1. Chapter 10 Measurement Section 10.2 Perimeter and Area

2. Measurements of Two-Dimensional Shapes Two-Dimensional shapes have many features to them. There are two of them in this section that we discuss measuring because it is useful to be able to calculate for many different reasons. Perimeter The perimeter of a simple closed plane figure is the length of its boundary. The difficulty is that the boundary might be made out of several segments or curves that need to be measured individually and added together. Perimeter of Polygons To find the perimeter of a polygon measure the lengths of the sides and add them together. 3 4 5 Perimeter = 3+ 4+5 = 12 2 5 5 2 Perimeter = 2+5+2+5 = 14 66 4 1 Perimeter = 4+6+1+6 = 17 2 2 2 7 7 Perimeter = 2+2+2+7+7=20

3. On a geoboard a unit is usually the horizontal or vertical distance between two consecutive dots. I can be other things but this is what is considered to be standard when no other unit is mentioned. 2 3 3 1 1 4 6 6 2+ 3 + 1 + 4+ 6 = 26 Find the perimeters of the two shapes pictured here. 5 7 4 4 4 + 5+ 7 + 4 = 20 Circumference: Perimeter of a Circle The perimeter of a circle (or the distance around the outside is called the circumference. It was discovered long ago the ratio between the circumference (C) and the diameter (d) (or twice the radius (2r)) is the number  (pi). We get the formulas: or

4. The length of a part of the circumference is split up just like the circle. If the circle is cut in half so is the circumference. If the circle is cut in quarters so is the circumference. Find the perimeter of the shape to the left. 4  4=  3 2  2=  1 4 8 4  + 3 +  + 1+ 4+ 8+ 4 = 20 + 2  Area The area of a two dimensional shape is a measure of how much space it takes up. This has very practical uses such as determining the amount of paint needed to paint a room or the amount of carpet needed to cover a floor. Area is usually measured in square units ( i.e. square inches, square feet, square meters etc.) although shapes other than squares can be used, but squares are the most common. The area is the number of non-overlapping squares that are required to cover up the shape. Here are some examples. 132456 798 101112 131415161718 Area = 18 square green units 123 567 4 8 Area = 8 square yellow units

5. A unit square on a geoboard is usually taken to be a square going over 1 unit and down 1 unit. Finding the area of a shape on a geoboard can be done by breaking the shape up into shapes you are more familiar with (such as rectangles) and computing the area of those shapes. This method utilizes van Hiele levels one and two to recognize one shape being made out of other less complicated shapes. First break this shape up into 3 rectangles. Area & Perimeter of Rectangles A rectangle is a shape for which the area and perimeter can be found by measuring just two distances we call the length ( l ) and width (w). A formula is given for each of the area and perimeter below. Area = (length)·(width) = l w Perimeter = 2·(length) + 2·(width) = 2 l + 2w w l Area = 3 2 1 1 3 6 2 · 3 + 1 · 1 + 3 · 6 = 6 + 1 + 18 = 25 square units

6. Another common method for finding the area of a shape besides cutting it apart is to fill in the missing part and remove it. For example in the shape to the left. Fill in the missing part to make a rectangle. The area of the rectangle is 3·4 =12, but the part that makes up the shape is only half of that which is 12  2=6. This gives a total area of: 4·4 + 6 = 22 square units. 4 4 1 15 3456 789101112 1314 2 161718 192021222324 What is the area of the rectangle below measured in blue rectangular units given to the right? The area is 24 blue rectangular units.