1. Section 10.4 Areas of Polygons and Circles Math in Our World

2. Learning Objectives  Find areas of rectangles and parallelograms.  Find areas of triangles and trapezoids.  Find circumferences and areas of circles.

3. Area Formulas The area of a rectangle is the product of the length and width. If l is the length and w is the width, then A = lw. In a square, the length and width are equal, so if s is the length of the sides, A = s 2 l w s

4. EXAMPLE 1 Finding the Cost of Installing Carpet A couple plans to carpet an L-shaped living room, as shown. Find the total cost of the carpet if it is priced at $25.00 per square yard. With a little bit of ingenuity, we can divide the room into two figures we know the area of: a rectangle and a square. SOLUTION 10 ft 15 ft

5. EXAMPLE 1 Finding the Cost of Installing Carpet A = lw = 15 x 10 = 150 square feet A = s 2 = 5 2 = 25 square feet SOLUTION 10 ft 5 ft 15 ft 5 ft The total area is 150 sq. ft + 25 sq. ft = 175 square feet. Now we can use dimensional analysis to finish the calculation: It will cost $486.11 to carpet the room.

6. Area Formulas If you “cut” the bottom left corner piece of a parallelogram off and attach it to the right side. This turns the parallelogram into a rectangle, which we can find the area of using length times width. The area of a parallelogram is the product of the base and the height. If b is the length of the base and h is the height, A = bh.

7. EXAMPLE 2 Finding the Area of a Parallelogram Find the area of the parallelogram. SOLUTION The base is 12 inches and the height 7 inches, so the area is

8. Area Formulas Now that we can find the area of a parallelogram, we can use that formula to develop one for triangles. Shown below, we see a triangle with base b and height h combined with another identical triangle to form a parallelogram. The area of that parallelogram is A = bh, and since it is built from two copies of the original triangle, the area of the triangle is half as much, or A=1/2bh.

9. Area Formulas The area of a trapezoid with parallel sides a and b and height h is A = 1/2h(a + b).

10. EXAMPLE 3 Finding the Area of a Triangle Find the area of the triangle shown. SOLUTION The base is 15 ft and the height is 10 ft: The area of the triangle is 75 square feet.

11. Circles By definition, polygons have sides that are line segments. The most common geometric figure that doesn’t fit that criterion is the circle—no part of a circle is straight. A circle is the set of all points in a plane that are the same distance from a fixed point, which we call the center of the circle. Center

12. Circles Based on the definition of circle, a line segment from any point on a circle to the center is always the same length. We call this length the radius of the circle. A line segment starting at a point on a circle, going through the center, and ending at a point on the opposite side is called a diameter of the circle, and its length is twice the radius.

13. Circles The distance around the outside of a circle is called the circumference (C) of the circle. The circumference of a circle is pi (  ) times the diameter, or 2  times the radius: C =  d or C = 2  r

14. EXAMPLE 4 Finding Distance Around a Track A dirt track is set up for amateur auto racing. It consists of a rectangle with half-circles on the ends, as shown. The track is 300 yards wide, and 700 yards from end to end. What is the distance around the track?

15. EXAMPLE 4 Finding Distance Around a Track From the diagram, we can see that the diameter of the circular ends is 300 yards, so the circumference is C =  d =  (300) ≈ 942 yards. (This is the total length of the curved portion, since the two half circles make one full circle.) SOLUTION

16. EXAMPLE 4 Finding Distance Around a Track The length of each straightaway is the total length of the track (700 yards) minus twice the radius of the circular ends, which is 150 yards So each straightaway is 700 – 2(150) = 400 yards. SOLUTION Now we can find the total length: L = 942 + 400 + 400 = 1,742 yards

17. Circles The area of a circle is pi times the square of the radius: A =  r 2.

18. EXAMPLE 5 Finding Area Enclosed by a Track Find the enclosed area by the track in Example 4.

19. EXAMPLE 5 Finding Area Enclosed by a Track The area is the sum of the area of a rectangle and the area of a circle (again because the two half-circles form one full circle). SOLUTION = 400 yd 300 yd The combined area is about 120,000 + 70,650 = 190,650 square yards.