How do we find the area of this pentagon?

Question

Homi · Accepted Answer

To find the area of an irregular pentagon, we divide it into simpler shapes like triangles.

Step 1 — Divide the pentagon into triangles

Let us label the vertices of the pentagon as A, B, C, D, and E in order. We can divide any polygon into triangles by drawing diagonals from one vertex. Let's choose vertex A. We draw diagonals AC and AD. This divides the pentagon into three triangles: $\triangle ABC$ , $\triangle ACD$ , and $\triangle ADE$ . The total area of the pentagon is the sum of the areas of these three triangles.

Diagram 1

To find the area of each triangle, we use its base and perpendicular height. The formula for the area of a triangle is:

$\text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

For example, to find the area of $\triangle ABC$ : We would measure the length of its base, say BC. Then, we would measure the perpendicular height from vertex A to the line segment BC. Let's call this height $h_1$ . So, Area( $\triangle ABC$ ) $= \frac{1}{2} \times \text{BC} \times h_1$ .

Similarly, we find the areas for $\triangle ACD$ and $\triangle ADE$ . For $\triangle ACD$ , we can use base AC and its perpendicular height from D. For $\triangle ADE$ , we can use base AD and its perpendicular height from E. The total area of the pentagon is the sum of these three areas.

$\boxed{\text{Area of Pentagon} = \text{Area}(\triangle ABC) + \text{Area}(\triangle ACD) + \text{Area}(\triangle ADE)}$

Step 2 — Measure the required lengths

In a real problem, the lengths of the sides and the necessary heights would be given. If not given, we would need to measure them directly from the diagram. We would use a ruler to measure the lengths of the bases. We would use a set square to draw and measure the perpendicular heights. Once all measurements are taken, we substitute them into the area formulas. Then we calculate the area of each triangle. Finally, we add these areas together to get the total area of the pentagon.

Answer

(i) To find the area of an irregular pentagon, we divide it into simpler shapes. (ii) The most common method is to divide the pentagon into triangles by drawing diagonals from one vertex. (iii) We then calculate the area of each triangle using the formula: Area $= \frac{1}{2} \times \text{base} \times \text{height}$ , and sum these areas. (iv) The necessary side lengths and perpendicular heights must be provided or measured from the diagram.

Question 14

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