Finding Complex Areas

1
Finding Complex Areas
Lesson 2.3.3
2
Lesson 2.3.3
Finding Complex Areas
California Standards Algebra and Functions
3.1 Use variables in expressions describing
geometric quantities (e.g., P 2w 2l, A 
½bh, C pd the formulas for the perimeter of
a rectangle, the area of a triangle, and the
circumference of a circle, respectively). Mathemat
ical Reasoning 1.3 Determine 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

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Lesson 2.3.3
Finding Complex Areas
Finding the area of a rectangle or a triangle is
one thing.
But once you can do that, you can start to find
out theareas 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.
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Lesson 2.3.3
Finding Complex Areas
Find Complex Areas by Breaking the Shape Up
Theres 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.
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Lesson 2.3.3
Finding Complex Areas
Example 1
Find the area of this shape.
Solution
Divide the shape into two rectangles, as shown.
Area of large rectangle 10 4 40 in2.
Now you need the dimensions of the small
rectangle, b and h.
b 10 6 4 in.
And h 8 4 4 in.
So the area of the small rectangle bh 4 4
16 in2.
So the total area of the shape is 40 16 56
in2.
Solution follows
6
Lesson 2.3.3
Finding Complex Areas
You 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.
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Lesson 2.3.3
Finding Complex Areas
Example 2
Find the area of the shape below.
Solution
Divide the shape into a rectangle and a triangle.
Area of rectangle 9 4 36 in2.
So the total area of the shape is 36 6 42 in2.
Solution follows
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Lesson 2.3.3
Finding Complex Areas
Guided Practice
Find the areas of the shapes below.
1.
2.
(3 4) (3 2) 18 in2
(6 2) (0.5 4 3) 18 cm2
3.
4.
(4 6) (12 6) (4 6) 120 cm2
(8 6) (8 18) (8 6) 240 in2
Solution follows
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Lesson 2.3.3
Finding Complex Areas
Complex Areas Can Involve Variables
Sometimes you have to use variables for the
unknown lengths.
But you can write an expression in just the same
way.
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Lesson 2.3.3
Finding Complex Areas
Example 3
Find the area of this shape.
Solution
Divide the shape into two rectangles.
Area of the large rectangle xy.
Area of the small rectangle ab.
So the total area of the shape is xy ab.
Solution follows
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Lesson 2.3.3
Finding Complex Areas
Guided Practice
Find the areas of the shapes below.
5.
6.
(x 2x) (x x) 3x2
7.
8.
bc ab bc ab 2bc
ab 3ab ab 5ab
Solution follows
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Lesson 2.3.3
Finding Complex Areas
You Can Subtract Areas As Well
Sometimes its easier to find the area of a shape
thats too big, and subtract a smaller area from
it.
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Lesson 2.3.3
Finding Complex Areas
Example 4
Calculate the area of the shape below.
Solution
This time its easier to work out the area of the
rectangle with the red outline, and subtract
the area of the gray square.
This time its easier to work out the area of the
rectangle with the red outline
Area of red rectangle p 2q 2pq.
Area of gray square q q q2.
So area of original shape 2pq q2.
Solution follows
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Lesson 2.3.3
Finding Complex Areas
Guided Practice
Use subtraction to find the areas of the shapes
below.
9.
10.
(16 10) (6 5) 130 in2
2ac bc
Solution follows
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Lesson 2.3.3
Finding Complex Areas
Independent Practice
Find the areas of the shapes below.
1.
2.
42 cm2
90 cm2
3.
4.
31 in2
8ab
Solution follows
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Lesson 2.3.3
Finding Complex Areas
Round Up
Remember that it doesnt matter whether your
lengths are numbers or variables you treat the
problems in exactly the same way.
Thats one of the most important things to learn
in algebra.