Subject: Re: Geometry Question
From: Martha Haehl (haehl@kcmetro.cc.mo.us)
Date: Sun Apr 02 2000 - 20:39:54 EDT
This may not be what you are looking for, but here are some ideas to help students estimate volumes in general.
1. If the students have the physical object, have them place the object in water and calculate water displacement--either by water spilled over, or water level change. Then have the students similary calculate the volume of a right circular cylinder of the same radius and height and a rectangular box with length and width = the diameter and the height = height of the cone and cylinder. Calculate the ratios to use in estimating the volume of a cone from the circumscribed cylinder or box. A similar activity could be done with prisms. I haven't done this activity but have talked to a technical instructor who uses this method.
2. Or have them weigh the objects (if they are solids of the same material). In either experiment, use ratios to estimate the volume of the cone from the volume of the cylinder or box. Comparison by this method would explain why the volume of a right circular cone is 1/3 of the volume of a right circular cylinder of the same radius and height. I know a shop teacher who did this.
3. The third would be more abstract but would not require the props. Start by comparing the area of a circle to the area of a circumscribed square. Have student draw the circles and squares on a grid and estimate the number of whole squares (including ones that have to be pieced together from parts) that lie within the circle. Have them calculate a ratio of area (squares within) the circle to the area of the circumscribed square. Compare their ratios with the ratio they get by calculating area of circle = pi*(r)^2 = pi*(d/2)^2 = (pi/4)*(d)^2 = (pi/4)*(area of square). This helps set the stage for estimations of volumes.
To move to 3-d, then the volume of the right circular cylinder would be (pi/4)*(volume of the circumscribed box of the same height). To build towards understanding why the right circular cone has 1/3 the volume of the right circular cylinder of the same height and radius, first have them see what happens to volume of the cylinder if the radius is halved. The volume of the smaller cylinder is about 79% of the larger one. (The exact ratio is pi/4, or about 0.79:1.) Discuss why the outside (discarded volume of the larger cylinder) with thickness 1/2 r has more volume than the inside cylinder of radius r. This would set the stage for observing why 2/3 of the cylinder gets "discarded" if a right circular cone of same radius and height is cut away from the cylinder.
In a world with perfect props, students could cut perfect right circular cones from right circular cylinders made of a pliable material (like modeling clay) and compare the volume of the cone and the volume left over and observe that the discarded materials are about double the materials in the cone.
Martha
----- Original Message -----
From: John Gann
To: MATHEDCC, Subscribers
Sent: Sunday, April 02, 2000 3:42 PM
Subject: Geometry Question
Anyone know of a way to estimate the volume and surface area of:
1) a right-circular cone
2) a pyramid
I'm trying to do some investigations with my class without resorting to looking up formulas in a text book. Thanks if you can help.
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