# 1 Lesson 2.3.3 Finding Complex Areas. 2 Lesson 2.3.3 Finding Complex Areas California Standards:...

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*Lesson 2.3.3Finding Complex Areas

*Lesson2.3.3Finding Complex AreasCalifornia Standards:Algebra and Functions 3.1Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A= bh, C = d the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively).Mathematical Reasoning 1.3Determine when and how to break a problem into simpler parts.What it means for you:Youll see how you can use the formulas for the areas of rectangles and triangles to find areas of much more complex shapes too.Key word:complex shape

*Finding Complex AreasFinding the area of a rectangle or a triangle is one thing. Lesson2.3.3But once you can do that, you can start to find out the areas of some really complicated shapes using those very same techniques.This is an important idea in math using what you know about simple situations to find out about more complex ones.

*Finding Complex AreasFind Complex Areas by Breaking the Shape UpTheres no easy formula for working out the area of a shape like this one.But you can find the area by breaking the shape up into two smaller rectangles.Lesson2.3.3

*Finding Complex AreasExample 1Lesson2.3.3Area of large rectangle = 10 4 = 40 in2.SolutionDivide the shape into two rectangles, as shown.Now you need the dimensions of the small rectangle, b and h.So the area of the small rectangle = bh = 4 4 = 16 in2.So the total area of the shape is 40 + 16 = 56 in2.b = 10 6 = 4 in.And h = 8 4 = 4 in.Find the area of this shape.Solution follows

*Finding Complex AreasYou dont always have to break a complicated shape down into rectangles.You just have to break it down into simple shapes that you know how to find the area of.Lesson2.3.3

*Finding Complex AreasExample 2Lesson2.3.3Divide the shape into a rectangle and a triangle.SolutionArea of rectangle = 9 4 = 36 in2.So the total area of the shape is 36 + 6 = 42 in2.Find the area of the shape below.Solution follows

*Finding Complex AreasGuided PracticeSolution followsLesson2.3.31.Find the areas of the shapes below.(4 6) + (12 6) + (4 6) = 120 cm22.3.4.(3 4) + (3 2) = 18 in2(6 2) + (0.5 4 3) = 18 cm2(8 6) + (8 18) + (8 6) = 240 in2

*Finding Complex AreasComplex Areas Can Involve VariablesSometimes you have to use variables for the unknown lengths.Lesson2.3.3But you can write an expression in just the same way.

*Finding Complex AreasExample 3Lesson2.3.3Divide the shape into two rectangles.SolutionArea of the large rectangle = xy.Area of the small rectangle = ab.So the total area of the shape is xy + ab.Find the area of this shape.Solution follows

*Finding Complex AreasGuided PracticeSolution followsLesson2.3.35.Find the areas of the shapes below.bc + ab + bc = ab + 2bc6.7.8.(x 2x) + (x x) = 3x2ab + 3ab + ab = 5ab

*Finding Complex AreasYou Can Subtract Areas As WellSometimes its easier to find the area of a shape thats too big, and subtract a smaller area from it.Lesson2.3.3

*Finding Complex AreasExample 4Lesson2.3.3This time its easier to work out the area of the rectangle with the red outline, and subtract the area of the gray square.SolutionArea of red rectangle = p 2q = 2pq.Area of gray square = q q = q2.So area of original shape = 2pq q2.This time its easier to work out the area of the rectangle with the red outlineSolution followsCalculate the area of the shape below.

*Finding Complex AreasGuided PracticeSolution followsLesson2.3.39.Use subtraction to find the areas of the shapes below.(16 10) (6 5) = 130 in22ac bc10.

*Finding Complex AreasSolution followsLesson2.3.31.Find the areas of the shapes below.2.3.4.42 cm290 cm231 in28abIndependent Practice

*Round UpRemember that it doesnt matter whether your lengths are numbers or variables you treat the problems in exactly the same way. Lesson2.3.3Finding Complex AreasThats one of the most important things to learn in algebra.