LESSON THIRTY-FIVE: ANOTHER DIMENSION

THREE-DIMENSIONAL FIGURES As you have certainly realized by now, objects in the real world do not exist in a two dimensional plane. The real world in its entirety exists in three dimensions.

THREE-DIMENSIONAL FIGURES You may remember from algebra and a bit from this class also that when we graph points, we generally do it on an x-y plane, hence the name “two dimensional”. we add a new dimension when dealing with three dimensional figures, logically named the z-plane.

THREE-DIMENSIONAL FIGURES Working with three dimensional figures is where the money is! Engineers, CAD Artists, Advertising Executives, Architects and especially Videogame Programmers, are constantly using 3D models in their work.

THREE-DIMENSIONAL FIGURES As all these professions know, there are many different perspectives to any 3D figure. You can think of a perspective as the angle from which you view an object.

THREE-DIMENSIONAL FIGURES

There are two main views we can examine of three-dimensional figures. The first is front view. This is where the front or face of an object is considered centered.

THREE-DIMENSIONAL FIGURES The second is isometric view. This is where the corner of an object is considered centered. You will need to make these on what is called isometric dot paper.

THREE-DIMENSIONAL FIGURES You may be asked to draw a rectangular prism with a length of 3, width of 4 and height of 5. The following slide shows this on isometric dot paper.

THREE-DIMENSIONAL FIGURES

We have two different definitions for the “sides” of a 3-dimensional figure. We call the bottom and top surfaces of a 3- dimensional figure, the bases. We call any flat surface of a 3-dimensional figure a face. A base is just a special type of face.

THREE-DIMENSIONAL FIGURES With that, we must discuss the various types of three dimensional figures. The first and simplest is a prism. Prisms are three-dimensional objects with two congruent and parallel faces.

THREE-DIMENSIONAL FIGURES There are many types of prisms. The easiest is a rectangular prism. In these… – There are six faces. – All faces meet at 90 . – Opposite faces are parallel.

THREE-DIMENSIONAL FIGURES A cube is a special type of this in which all sides are equal in area and are squares.

THREE-DIMENSIONAL FIGURES The next is a triangular prisms. Simply enough, this is a prism in which the bases are both triangles.

THREE-DIMENSIONAL FIGURES Beyond that, lie the polygonal prisms. These are prisms with regular or irregular polygons for their bases. We name each of these by the figure that makes its base. – For example we would call the prism below hexagonal and pentagonal respectively.

THREE-DIMENSIONAL FIGURES Prisms can be what we call right or oblique. A right prism is one in which the base edges and the lateral edges all form right angles. An oblique prism is one in which not all the base edges and lateral edges form right angles.

THREE-DIMENSIONAL FIGURES After the prisms, come the pyramids. Pyramids are three-dimensional objects with a polygon for the base, and triangles for faces.

THREE-DIMENSIONAL FIGURES A triangular pyramid is a pyramid whose base is a triangle. A triangular pyramid that’s made up of four equilateral triangles is called a tetrahedron.

THREE-DIMENSIONAL FIGURES Obviously, a square pyramid is a pyramid with a square for its base. And a rectangular pyramid is one with a rectangle for it’s base.

THREE-DIMENSIONAL FIGURES Polygonal pyramids are ones with polygons for their bases. We will be dealing primarily pyramids that have regular polygons for their bases.

THREE-DIMENSIONAL FIGURES Much like prisms, pyramids can be slanted or what we would call “straight”. These are called simply regular and non- regular respectively. We’ll come back to this in a later lesson.

THREE-DIMENSIONAL FIGURES Next are the cylinders. Cylinders are three-dimensional objects with two parallel, circular bases. Since these can only have a circle for the base, these don’t really have “types” like our previous figures.

THREE-DIMENSIONAL FIGURES The same can be said four our next figure, the cones. Cones are a three-dimensional figure with a circular base and a curved face that comes to a point.

THREE-DIMENSIONAL FIGURES Finally, one of the most common real-world objects and yet one of the most difficult is the sphere. Our only definition of this is that it is a three- dimensional figure in which all points are equidistant from a center point.

THREE-DIMENSIONAL FIGURES In this unit, we will be learning about all the figures just mentioned. Today, we will be focusing on prisms.

THREE-DIMENSIONAL FIGURES Prisms have many measures and many which you will have worked with before. First is the surface area. This is the sum area of all the bases and faces of a prism.

THREE-DIMENSIONAL FIGURES Most of these will be rectangles. Some may be regular polygons so remember that A= ½ nsa or A = ½ Pa.

THREE-DIMENSIONAL FIGURES Second is the lateral area. This is the area of all the lateral faces Basically, think of this as the surface area minus the area of the bases.

THREE-DIMENSIONAL FIGURES So let’s try a couple! What is the surface area and lateral area of the prism below? 10 cm 12 cm 30cm

THREE-DIMENSIONAL FIGURES For the surface area we simply add the area of all the faces. 2(30 x 10) + 2(12 x 10) + 2(30 x 12) =1560 cm² 10 cm 12 cm 30 cm

THREE-DIMENSIONAL FIGURES For the lateral area we simply don’t count the two bases. You can say the bases are the top and bottom. 2(30 x 10) + 2(12 x 10) + 2(30 x 12) = 960 cm² 10 cm 12 cm 30 cm

THREE-DIMENSIONAL FIGURES What about this one? Well the area of the regular hexagons is ½nsa, in this case ½(6)(5)(5) which equals 75 cm² 5 cm 7 cm 5 cm

THREE-DIMENSIONAL FIGURES There are two of these so the sum area of the bases is 150 cm². All the rectangles have a dimensions of 5 x 7 and there are six of them, so their total area is 210cm² (this is the lateral area!) So the surface area is 360 cm². 5 7 cm 5 cm

THREE-DIMENSIONAL FIGURES We could have also figured the lateral area by taking the perimeter of the polygon an multiplying by the altitude of the prism. 30 cm x 7 cm = 210 cm²

THREE-DIMENSIONAL FIGURES You’ll find that the volume of prisms is much easier to find. You all have heard length x width x height. This works for rectangular prisms but what of the others?

THREE-DIMENSIONAL FIGURES A more general formula is Ba or base x altitude. For this, we take the area of the base and multiply by the altitude. This will work for all prisms.

THREE-DIMENSIONAL FIGURES Take this example. I find the area of the hexagon ½ (6)(5)(15) which equals 225 ft² Then I multiply by the altitude to get 2250 ft³. 10 ft 15 ft

THREE-DIMENSIONAL FIGURES You will be asked to solve for lateral area, surface area and volume of prisms today. In the coming days and weeks, we will cover each of these for cones, cylinders, pyramids and spheres as well.