Three-Dimensional Geometry Spatial Relations

Many jobs in the real-world deal with using three-dimensional figures on two-dimensional surfaces. A good example of this is architects use drawings to show what the exteriors of buildings will look like.

How many faces do most three-dimensional figures have? Three-dimensional figures have faces, edges, and vertices. A face - is a flat surface, and edge - is where two faces meet, and a vertex - is where three or more edges meet. Volume is measured in cubic units. See the example below. Isometric dot paper can be used to draw three-dimensional figures. How many faces do most three-dimensional figures have?

Measured in cubic units3 Volume of Cylinders Measured in cubic units3

Cylinder: a cylinder is a three-dimensional figure with two circular bases. The volume of a cylinder is the area of the base B times the height h. V = Bh or V = (πr²)h

Find the volume of the cylinder V = Bh or V = πr2h V = (π · 42) · 10 V = 502.4 cm3 The volume of the cylinder is 502.4 cm3. Volume is measured in cubic units.

Effects of Changing Dimensions By changing the dimensions of a figure, it can have an effect on the volume in different ways, depending on which dimension you change. Lets look at what happens when you change the dimensions of a cylinder.

A juice can has a radius of 1. 5 in. and a height of 5 in A juice can has a radius of 1.5 in. and a height of 5 in.. Explain whether doubling the height of the can would have the same effect on the volume as doubling the radius Original V = πr²h V = π·1.5²·5 V = 11.25π cu.in. Double V = πr²h radius V = π·3²·5 V = 45π cu.in. height V = π·1.5²·10 V = 22.5π cu.in.

Volumes of Pyramids and Cones 1/3 of prisms and cylinders

A pyramid is named for the shape of its base A pyramid is named for the shape of its base. The base is a polygon, and all the other faces are triangles. A cone has a circular base. The height of a pyramid or cone is a perpendicular line measured from the highest point to the base.

A cone has a circular base A cone has a circular base. The height of a pyramid or cone is perpendicular line measured from the highest point to the base. In the cone to the left the height is h and the radius of the circular base is r. The s is the slant height which is used to measure surface area of a cone or pyramid. The volume formula for a cone is V = 1/3Bh or V = 1/3πr²h

A pyramid is named for its base A pyramid is named for its base. The base is a polygon, and all the other faces are triangles that meet at a common vertex. The height is a perpendicular line from the base to the highest point. The volume formula for a pyramid is V = 1/3Bh V = 1/3(lw)h

The volumes of cones and pyramids are related to the volumes of cylinders and prisms. V = πr²h V = Bh V = 1/3πr²h V = 1/3Bh A cone is 1/3 the size of a cylinder with the same base and height. Also, a pyramid is 1/3 the size of a prism with the same height and base.

A practical application Finding Volumes A practical application

Find the volume of the cylinder to the nearest tenth Find the volume of the cylinder to the nearest tenth. V = Bh V = πr2 · h V = 3.14 · 32 · 8.6 V = 243.036 cm3 V = 243 cm3

Example 2 Calculate the volume of: V= 1/3 pr2h V= 1/3 (p)(7)2(9) V = 147pm3

Example 3 Calculate the volume of: V = 1/3 Bh V = 1/3 (102)(15) V = 500in3 15” 10”

Example 4 Find the volume. V = 4/3 pr3 = p(3)3 =36p cm3  113.0 m3

Example 5 Dixie cups are cones with a 3 inch height and a 2 inch radius. How much water fits in one Dixie cup?

Example 6 The top of the Washington Monument in Washington, D.C., consists of a regular square pyramid with a height of 55 ft. The length of a side of the base of the pyramid is about 34.4 ft. Find the volume.

Example 7 Find the volume of a sphere with radius 9 m, both in terms of p and to the nearest tenth of a unit.

Cavalieri’s Principle for Volmue If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal .