Perimeter & Area

1. Perimeter & Area MATH 102 Contemporary Math S. Rook

2. Overview • Section 10.3 in the textbook: • Perimeter & area • Pythagorean theorem

3. Perimeter & Area

4. Perimeter & Area in General • Perimeter: the sum of the lengths of the sides of a polygon • i.e. the length around (“rim”) the polygon • Measured using the same units as the sides • Area: the amount of space inside of the polygon • Measured in square units of the sides • On the next few slides are formulas for perimeter and area for common polygons

5. Parallelogram • Recall the shape and characteristics of a parallelogram • No special formula for perimeter • Just add the lengths of all the sides • For a parallelogram with height h and base b, area = h x b • The height is a straight vertical line • See page 471 for theory

6. Trapezoid • Recall the shape and characteristics of a trapezoid • No special formula for perimeter • Just add up the sides • For a trapezoid with lower base b1, upper base b2, and height h, area = ½ (b1 + b2) x h • b1 and b2 are parallel to each other • The height h is a straight vertical line

7. Triangle • No special formula for perimeter • Just add up the lengths of all the sides • For a triangle with a height h and a base b, area = ½ b x h • The height h is a straight vertical line

8. Rectangle & Square • Recall that a rectangle has two pairs of corresponding sides each equal in length • Given a rectangle with length l and width w: • Perimeter = 2l + 2w • Area = l x w • Recall that a square is a rectangle, but with four equal sides • i.e. l and w are the same length • Given a square with a sides of length l: • Perimeter = 4l • Area = l2

9. Circle • Recall that the line segment with endpoints at the center of the circle and on the outside of the circle is known as the radius • Given a circle with radius r: • Circumference = 2πr • Circumference is the circle’s equivalent to perimeter • Area = πr2 π (pi) is a special constant in mathematics related to the circumference of a circle and its diameter • π is infinite so we often approximate it as 3.14

10. Perimeter & Area (Example) Ex 1: Susan wishes to plant Black-Eyed Susans in her circular garden which has a radius of 5 feet. If a package of seeds will cover 12 square feet of her garden, how many whole packages of seeds must Susan buy in order to cover the entire garden?

11. Perimeter & Area (Example) Ex 2: The owners of a 50 foot by 20 foot rectangular field have decided to install a walkway which will border the field. If the walkway extends 5 feet in all directions, find the perimeter of the walkway.

12. Perimeter & Area (Example) Ex 3: Find the area of the shaded region: a) b)

13. Pythagorean Theorem

14. Pythagorean Theorem • Pythagorean Theorem: given a right triangle with legs a & b and hypotenuse c, the following relationship exists: a2 + b2 = c2 • It does not matter which of the legs is a and which is b • The hypotenuse, c, is the longest sideAND is ALWAYSopposite the 90°-angle • When solving problems with right triangles, it is often helpful to draw a picture

15. Pythagorean Theorem (Example) Ex 4: A 25-foot ladder leans on the roof of a 20 foot-tall building. How far from the building does the base of the ladder extend?

16. Pythagorean Theorem (Example) Ex 5: Find the area of the parallelogram:

17. Summary • After studying these slides, you should know how to do the following: • Solve problems involving area & perimeter of common polygons and circles • Understand and apply the Pythagorean Theorem • Additional Practice: • See problems in Section 10.3 • Next Lesson: • Volume & Surface Area (Section 10.4)