Areas and perimeters:
The perimeter of a two dimensional ¯gure is the sum of the lengths of the sides. Here are some of the
formulas you should know:
¯gure
rectangle
parallelogram
triangle
trapezoid
circle
area
base £ height
base £ height
(1/2)£ base £ height
(1/2) £(b1 + b2 ) £ height
¼r 2
Note that a rectangle is a special case of a parallelogram. Also you should know that the circumference
of a circle is 2¼r where r is the radius.
Heron's Formula: Given a triangle with sides a,b,c, the area of the triangle is
p
a = s(s ¡ a)(s ¡ b)(s ¡ c)
where s =
a+b+c
is the semi-perimeter.
2
Here is a link to a proof of the Pythagorean Theorem. Before you look at the proof, see if you are as
clever as Pythagoras. Can you prove the Pythagorean Theorem, which states that
c2 = a2 + b2
for a right triangle with sides a and b and hypotenuse c?
Click here for a clear and elegant proof. This is a short Java program. It will work if you use Netscape
as your browser, and it might not work if you use an older version of Internet Explorer.
Pythagoras introduced the concept of giving a proof in mathematics. He invented a musical scale, not
the equal temperment, but one based on mathematics. He was a philosopher and mixed religion and
mysticism. There are cults even today based on his philosophies. If you have some time to spend, enter
Phythagoras as a search word using Yahoo or some other search engine. You will ¯nd other proofs of the
Pythagorean Theorem. Eventually you will come to some weird stu®. (Not endorsed or recommended
by me. It's just that you might ¯nd it interesting.)
Volume and Surface Area:
¯gure
rectangular box
right cylinder
right circular cylinder
sphere of radius R
cone
right circular cone
volume
l£w£h
(area of base )£ height
¼r 2 £ h
(4=3)¼R3
(1/3) £ area of base £ height
(1=3)£ area of base £ height
surface area
2w£h+2l£h+2w£l
2¼rh + 2¼r 2
4¼R2
¼rs, s= slant height
Scaling: if you multiply the linear dimension by a factor of f, the area changes to f 2 £ area, and the
volume changes to f 3 £ volume. Think of the original ¯gure or solid as being made up of little squares
or cubes, and the new ¯gure or solid made up of little squares or cubes that have been expanded or
contracted according to the scale factor f.
CAVALIERI'S THEOREM:
Two solids with equal heights and identical parallel cross-sectional areas have the same volume.
Example 1: A twisted solid: A square of side length s lies in a plane perpendicular to a line L. One vertex
of the square lies on L. As this square moves a distance h along L, the square turns one revolution about
L to generate a corkscrew-like column with square cross sections. Find the volume of the column.
Answer: the cross section area at each level of the twisted column is the same as the cross section area
of the original column, so they both have the same volume: s2 £ h
Example 2: A solid lies between planes perpendicular to the x-axis at x=0 and x=12. The cross sections
by planes perpendicular to the x-axis are circular disks whose diameters run from the line y=x/2 to the
line y=x. Find the volume of the solid.
Answer: This solid has the same cross section area as a cone with height 12 and diameter of base 6, hence
the volume is (1=3) £ ¼ £ 32 £ 12 = 36¼: