Justify area formulas for rectangles, parallelograms, triangles, and trapezoids

Triangles

First we are going to add an identical triangle onto our existing triangle to create a parallelogram. Once we do this we are able to use the parallelogram formula of b x h to find the area of the parallelogram. Since a triangle is 1/2 of the parallelogram we are going to divide our formula in half. We then find the triangle formula of A= 1/2(b x h).

Trapezoids split into 2 triangles

First we are going to split the trapezoid into 2 triangles. Then we are going to find the area of both of the triangles to find the area of them. The area formula for a triangle is 1/2(b x h). So the formulas for the two triangles are Triangle A: 1/2(b1 x h) and Triangle B: 1/2(b2 x h). The next step that happens is that we add those together to get the area of the trapezoid. We get the answer of 1/2 (b1 x h) + 1/2(b2 x h). We can then pull out comon multiples of 1/2 and h to simplify our answer. We then get the Trapezoid area formula of 1/2h(b1 x b2).

Circles

First we have to cut the circle into 8 equal parts. Then we have to put them together to look like a parallelogram. The height of the parallelogram like shape we created is going to have a height of the radius or r. We also know that the base is going to be 1/2 of the circumference. We know that the formula for circumference is C= 2 πr. So one half of the circumference or the base of our parallelogram like shape will be πr. We know now the base and the height of the parallelogram like object we have created. Now we multiple the base of πr by the height of radius or r to get the formula for the area of a circle of A= π r^2

Trapezoid split into 3 triangles

First we split the trapezoids into 3 triangles. Then we have to find the area of the three traingles A,B,C. The bases of our 3 triangles are going to be b1, c, d, the bases c and d together make the base of b2. Triangle A area is 1/2(c x h) Triangle B area is 1/2(b1 x h) and Triangle C area is 1/2(d x h). Next we are going to add the three triangle areas to find the area of the trapezoid. Triangle A 1/2( c x h) + Triangle B 1/2(c x h) + Triangle C 1/2(d x h) = Trapezoid Area. After we add them together we can pull out the common multiples of 1/2 and h. We are then left we (c x b1 x d). Then we can combine the base lengthen of c and d to create b2. We then get the fomrula of 1/2h(b1 x b2)

Rectangles

The area is the number square units that make up the rectangle. We could just count the number of square units but it doesn't always work out this way. By multiplying the number of rows or the base by the number of columns the height we can obtain the number of number of Square units or the area.

Adding an identical trapezoid

The first thing we are going to do is add an identical trapezoid onto our existing trapezoid to create a parallelogram. We are then able to use the parallelogram formula of A= b x h to find the area of our new shape. The base of our shape is going to be (b1 + b2). Both of our parallelograms bases are going to be the length of (b1 + b2). The height is going to be the height of our parallelogram. The area formula is going to be (b1 x h) + (b2 x h)and is going to find us the area of our parallelogram. We are then going to pull out the common multiplie which is h. We then get the formula of h(b1) + h(b2). Since the trapezoid is 1/2 the the area of the parallelogram we are going to divide the area by 2. We then get the answer of 1/2h(b1 x b2) which is our final answer.

Parallelogram

The law of conversations allow for us to re arrange the make up of a shape. The first thing we are going to do is cut a triangle off of the Parallelogram and move to the other side of the shape. This is going to turn the parallelogram into a rectangle. We can use the rectangle formula to figure out the area of the parallelogram that we transformed into a rectangle. Multiply the b x h and we have found the area.