Formulas… They help us find the area. They did not fall out of the sky! In Exploration 10.7, you will develop the formulas for the area of a triangle, rectangle, and parallelogram. Now, let’s develop the formula for the area of a trapezoid.

Area of Trapezoids First method: draw a diagonal, and find the area of 2 triangles. Base 1 Base 2 Height

Area of Trapezoids Method 2: make a 180˚ rotated image; find the area, and cut it in half. Base 1 Base 2 Height Base 1 Base 2 Height

Area of a circle If you like, read Exploration It explains in more detail why the area of a circle is πr 2.

Take any circle. Subdivide it into many congruent sectors--in this case, we made 16.

Cut out each sector. Rearrange them. What shape does this remind you of? –What is the formula for finding the area of this shape? Find it!

Pythagorean Theorem The most proved theorem ever--over 300 proofs! One was done by James Garfield, before he was president of the United States. If you have a right triangle with hypotenuse of length “c”, then a 2 + b 2 = c 2.

It looks like this! a 2 + b 2 = c 2.

But we use it like this. Find the perimeter and area of this triangle. 5 feet x feet 13 feet

Other ways to make our life easy. Compare the circumference and area. r 2r

Try this--find perimeter and area 13 “ 10 “ 20 “

P = tri + rect + sem sem (.5 2π 5) A = tri + rect + sem x 2 = 13 2 x = π “ 10 “ 20 “

Try to find the shaded area Assume the trapezoid is isosceles. 24 cm 38 cm--whole base 7 cm 4 cm

Area of trapezoid - area of parallelogram Trap:.5 24 ( ) Para: 7 4 Did not need Pythagorean Theorem! 24 cm 38 cm--whole base 7 cm 4 cm

Find the perimeter and area… If it looks right or congruent, it is. (1)(2) 9 in. 18 in. 10 m 14 m4 m m 2 m

One Perimeter –Sides of large triangle: = x 2 x = = 68.6 in. Area: Note that the large triangle can be moved to make a rectangular figure. –9 18 = 162 in. 2 9 in. 18 in.

Two Perimeter: – = 35.2 m Area: –Two trapezoids and a rectangle –(.5)(2)( ) + (.5)(2)( ) –84 m 2 10 m 14 m4 m m 2 m