3-D Objects

Spheres, Cubes, and more ...

Contents

  • Sphere: the most common 3-D object in Astronomy

  • Disks

  • Cubes and Blocks

    Spheres: The most Common 3-D Objects in Astronmy

    Spheres are the most basic three dimensional objects that you will encounter in Astronomy. A sphere is defined much like a circle , except it is in 3 dimensions. Therefore, it is defined as the set of all points in space that are equally distant from a center. As such, one of the characteristic paramters of a sphere is its radius.


    The definition of the sphere and the picture it invokes implies that it is a bounded, closed geometric figure in 3-D. Thie sllows us to define a surface area and a volume for the sphere.

    Surface Area of A Sphere
    A sphere will have a surface. We can imagine the Earth as being similar to the general shape of a sphere; it has a surface area and we know that we can go around it and come back to the location we started at. The surface area of a perfect sphere depends on its radius :

    Surface Area of a Sphere of radius r: A = 4 × p × r2

    Notice that this surface area depends on the radius squared. In terms of the units and dimentions this makes sense because the area should be given in terms of a unit of length squared. You can read more about powers and units to find out why this makes sense>

    Volume of a Sphere
    The sphere, by virute of its definition also encloses a certain amount of volume of space. The larger the radius of a sphere the more volume it will enclose, so we also expect the volume to to depend on radius. For a perfectly round sphere the volume is given by:

    Volume of a Sphere of radius r: V = 4/3 × p × r3

    Note now that the volume depends on the radius cubed. Again, in terms of units, this makes sense (length3 gives volume of an object). Also, it is probably interesting for to go back and compare and contrast the circumference and area of a circle with the surface area and volume of a sphere.

    Discs

    Another common 3-D geometrical figure that shows up in Astronomy (when one studies galaxies for instance) are disk shaped objects. To picture a disk you can imagine a penny or a any coin. It has a circular shape, but it is 3-D because it has some - albeit a rather small - thickness. Thus, a disk looks like:

    As you can see from the figure, the circular base of a disc is defined excatly like a circle was defined: it has a radius r and one can talk about the circumference and the area of this base. The thickness of the disk is given in by specifying some value of, as is shown by h in the figure.

    Volume of a Disk
    One can imagine, based on the figure above, that the larger the thickness of a disk the larger of a volume it will enclose. Also, if the disk has a larger circular base, then it will occupy more volume as well. Thus, the volume of a disc is given by:

    Volume of a disk of base radius r and thickness h : V = p × r2× h

    Cubes, and Blocks

    You are most likely already familiar, or have a picture in your own mind, of what a cube is is like. A cube is basically the 3-D version of a square .... Recall that a square is a close geometrical figure with straight sides in which all sides are equal to each other. Well, a cube has a base that is square and then it has an extension into the 3rd dimension of speace, which we can call its hight. A cube is defined as one such figure in which all sides, the base sides and the height, are all of equal length. Here is a picture of a cube with each side of length L:

    Volume of a Cube

    The volume of a cube - in the light of the definition of a disk - is equal to the base area (i.e., the area of a square) multiplied by the height. Well, since the area of a square of length L is equal to L2, and since the height of the cube is equal to L, then the volume of the cube is given by:

    Volume of a disk of base radius r and thickness h : V = L×L×L = L3

    To emphasize the point even further, notice that once again the volume is a quantity that is given by length3 and has units that are in units of length cubed.

    How Does This Relate to Astronomy?

    You might be curious as to why we need to worry about cubes ... afterall, every star and planet that we encounter is practially spherical in shape, so why worry about cubes. Cubes are actually a good picture to develop and a good way of understanding the basic notion of volume. In fact, the term for a quantity that is raised to third power is "cubed", and this is not a coincidence because a cube has a volume that is equal to the length of one side raised to the third power!

    Once you have the notion of volume in hand you will encounter terms like unit volume. When we encounter a large extended body, we often ask ourselves questions that relate to its size, and the "size" of a larged 3-D object is best envisioned in terms of its volume. Furthermore, we also ask questions about the mass of such objects, and over how much space is this mass distributed. This quantitiy is called the average density and is related to the total mass and total volume ... and the value is given as "mass per unit volume".