Class 8 - Area Of A Polygon

Area of a polygon means the space inside a closed shape with straight sides. To find it, use the correct formula based on the shape. For regular polygons, the area can be found with perimeter and apothem. For irregular polygons, split the shape into smaller parts and add their areas. This topic is useful in geometry, maps, buildings, and design. Learning it helps students solve shape problems easily and understand real life measurements better. Shapes are everywhere around us the floor tiles in your house, the field where you play cricket, the stop signs on roads. Many of these shapes are polygons, and knowing how to find their area is super useful in real life. Different polygons need different methods to find their area. A triangle needs one formula, a rectangle needs another, and irregular polygons need special techniques. 

Table of Contents

• What is the Area of a Polygon?
• Area of Polygon Formulas
• Area of Polygon Formulas Table
• Area of Regular Polygons
• Area of Irregular Polygons
• Area of Polygons with Coordinates
• Important Properties Area of a Polygon
• Area Of A Polygon Solved Example

What is the Area of a Polygon?

Have you ever wondered how much space a shape takes up That's where the concept of area comes in. The area of a polygon is simply the amount of space that sits inside the shape's boundaries. Think of it like measuring how much paint you need to cover a wall or how much grass seed you need for your yard. We measure area in square units. These could be square centimeters, square meters, square inches, or even square miles. The unit depends on what you're measuring. A polygon can be any shape with straight sides, from a simple triangle to a more complex octagon. Well, architects use polygon areas to design buildings. Farmers calculate field areas to plan crops. Engineers use these calculations for construction projects. In fact, nearly every profession that deals with physical space uses polygon area calculations. Different types of polygons have different formulas.

Area of Polygon Formulas

Let us discuss here some formulas for different types of polygons.

Triangle

For a triangle, you take half of the base times the height. The base is any side you choose, and the height is the perpendicular distance from that side to the opposite corner.

Area = ½ × Base × Height.

Rectangles

Rectangles are super easy. Just multiply length by width. Since all angles in a rectangle are right angles, this multiplication gives you the exact space inside.

Area = Length × Width

Square

A square is just a special rectangle where all sides are equal. So instead of multiplying different numbers, you square one side.

Area = Side × Side Or  Area = Side²

Parallelograms

Parallelograms look like rectangles that got pushed over. You can't just multiply length and width because of the angle. Instead, multiply the base by the perpendicular height (not the slanted side). 

Area = Base × Height

Trapezoids

Trapezoids have one pair of parallel sides. This makes them interesting because you need both parallel sides to find the area. You add both parallel sides, divide by two, then multiply by the height.

Area = ½ × (Base₁ + Base₂) × Height.

Area of Polygon Formulas Table

Area of Regular Polygons

Regular polygons are the organized, symmetrical ones. All sides are the same length, and all angles are identical. This regularity actually helps us calculate their area pretty efficiently. The method to calculating regular polygon areas is something called the apothem. It sounds fancy, but it's just the distance from the center point of the polygon to the midpoint of any side. Once you have the apothem, the calculation becomes straightforward.

You can use this formula: Area = ½ × Perimeter × Apothem Or Area = ½ × n × s × a

The "n" is how many sides your polygon has. The "s" is the length of one side (remember, they're all equal). The "a" is the apothem measurement.

Let's work through a regular hexagon to see how this plays out. A hexagon has six sides. each side is 4 centimeters long. First, calculate the perimeter: 6 sides times 4 cm equals 24 cm. For a hexagon with 4 cm sides, the apothem works out to about 3.46 cm. Now plug these into the formula: ½ × 24 × 3.46 = 41.52 cm². That's your area. This method works for any regular polygon, whether it's a pentagon, octagon, or anything else.

Area of Irregular Polygons

Not all polygons are nice and regular. Many shapes you'll encounter in the real world have sides of different lengths and angles that don't match. These are irregular polygons, and they're trickier to measure, but definitely not impossible. There's a clever method called the Shoelace Formula that works for any irregular polygon. The formula got its name because the process of multiplying and subtracting coordinates looks like you're lacing up a shoe. Weird name, but it actually works great. Here's how you use it step by step. First, write down all the corner points of your polygon using their coordinates. Make sure you go around the polygon in order, either clockwise or counterclockwise. Second, you multiply the coordinates in a specific pattern and add those products up. Then you do the same process backwards and subtract. Finally, divide everything by 2 and take the absolute value.

Let us see that how it looks like with an irregular quadrilateral. the Shoelace Formula is that it works for absolutely any polygon shape, whether it's regular or irregular, whether it has 5 sides, 10 sides, or even more. You just need to know the coordinates of each corner point.

Area of Polygons with Coordinates

When you have the actual coordinates of each corner of a polygon, calculating the area becomes pretty straightforward using the Shoelace Formula. You might be wondering why this is useful. Well, if you're working with maps, computer graphics, or surveying land, you often get data as coordinates rather than measurements of sides and heights.

The Shoelace Formula is: Area = ½ |Σ(x₁y₂ - x₂y₁)|

Step 1: Write out all your coordinates in order around the polygon.

Step 2: Multiply each x-value by the next y-value and add all those products together. This is the "forward" direction.

Step 3: Multiply each y-value by the next x-value and add those together. This is the "backward" direction.

Step 4: Subtract the backward total from the forward total.

Step 5: Take the absolute value (make it positive if it's negative) and divide by 2.

Let me show you this with an actual triangle with coordinates: A(0,0), B(5,0), C(2,4)

x	y
0	0
5	0
2	4

Here's the actual calculation:

• Forward direction: (0×0) + (5×4) + (2×0) = 0 + 20 + 0 = 20
• Backward direction: (0×5) + (0×2) + (4×0) = 0 + 0 + 0 = 0
• Take the difference: 20 - 0 = 20
• Divide by 2: 20 ÷ 2 = 10 square units

That's the area: 10 square units. 

Important Properties Area of a Polygon

Polygons have several key features that affect area calculation:

• Sides: Straight lines forming the polygon's boundary
• Vertices: Points where two sides meet
• Angles: The space between adjacent sides
• Perimeter: The total distance around the polygon
• Regularity: Equal sides and angles (regular polygons) or unequal (irregular polygons)

Regular polygons have symmetry, making area calculations simpler. Irregular polygons have different side lengths and angles, requiring different methods.

Area Of A Polygon Solved Example

1: Area of a Regular Pentagon

Question: Find the area of a regular pentagon with side length 6 cm and apothem 4.1 cm.

Solution: Use the formula for a regular polygon:

A=12×p×a

Where:

• p= perimeter

•  a = apothem

Step 1: Find perimeter p=5×6=30cm

Step 2: Apply formula A=12×30×4.1

Step 3: Calculate  A=61.5 cm2

Final Answer: Area = 61.5 cm²

2: Area of a Triangle (Polygon with 3 sides)

Question: Find the area of a triangle with base 10 cm and height 5 cm.

Solution: Use the triangle area formula:

Step 1: Substitute values A=12×10×5

Step 2: Calculate A=25 cm2

Final Answer: Area = 25 cm²

3: Area of a Rectangle (4-sided polygon)

Question: Find the area of a rectangle with length 8 cm and width 3 cm.

Solution: Use the formula: A=l×w

Step 1: Substitute valuesA=8×3

Step 2: CalculateA=24 cm2

Final Answer: Area = 24 cm²

4: Area of an Irregular Polygon (Using Division Method)

Question: Find the area of an irregular polygon by dividing it into two rectangles:

• Rectangle 1: 6 cm × 4 cm

• Rectangle 2: 3 cm × 2 cm

Solution:

Step 1: Find area of each part

• Area₁ =6×4=24cm²

• Area₂ =  3×2 = 6 cm²

Step 2: Add areas

Total Area =24+6=30 cm2

Final Answer: Area = 30 cm²

5: Area Using Coordinates (Advanced)

Question: Find the area of a polygon with vertices: (0,0), (4,0), (4,3), (0,3)

Solution: This forms a rectangle.

Step 1: Length = 4, Width = 3

Step 2: Area =4×3=12 cm2

Final Answer: Area = 12 cm²