Area of any Polygon How do we find the area of this quadrilateral? What measurements do we need for this?

Question

Homi · Accepted Answer

To find the area of any quadrilateral, we can split it into two triangles using a diagonal.

Step 1 — Required Measurements

Let us draw a diagonal, for example, AC, to divide the quadrilateral ABCD into two triangles. These triangles are triangle ABC and triangle ADC. We need to measure the length of this diagonal AC. Let us call this length $d$ . We also need to measure the perpendicular height from vertex B to the diagonal AC. Let us call this height $h_1$ . Finally, we need the perpendicular height from vertex D to the diagonal AC. Let us call this height $h_2$ .

$\boxed{\text{Measurements needed: diagonal length } d \text{, and perpendicular heights } h_1, h_2}$

Diagram 1

Step 2 — Calculating the Area

The area of a triangle is found by multiplying half of its base by its height. For triangle ABC, the base is AC and the height is $h_1$ . The area of triangle ABC is:

$\text{Area}_{\text{ABC}} = \frac{1}{2} \times \text{AC} \times h_1$

$= \frac{1}{2} \times d \times h_1$

For triangle ADC, the base is AC and the height is $h_2$ . The area of triangle ADC is:

$\text{Area}_{\text{ADC}} = \frac{1}{2} \times \text{AC} \times h_2$

$= \frac{1}{2} \times d \times h_2$

The total area of the quadrilateral ABCD is the sum of the areas of these two triangles.

$\text{Area}_{\text{ABCD}} = \text{Area}_{\text{ABC}} + \text{Area}_{\text{ADC}}$

$= \left( \frac{1}{2} \times d \times h_1 \right) + \left( \frac{1}{2} \times d \times h_2 \right)$

We can take out the common factor $\frac{1}{2} \times d$ .

$= \frac{1}{2} \times d \times (h_1 + h_2)$

$\boxed{\text{Area} = \frac{1}{2} \times \text{diagonal length} \times (\text{sum of perpendicular heights})}$

Answer

(i) We need the length of one diagonal and the perpendicular heights from the other two vertices to that diagonal. (ii) The area is calculated by multiplying half the diagonal length by the sum of the two perpendicular heights.

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Area of any Polygon

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