unit 1.3  measuring line segments using squares
what do we need to know?
 The area of a triangle: (Base x Height)/2
 The area of a square: Base x Height
 The area of a "bigger" shape is equal to the sum of the areas of all the shapes that make up said shape.
 The area of a square: Base x Height
 The area of a "bigger" shape is equal to the sum of the areas of all the shapes that make up said shape.
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To measure line segments in this part of the unit, we will use two types of squares (all within a grid).
1. A square which corners are perfectly lined up:
1. A square which corners are perfectly lined up:
2. A square which corners do not linedup (tilted):
The two methods to measure the square's side length
the "inside" Method
 Figure out all the shapes inside the square.
 Calculate the area of each of those shapes.
 Add all the areas together.
 Once you have the greater area, remember that:
 Calculate the area of each of those shapes.
 Add all the areas together.
 Once you have the greater area, remember that:
linedup corners
Look at this example:
 As you can see, this type of squares is made up of 4 IDENTICAL TRIANGLES.
 That means, the area of the square is equal to the area of 4 triangles added together.
 The area of ONE TRIANGLE = (Base x Height) / 2 = (5 x 5) / 2 = 12.5 units square. Each of the triangles has an area of 12.5 units square
 The area of the square = 4 x (area of one triangle) = 4 x 12.5 = 50 units square.
So, the area of the square is 50 UNITS SQUARE
and the side length is:
 That means, the area of the square is equal to the area of 4 triangles added together.
 The area of ONE TRIANGLE = (Base x Height) / 2 = (5 x 5) / 2 = 12.5 units square. Each of the triangles has an area of 12.5 units square
 The area of the square = 4 x (area of one triangle) = 4 x 12.5 = 50 units square.
So, the area of the square is 50 UNITS SQUARE
and the side length is:
nonlined up corners
 From each of the corners, project and draw a line towards the middle until that lines intercepts another line.  When you do this, you'll notice that the inside of this square is made up of one smaller square in the middle, and four triangles.  Calculate the area of the smaller middle square.  In this example, the area is = Base x Height = 4 x 4 = 16 units squared  Notice that the 4 triangles inside the square are identical.  If we calculate the area of 1 triangle, and multiply it times 4, we will get the total triangular area.  Area of one triangle = (Base x Height)/2 = (3 x 7)/2 = 21 / 2 = 10.5 units squared.  Thus, the TOTAL triangular area is = (10.5 units squared) x 4 = 42 units squared  Now that we have the areas of all the shapes inside the big square, all we need to do is add them all together.  Total Area = Area of smaller square + Area of 4 triangles = 16 units squared + 42 units squared = 58 units squared 
The side length of this square is:
The "outside" method
THIS METHOD CAN ONLY BE USED WHEN THE SQUARE HAS ALL CORNERS LINED UP!!!
 As it works out, the area of a linedup square inside a grid is equal to the sum of the area of the 4 triangles in the outside corners.
 These triangles are identical to one another.
 To calculate the area, get the area of one triangle and multiply it times 4.
 These triangles are identical to one another.
 To calculate the area, get the area of one triangle and multiply it times 4.
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