Chapter 18: 2D and 3D Objects

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18.1 Drawing and Constructing 3D Objects

Isometric drawings are used so all dimesions are drawn to scale but all angles are distorted to give the 3D effect.

Plans and evalations show lengths and angles to scale, but it is harder to visualise the object in 3D

(http://cribbd.com/data/911352832057.6959_2f3c9e8a7fd53e3e5fcaf00dcda3cce4_500x306.jpg)

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18.1 Nets

To make a model of a 3D object, draw a net, where each of its faces are connected to at least one other, so the cut out will fold up to make the 3D solid.

(http://www.bbc.co.uk/staticarchive/d4203c2bcf1726751621daf4d3b30d7e586265a7.gif)

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18.2 Coordinates in 3D

As well as the usual x and y axes, a z axis specifies height above or below the horizontal grid.

(http://www.csee.umbc.edu/~rheingan/435/pages/res/3dt/3dt1-1a.gif)

To find the midpoint of a line segment, take the average of the coordinates at each end. The midpoint of the line joining (x1,y1,z1) to (x1,y2,z2) is  a

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18.3 Volume and Surface Area

Square and triangular based pyramids and cones fill exactly one third of the enveloping prism or cylinder that would contain them, therefore, the volume of a pyramid is:

Pyramid: V = 1/3 x area of base x perpandicular height

For a cone of base radius r and height h:

Cone: V = 1/3 x (pi) x r(squared) x h

To find the surface area:

Pyramid: SA =(area of face)+(area of face)+(area of face)+(area of face)

Cone: Curved SA = (pi) x r x l

Sphere: SA = 4 x (pi) x r(squared)

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18.4 Converting Between Units & 18.5 Dimensional A

Areas must be multiplied by the square of the conversion factor, and volumes by its cube.

e.g. 1cm = 10mm, 1cm(squared) = 100mm(squared), 1cm(cubed) = 1000mm(cubed)

When analysing a mathematical formula, regard numbers as dimensionless.

e.g. (pi) x r(squared) = (dimensionless) x (length) x (length)

This is length(squared) which is an area.

For 1/3 x (pi) x r(squared) x h, both 1/3 and pi are dimensionless. Therefore, the formula has dimensions: length(squared) x length = length(cubed) which is a volume

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