LESSON THIRTYFIVE ANOTHER DIMENSION THREEDIMENSIONAL FIGURES As you

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 3 D models in their work.

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

THREE-DIMENSIONAL FIGURES

THREE-DIMENSIONAL FIGURES

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

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 nonregular 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 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 threedimensional 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? 30 cm 10 cm 12 cm

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 30 cm 12 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 30 cm 12 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 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 210 cm² (this is the lateral area!) • So the surface area is 360 cm². 7 cm 5 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 10 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.