Area Of Polygons Worksheet With Answers

Area of Polygons Worksheet with Answers: A Comprehensive Guide

Finding the area of polygons is a fundamental concept in geometry with applications across various fields, from architecture and engineering to computer graphics and game development. Mastering this skill requires understanding the formulas for different polygon types and practicing their application through worksheets. This comprehensive guide provides a detailed explanation of how to calculate the area of various polygons, along with example problems and answers, effectively serving as a complete area of polygons worksheet with answers.

Understanding Polygons and Their Areas

A polygon is a closed two-dimensional figure with straight sides. Different polygons are classified by the number of sides they possess. Some common polygons include:

• Triangles: Three sides
• Quadrilaterals: Four sides (squares, rectangles, parallelograms, trapezoids, rhombuses)
• Pentagons: Five sides
• Hexagons: Six sides
• Heptagons: Seven sides
• Octagons: Eight sides
• Nonagons: Nine sides
• Decagons: Ten sides

The area of a polygon represents the amount of two-dimensional space enclosed within its boundaries. The formula for calculating the area varies depending on the type of polygon.

Calculating the Area of Common Polygons

Let's explore the formulas and methods for calculating the area of several common polygons:

1. Triangles

The most basic polygon, the triangle, has three sides and three angles. The area of a triangle can be calculated using the following formula:

Area = (1/2) * base * height

Where:

• base: The length of one side of the triangle.
• height: The perpendicular distance from the base to the opposite vertex (the highest point).

Example: A triangle has a base of 6 cm and a height of 4 cm. Its area is (1/2) * 6 cm * 4 cm = 12 cm².

2. Rectangles

A rectangle is a quadrilateral with four right angles. Its area is simply the product of its length and width:

Area = length * width

Example: A rectangle with a length of 8 cm and a width of 5 cm has an area of 8 cm * 5 cm = 40 cm².

3. Squares

A square is a special type of rectangle where all four sides are equal in length. Therefore, its area is:

Area = side * side = side²

Example: A square with a side length of 7 cm has an area of 7 cm * 7 cm = 49 cm².

4. Parallelograms

A parallelogram is a quadrilateral with opposite sides parallel and equal in length. Its area is given by:

Area = base * height

Where:

• base: The length of one side.
• height: The perpendicular distance between the base and the opposite side.

Example: A parallelogram with a base of 10 cm and a height of 6 cm has an area of 10 cm * 6 cm = 60 cm².

5. Trapezoids

A trapezoid is a quadrilateral with at least one pair of parallel sides (called bases). The area of a trapezoid is calculated using:

Area = (1/2) * (base1 + base2) * height

Where:

• base1 and base2: The lengths of the two parallel sides.
• height: The perpendicular distance between the two parallel sides.

Example: A trapezoid with bases of 8 cm and 12 cm and a height of 5 cm has an area of (1/2) * (8 cm + 12 cm) * 5 cm = 50 cm².

6. Rhombuses

A rhombus is a quadrilateral with all four sides equal in length. Its area can be calculated using:

Area = (1/2) * diagonal1 * diagonal2

Where:

• diagonal1 and diagonal2: The lengths of the two diagonals of the rhombus.

Example: A rhombus with diagonals of 6 cm and 8 cm has an area of (1/2) * 6 cm * 8 cm = 24 cm².

Irregular Polygons and Area Calculation

Calculating the area of irregular polygons, those without easily identifiable formulas, requires breaking them down into smaller, simpler shapes like triangles and rectangles. This process involves:

1. Dividing the polygon: Split the irregular polygon into several triangles or rectangles.
2. Calculating individual areas: Find the area of each smaller shape using the appropriate formula.
3. Summing the areas: Add up the areas of all the smaller shapes to get the total area of the irregular polygon.

Area of Polygons Worksheet: Practice Problems

Here are some practice problems to solidify your understanding:

Problem 1: Find the area of a triangle with a base of 15 cm and a height of 8 cm.

Answer: (1/2) * 15 cm * 8 cm = 60 cm²

Problem 2: Calculate the area of a rectangle with a length of 12 m and a width of 7 m.

Answer: 12 m * 7 m = 84 m²

Problem 3: Determine the area of a square with a side length of 9 in.

Answer: 9 in * 9 in = 81 in²

Problem 4: Find the area of a parallelogram with a base of 20 ft and a height of 11 ft.

Answer: 20 ft * 11 ft = 220 ft²

Problem 5: Calculate the area of a trapezoid with bases of 6 cm and 10 cm and a height of 4 cm.

Answer: (1/2) * (6 cm + 10 cm) * 4 cm = 32 cm²

Problem 6: Find the area of a rhombus with diagonals of 14 cm and 18 cm.

Answer: (1/2) * 14 cm * 18 cm = 126 cm²

Problem 7: An irregular pentagon can be divided into a rectangle (length 4 cm, width 3 cm) and a triangle (base 4 cm, height 2 cm). Find the total area of the pentagon.

Answer: Area of rectangle: 4 cm * 3 cm = 12 cm²; Area of triangle: (1/2) * 4 cm * 2 cm = 4 cm²; Total area: 12 cm² + 4 cm² = 16 cm²

Problem 8: A composite shape consists of a semicircle with a diameter of 10 cm and a rectangle with a length of 10 cm and width of 5 cm. Find the total area. (Use π ≈ 3.14)

Answer: Area of semicircle: (1/2) * π * (5 cm)² ≈ 39.25 cm²; Area of rectangle: 10 cm * 5 cm = 50 cm²; Total area: 39.25 cm² + 50 cm² ≈ 89.25 cm²

Problem 9: A regular hexagon can be divided into six equilateral triangles. If each side of the hexagon is 5 cm, find the area of the hexagon. (Hint: Use the formula for the area of an equilateral triangle: Area = (√3/4) * side²)

Answer: Area of one equilateral triangle: (√3/4) * (5 cm)² ≈ 10.83 cm²; Area of hexagon: 6 * 10.83 cm² ≈ 64.98 cm²

Advanced Concepts and Applications

The principles of area calculation extend beyond simple polygons. More advanced concepts include:

• Area of complex shapes: Combining different shapes and techniques to find the area of irregular and composite figures.
• Cavalieri's principle: Determining the area of figures based on cross-sectional areas.
• Integration: Using calculus to calculate areas of irregular shapes defined by curves.
• Coordinate geometry: Finding areas using the coordinates of vertices.

Conclusion: Mastering Area of Polygons

Understanding how to calculate the area of polygons is a crucial skill in mathematics and various related fields. This comprehensive guide, acting as a detailed area of polygons worksheet with answers, provided a thorough explanation of the formulas and methods for different polygons. Through practice problems and the exploration of advanced concepts, this resource equips learners with the knowledge and skills to confidently tackle area calculations in diverse contexts. Remember, consistent practice is key to mastering this fundamental geometrical concept. Continue practicing with different shapes and variations to strengthen your understanding and improve your problem-solving abilities.