Question

There are two formula to find the area of a parallelogram.

1. Base x Altitude
2. Base x Length

But when we use both formula, the area is different. why ?
Please send answer at my email address as well.

Answer 1

Conditions

There are two formulas to find the area of a parallelogram.

1. Base x Altitude

2. Base x Length

But when we use both formulas, the area is different. Why?

Please send answer at my email address as well.

Solution

There are several formulas, which are defined the area of parallelogram:

Consider a graph:

The area K of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles.

The area of the rectangle is

$$A _ {\text {rect}} = (B + A) \times H$$

and the area of a single orange triangle is

$$A _ {\text {tri}} = \frac {1}{2} A \times H.$$

Therefore, the area of the parallelogram is

$$\begin{array}{l} K = A _ {\text {rect}} - 2 \times A _ {\text {tri}} \\ = \left(\left(B + A\right) \times H\right) - \left(A \times H\right) \\ = B \times H \\ \end{array}$$

Another area formula, for two sides $B$ and $C$ and angle $\theta$ , is

$$K = B \cdot C \cdot \sin \theta .$$

The area of a parallelogram with sides B and C ( $B \neq C$ ) and angle $\gamma$ at the intersection of the diagonals is given by

$$K = \frac {| \tan \gamma |}{2} \cdot \left| B ^ {2} - C ^ {2} \right|.$$

These formulas are well-known in all mathematical societies and **always give equal results**.