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Geometry (Chapter 11): Measuring Length & Area
Transcript of Geometry (Chapter 11): Measuring Length & Area
Trapezoids A=(1/2)bh A = lw A = (a+b) ______ 2 x h The (Quite Obvious) Area Congruence Postulate and More 5 5 5 5 Those 2 squares' areas are the same because their side lengths are the same. The Area Congruence Postulate is basically this. 25 25 ALSO: The Area Addition Postulate states that the area of a region is the sum of the areas of its nonoverlapping parts. For example, a trapezoid could be split into a rectangle and 1-2 triangles. The sum of the areas of those shapes = the area of the trapezoid. Trapezoid split into a rectangle and triangles. Red line shows where the trapezoid was cut. Now it's time for the more advanced stuff... Area of Kites We're only going to go over this briefly, but I feel that it must be mentioned and given it's own frame. To find the area of a kite, you need to first find the lengths of its diagonals. Let's say that d1 = 5 and d2 = 12.
After you've found the values for the diagonal lengths, plug it into this equation. A = 1 2 _ (d d ) 1 2 And the answer we get for the example is 30 units squared. Ratio Time! Ratios, that's what the Area of Similar Polygons Theorem is all about. The theorem states that if two polygons are similar, in which case the ratio of their corresponding sides is a:b, than the ratio of their areas is a :b . Now, how's about we go over what a similar polygon is? 2 2 5 5 25 10 10 100 Similar polygons are polygons that have the same shape, but are not congruent. All of the angle measurements are equal to their corresponding angles. In the example above, square A (the small one) is similar to square B (the larger one). Notice the congruent angles. Square A's perimeter is 20 and Square B's perimeter is 40. Perimeter ratio is equal to side length ratio. As stated before, the ratio of the areas of similar polygons is a :b . If the side length ratio for our squares is 5:10 (simplifies to 1:2), than the area ratio is 25:100 (simplifies to 1:4). 2 2 Circles Galore! This is where I'd like to spend the most time on - circles. First, let's cover circumference (the easy part). Circumference is a circle's perimeter. We calculate the circumference of a circle with this equation: C = d ...where d is the diameter of the circle. Oh, and by the way, here's the value of PI: 3.1415926535897....... You know, this?: You can go ahead
and round that to 3.14. Up next, we're going to talk about the Arc Length Corollary. Circles Galore! (cont.) Alright, time for the really confusing equations. The Arc Length Corollary is used to find the actual length of an arc, whereas before we were only dealing with measurements in degrees. The equation goes like this if trying to find the length of an arc: Arc length of AB mAB 2 r 360 o = 5 That's an arc REMEMBER: Degrees does NOT equal length. You would not believe how often kids screw that up. OK, up next is circle area. Circles Galore! (cont.) Finding circle area is very easy, so easy that I'm surprised that I'm dedicating a whole frame for it. The equation is very simple; it goes like this: 5 A = d Where A is area, and d is diameter. Finding circle sector areas isn't much harder. Just use this equation, where n is the # of degrees of the central angle: Lastly, we have... the ending! The End, Fellas! - I would've done more, but I was pressed for time (each frame takes about 25 minutes to complete) -