately)

The area is then as above:

A=((1+\sqrt{2})a)^2-a^2=2(1+\sqrt{2})a^2.

Another simple formula for the area is

\ A=2ad

where d is the distance between parallel sides (the same as span S in the diagram).

Standard coordinates

The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are:

  • (±1, ±(1+√2))
  • (±(1+√2), ±1).

Uses of octagons

Derived figures

Petrie polygons

The octagon is the Petrie polygon for eight higher dimensional polytopes, shown in these skew orthogonal projections:

Construction

A regular octagon is constructible using compass and straightedge:



From Yahoo Answers

Question:can nyone help with derivation of formula of following 3d objects: cube cuboid cylinder cone sphere

Answers:i denote pi with P. volume area cube a^3 6a^2 cuboid abc 2(ab+bc+ca) cylinder Pr^2h 2Pr^2+h(2Pr) cone (1/3)bh Pr^2+Prs sphere (4/3)Pr^3 4Pr^2

Question:Please............I need it right away. Please make it simple and easy to understand. Thanks :) volume-sphere and square even if you just answer one of the above it will really help me a lot.

Answers:You haven't specified in what grade you are, so I will just be as simple as possible: Formula for acceleration: This is according to Newton's second law: F= m*a Where a is equal to the acceleration, F is equal to the force in Newton applied on the object which you are moving, and m = the mass of the object in kilograms. This is the second law of Newton. Newton's unit is the measurement for force. 1 Newton = 1 kg*1m/second square. This is how force is measured. I will give you a simple example: A car has a mass of 1000 kilograms; a force of 1000 Newtons is applied upon the car. What is the acceleration of the car as a result of this applied force neglecting the resistance coming from friction of the wheels or from the wind if it were windy that day! 1000 = 1000 *a a = 1000/1000 = 1 Newton/kilogram. This is equivalent to: 1 meter per second square acceleration unit is = 1 meter/second square. One Newton =1 kg.1m/second square. This is the unit of Force. Now, the equation of velocity which is the same as speed accept that velocity denotes that an object is moving with a certain speed and towards a certain direction, which we call it in Physics a "vector". To calculate the velocity to any given object, the equation of motion would be: V final = v initial + 1/2 a*t^2 where t is time and a is acceleration and v final is the final velocity and v initial is the initial velocity; the unit is meter per second. This can also be written as: V2^2 = v1^2 +2as, where s is the distance in meter attained, a is the acceleration. Now what else you need: Volume of course is the space that any object occupies. I will give you the volume of few objects: The volume of a sphere is 4/3*pi*r^3, where pi is 3.14, and r is the radius of the sphere. For a cube, the volume is = the length of its size to the third power, or a^3 For a cylinder, the volume is pi*r^2*h, where h is the height of the cylinder. For a pyramid, V= 1/3( area of the base * height) For a cone, the volume is 1/3*pi*r^3*h

Question:I've got to try and finish this calculus homework but I can't remember the formula.

Answers:Volume of a hemi-sphee is 2*pi*r^3/3 If it is thesurface area of a solid hemisphere that you want to know,it is 3*pi*r^2 Perimeter of a semi-circle is pi*r+2r

Question:It's for my maths home work

Answers:V = 1/3 * pi * r * h r = radius of the bottom of the cone h = height of the cone (from top to center of bottom)

From Youtube

Build a Gazebo : Calculate Octagonal Gazebo Dimensions :Calculate the octagonal dimensions when building a gazebo, connect points and learn formulas to figure out the square footage in this free construction video. Expert: Charlie Folkman Bio: Charlie has been a general partner for NorAz Outdoor Furniture since 1998. Filmmaker: Dixon Gillette

Archimedes derives the volume of a sphere formula :Gary Rubinstein teaches how Archimedes in 'The Method,' a manuscript which was lost between 900 AD and 1900 AD (and then lost again until 1998) first derived the formula for the volume of a sphere using the law of the lever.