a. A square. This can be done by very young learners. Start with a square piece of paper and fold as follows:
1. Introduction
a. A square. This can be done by very young learners. Start with a square piece of paper and fold as follows:
b. A rhombus. Find the midpoints of the four sides, and fold:
c. A rectangle. The construction is given below.
d. A parallelogram. The construction is similar to that of a rectangle. See below.
Construct a circle (or a part of a circle) with center at the midpoint of the median and radius one half the length of the median. The circle will cut the rectangle at four points. Choose two opposite points as corners of your envelope:
Connect the ends of the median with the intersection points to form a rectangle:
Check with an index card that the marked angle really measures 90°.
Lightly score the lines outlining the rectangle so that they will fold easily, and fold:
You have a rectangular envelope made from a rectangle!
Here is an animation showing how to make it:
The procedure is the same when you start with a parallelogram. Draw the long median and a circle as you did with the rectangle, and connect the ends of the median with the intersection points.
3. Two questions:
(a) Suppose I want to make an envelope with side lengths x and y. What should be the dimensions a and b of my rectangle?
(b) Suppose I have a rectangle with side lengths a and b. What are the dimensions of the envelope that I will make?
(Notice that the area of the envelope is always half of the area of the rectangle.)
See the picture above. We have these equations:
These equations also hold for a parallelogram with base a and height b:
4. We found that designing the envelopes and how they fold, using Geometer's Sketchpad, is a nice exercise that teaches about mirror symmetry, among other concepts.