Difference between revisions of "3D Geometry"
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| − | '''3D Geometry''' deals with objects in 3 [[dimension]]s. For example, a drawing on a piece of paper is 2 dimensional since it has | + | '''3D Geometry''' deals with objects in 3 [[dimension]]s. For example, a drawing on a piece of paper is 2-dimensional since it has length and width. A baseball, on the other hand, is three-dimensional because it not only has length and width, but also depth. |
| − | + | = Making 3D Problems 2D = | |
| − | A very common technique for approaching 3D Geometry problems is to make it 2D. We can do this by looking at certain cross- | + | A very common technique for approaching 3D Geometry problems is to make it 2D. We can do this by looking at certain cross-section(s) of the diagram one at a time. Another common solution is to try the 2D analogous problem first. |
| − | === | + | == Example Problem == |
| + | === Problem === | ||
On a sphere with a radius of 2 units, the points <math> A </math> and <math> B </math> are 2 units away from each other. Compute the distance from the center of the sphere to the line segment <math> AB. </math> | On a sphere with a radius of 2 units, the points <math> A </math> and <math> B </math> are 2 units away from each other. Compute the distance from the center of the sphere to the line segment <math> AB. </math> | ||
| − | + | === Solution === | |
| − | First we note that the distance of a point to a line is | + | First, we note that the distance of a point to a line is the ''shortest'' distance between the point and the line. This occurs when the perpendicular to the line segment through the point is drawn. |
| − | Now we know what we are looking for, we can choose an appropriate cross-section to look at. We choose to look at the cross-section containing <math> A, B </math> and the center of the sphere as shown in the following diagram: | + | Now that we know what we are looking for, we can choose an appropriate cross-section to look at. We choose to look at the cross-section containing <math>A</math>, <math>B</math>, and the center of the sphere as shown in the following diagram: |
<center>[[Image:sphere3d.PNG]]</center> | <center>[[Image:sphere3d.PNG]]</center> | ||
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<center>[[Image:sphere3dtriangle.PNG]]</center> | <center>[[Image:sphere3dtriangle.PNG]]</center> | ||
| − | From here, we can note the 30-60-90 triangle or the Pythagorean Theorem to find that <math> x = \sqrt{3} </math> units. | + | From here, we can note the 30-60-90 triangle, or the Pythagorean Theorem, to find that <math> x = \sqrt{3} </math> units. |
== See also == | == See also == | ||
| − | * [[ | + | * [[Solid Angle]] |
| + | * [[Sphere]] | ||
| + | * [[Cylinder]] | ||
| + | * [[Cone]] | ||
| + | * [[Cube (geometry) | Cube]] | ||
| + | * [[Platonic solids]] | ||
| + | * [[Tetrahedron]] | ||
| + | * [[Octahedron]] | ||
| + | * [[Dodecahedron]] | ||
| + | * [[Icosahedron]] | ||
| + | * [[Rhombic dodecahedron]] | ||
Latest revision as of 21:45, 27 August 2024
3D Geometry deals with objects in 3 dimensions. For example, a drawing on a piece of paper is 2-dimensional since it has length and width. A baseball, on the other hand, is three-dimensional because it not only has length and width, but also depth.
Making 3D Problems 2D
A very common technique for approaching 3D Geometry problems is to make it 2D. We can do this by looking at certain cross-section(s) of the diagram one at a time. Another common solution is to try the 2D analogous problem first.
Example Problem
Problem
On a sphere with a radius of 2 units, the points 
and 
are 2 units away from each other. Compute the distance from the center of the sphere to the line segment 
Solution
First, we note that the distance of a point to a line is the shortest distance between the point and the line. This occurs when the perpendicular to the line segment through the point is drawn.
Now that we know what we are looking for, we can choose an appropriate cross-section to look at. We choose to look at the cross-section containing , , and the center of the sphere as shown in the following diagram:
We now draw in the perpendicular to 
:
From here, we can note the 30-60-90 triangle, or the Pythagorean Theorem, to find that 
units.
