How do we find the area of this pentagon?

Question

Homi · Accepted Answer

Understand the Question

To find the area of an irregular polygon such as a pentagon, we divide it into simpler, non-overlapping geometric shapes (such as triangles or trapeziums).
By drawing diagonals from one vertex to all non-adjacent vertices, an $n$ -sided polygon is split into $(n - 2)$ triangles.
The total area of the pentagon is obtained by calculating the area of each individual triangle using the formula $\text{Area} = \dfrac{1}{2} \times \text{base} \times \text{height}$ and adding them together.

Step 1 · Divide the Pentagon into Triangles

Let the vertices of the pentagon be labeled $\text{A, B, C, D,}$ and $\text{E}$ in order. By drawing diagonals $\text{AC}$ and $\text{AD}$ from vertex $\text{A}$ , the pentagon is divided into three non-overlapping triangles: $\triangle \text{ABC}$ , $\triangle \text{ACD}$ , and $\triangle \text{ADE}$ . Diagram 1

The area of any triangle is given by: $\text{Area of a triangle} = \dfrac{1}{2} \times \text{base} \times \text{height}$

For $\triangle \text{ABC}$ with base $\text{BC}$ and perpendicular height $h_1$ from vertex $\text{A}$ : $\text{Area}(\triangle \text{ABC}) = \dfrac{1}{2} \times \text{BC} \times h_1$

Similarly, calculate $\text{Area}(\triangle \text{ACD})$ and $\text{Area}(\triangle \text{ADE})$ using their respective bases and perpendicular heights.

The total area of the pentagon is the sum of these three areas: $\text{Area of Pentagon} = \text{Area}(\triangle \text{ABC}) + \text{Area}(\triangle \text{ACD}) + \text{Area}(\triangle \text{ADE})$

Step 2 · Measure and Calculate the Total Area

Measure the length of each chosen base (using a ruler).
Measure the corresponding perpendicular height for each triangle (using a set square).
Substitute the values into the formula for each triangle and sum the results to obtain the total area of the pentagon.

Answer

Divide the pentagon into three triangles by drawing diagonals from one vertex, find the area of each triangle using $\text{Area} = \dfrac{1}{2} \times \text{base} \times \text{height}$ , and sum their areas: $\text{Area of Pentagon} = \text{Area}(\triangle \text{ABC}) + \text{Area}(\triangle \text{ACD}) + \text{Area}(\triangle \text{ADE})$

Common Mistakes

Overlapping Regions: Drawing intersecting diagonals from multiple different vertices creates overlapping triangles, leading to double-counted areas. Always draw all diagonals from a single common vertex.
Slant Height vs. Perpendicular Height: Using the side length of the triangle instead of the true perpendicular altitude to the base when applying the area formula $\dfrac{1}{2} \times \text{base} \times \text{height}$ .

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