Write a single formula for area that will work for a rectangle, a parallelogram, a trapezoid, a triangle, and a square. Explain how it works.

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Megan and Rebbekah's solution

PROBLEM : To find a single formula for area that will work for a rectangle, a parallelogram, a trapezoid and a square.

FORMULA : 1/2 h(a+b)

KEY:

h = Perpendicular line rising from the base.
a = The length of the line parallel to the base.
b = The base of the shape.

SQUARE = As a can equal h, and b can also equal h, because the shape is a square. You then add h+h together, then halve them to gat the area.

a=h
b=h
area=1/2 h(h+h)
area=1/2 h(h^2)
area=h^2

RECTANGLE = because you multiply 1/2 h which is 2 (from the example on the shape) by a+b which is 20, you get 2 * 20 which equals 40 so that is the area of the example rectangle.

1/2 h = 2
a+b = 20
2*20 =40

TRIANGLE = With the triangle point a = 0 so (a+b) is really only (b); then you halve the h and multiply the half h by the b to get the area.

a = 0
(a+b) = (b)
1/2 h *(b) = area

PARALLELOGRAM = a parallelogram is worked with the same method as the rectangle, because you move the triangle on the left and place it so it makes a rectangle on the right.

TRAPEZOID = you first work out the rectangle that has been sectioned off. You work it out by using the method for the rectangle. Then join the 2 triangles together so you get one triangle.Work this triangle out with the method for triangles.Then add the two shapes answers together to get the area.

     Rectangle          Triangle

     a = h              a = 0
     b = h              (a+b) = (b)
     A = 1/2 h (h+h)    A = 1/2 hb
     A = 1/2 h (h^2)
     A = h^2         

Area = h^2 + 1/2 hb
Area = 1/2 h (a+b)

CONCLUSION = We found the formula, 1/2 h (a+b) from working out the area for the trapezoid. Then we applied this rule for the trapezoids to the other shapes. You could work out the area of the other shapes so it must have been one of the correct formulas.