Height of larger triangle is twice the of the smaller triangle. 5∗ 2
- The ratio of the lengths of two similar rectangles is 2 : 3. The larger rectangle has a width of 18 cm. What is the
width of the smaller rectangle?
Answer:
2
3
=
x
18The width of the smaller rectangle is 12 cm.x= 12Circumference and Arc Length
Pi is an Irrational Number –Many students can give the definition of an irrational number. They know that an
irrational number has an infinite decimal that has no pattern, but they have not really internalized what this means.
Infinity is a difficult concept. A fun way to help the students develop this concept is to have a pi contest. The students
can chose to compete by memorizing digits of pi. They can be given points, possible extra credit, for ever ten digits
or so, and the winner gets a pie of their choice. The students can also research records for memorizing digits of pi.
The competition can be done on March 14th, pi day. When the contest is introduced, there is always a student who
asks “How many points do I get if I memorize it all?” It is a fun way to reinforce the concept of irrational numbers
and generate a little excitement in math class.
How is Pi Calculated? -Students frequently ask how mathematicians calculate pi and how far they have gotten.
One method is by approximating the circumference of a circle with inscribed or circumscribed polygons. Inspired
students can try writing the code themselves, and possibly sharing it with the class. There are many other more
commonly used methods, but they involve calculus or other mathematics that is beyond geometry students.
There AreTwo Values That Describe an Arc –The measure of an arc describes how curved the arc is, and the
length describes the size of the arc. Whenever possible, have the students give both values with units so that they
will remember that there are two different numerical descriptions of an arc. Often student will give the measure of
an arc when asked to calculate its length.
Arc Length Fractions –Fractions are a difficult concept for many students even when they have come as far as
geometry. For many of them putting the arc measure over 360 does not obviously give the part of the circumference
included in the arc. It is best to start with easy fractions. Use a semi-circle and show how 180/360 reduces to^12 ,
then a ninety degree arc, and then a 120 degree arc. After some practice with fractions they can easily visualize, the
students will be able to work with that eighty degree arc.
Exact or Approximate –When dealing with the circumference of a circle there are often two ways to express the
answer. The students can give exact answers, such as 2πcm or the decimal approximation 6.28 cm. Explain the
strengths and weaknesses of both types of answers. It is hard to visualize 13πfeet, but that is the only way to
accurately express the circumference of a circle with diameter 13 ft. The decimal approximations 41, 40. 8 , 40 .84,
can be calculated to any degree of accuracy, are easy to understand in terms of length, but are always slightly wrong.
Let the students know if they should give one, the other, or both forms of the answer.
Circles and Sectors
Reinforce –This section on area of a circle and the area of a sector is analogous to the previous section about
circumference of a circle and arc length. This gives students another chance to go back over the arguments and
logic to better understand, remember, and apply them. Focus on the same key points and methods in this section,
2.10. Perimeter and Area