and Perimeter
Area
Area Arc Length of Circular Sectors
Conclusion
MATH 113 Section 10.2: Area and Perimeter
Prof. Jonathan Duncan
Walla Walla University
Winter Quarter, 2008
Perimeter
Area
Area and Arc Length of Circular Sectors
Conclusion
Outline
1
Perimeter Area Area and Arc Length of Circular Sectors Conclusion
2
3
4
Perimeter
Area
Area and Arc Length of Circular Sectors
Conclusion
Perimeter Defined
In this section we will examine two measurements common to two-dimensional figures. Perimeter Perimeter is the distance around an object. In a polygon, this can be determined by adding the length of the sides. Example Find a formula for the perimeter of each figure:
1 2 3
A square of side length s. An isosceles triangle with base length b and side length s. A rectangle of width w and height h.
Perimeter
Area
Area and Arc Length of Circular Sectors
Conclusion
Perimeter and the Pythagorean Theorem
One of the most well-known theorems of elementary geometry has to do with side lengths (perimeter) for certain types of triangles. Pythagorean Theorem In a right triangle as shown to the right, a2 + b 2 = c 2 Proof of the Pythagorean Theorem
b a c a b
c b
a
c
b
c c a b a
Perimeter
Area
Area and Arc Length of Circular Sectors
Conclusion
A Pythagorean Example
While it may seem like the pythagorean theorem has limited applicability, it can be useful in many different situations. Example Use the pythagorean theorem to determine the length of the lake pictured below.
B
A
x
D
C
Perimeter
Area
Area and Arc Length of Circular Sectors
Conclusion
Perimeter on Geoboards
The Pythagorean theorem is also useful for finding the perimeter of polygons as they can often be broken into right triangles. Example Find the perimeter if each geoboard figure below.
Perimeter
Area
Area and Arc Length of Circular Sectors
Conclusion
Definition of Area
While perimeter is a measure of length around a figure, area is the measure of the enclosed region of the figure. Area Area is a physical quantity expressing the size of a part of a surface. Example Find a formula for the area of each figure:
1 2 3
A square of side length s. An isosceles triangle with base length b and side length s. A rectangle of width w and height h.
Perimeter
Area
Area and Arc Length of Circular Sectors
Conclusion
Area of Irregular Shapes
Area formulas for polygons are relatively simple as we can break these figures down into squares or triangles. How do we find the area of irregularly shaped figures? Example How could one measure this irregularly shape?
1 2
Use a grid or graph paper. Use triangles and rectangles to estimate. Use trapezoids to estimate. Find the area of a rectangle with the same perimeter. Weigh it.
3 4
5
Be Careful! Does number 4 above really work?
Perimeter
Area
Area and Arc Length of Circular Sectors
Conclusion
Relation Between Area and Perimeter
While there is a relationship between area and perimeter (which we have explored in lab), figures with the same perimeter need not have the same area. Example Find the area and perimeter of rectangles with the following lengths and widths. Length 80 70 60 Width 20 30 40
Perimeter
Area
Area and Arc Length Circular of Sectors
Conclusion
Square Units
It is important to remember that since area is a two dimensional measurement, the units will be different. Example How man feet are there in a square yard? Example Give the number of square yards in a rectangle that is 4 feet by 8 feet. Dimensional Analysis You can keep track of the correct units in an area measurement by multiplying the linear units together. For example, a rectangle which is w feet by l feet will have area (w l)feet feet = (w l) square feet.
Perimeter
Area
Area and Arc Length of Circular Sectors
Conclusion
Circumference and Arc Length
The perimeter of a circle has a special name. Circumference The circumference (perimeter) of a circle is given by the formula C = d = 2r What is ? is defined to be the ratio of the circumference of a circle to its diameter. This is constant for all circles. That is, = C . Solving d this for C yields the above equation. Example Find the circumference of a circle of radius 4 and a circle of diameter 7.
Perimeter
Area
Area and Arc Length of Circular Sectors
Conclusion
An Arc Length Example
Using proportions, we can find the arc length, or perimeter, of a piece of a circle. Example Find the arc length of an arc of 40 from a circle of radius 6. Solution Find the circumference of the entire circle. The first ratio is the arc length to this circumference. The second ratio is the angle measure to the entire 360 in the circle. Solving this proportion yields the arc length.
Perimeter
Area
Area and Arc Length of Circular Sectors
Conclusion
How Eratosthenes Measured the Earth
Around 240 B.C. a Greek mathematician named Eratosthenes calculated the circumference of the earth to within 2% accuracy. Calculating the Circumference of the Earth The following is the procedure Eratosthenes used to calculate the circumference of the earth. On a certain day the sun was directly overhead in the town ...