What Is The Difference Between The Area And The Perimeter

The area and the perimeter are two fundamental concepts in geometry that describe different attributes of a two-dimensional shape. While both are measurements associated with shapes, they quantify distinct properties and are used for different purposes. Understanding the difference between area and perimeter is crucial for various applications, from calculating the space needed for a garden to determining the amount of fencing required for a yard.

Defining Area

Area is the measure of the two-dimensional space inside a closed shape. It is often described as the amount of surface a shape covers. Area is measured in square units, such as square inches, square feet, square meters, or square kilometers. The choice of unit depends on the size of the shape being measured.

Formulas for Calculating Area

The formula for calculating area varies depending on the shape:

• Square: Area = side × side = s²
• Rectangle: Area = length × width = l × w
• Triangle: Area = ½ × base × height = ½ × b × h
• Circle: Area = π × radius² = πr²
• Parallelogram: Area = base × height = b × h
• Trapezoid: Area = ½ × (base1 + base2) × height = ½ × (b1 + b2) × h

Examples of Area Calculations

1. Rectangle: Consider a rectangle with a length of 10 inches and a width of 5 inches. The area would be calculated as follows:

   Area = length × width Area = 10 inches × 5 inches Area = 50 square inches

   This means that the space inside the rectangle covers 50 square inches.

2. Circle: Suppose we have a circle with a radius of 7 meters. The area would be calculated as:

   Area = π × radius² Area = π × (7 meters)² Area = π × 49 square meters Area ≈ 153.94 square meters

   Thus, the surface of the circle covers approximately 153.94 square meters.

3. Triangle: Let's calculate the area of a triangle with a base of 8 cm and a height of 6 cm.

   Area = ½ × base × height Area = ½ × 8 cm × 6 cm Area = ½ × 48 square cm Area = 24 square cm

   The area inside the triangle is 24 square cm.

Practical Applications of Area

Area calculations are useful in many real-world scenarios:

• Construction: Architects and builders use area to determine the amount of flooring needed for a house, the size of a room, or the amount of roofing material required.
• Gardening: Gardeners use area to calculate how much space they have available for planting, how much fertilizer to use, or how much mulch to spread.
• Painting: Painters use area to determine how much paint is needed to cover a wall or a room.
• Real Estate: The value of a property is often determined by its area.

Defining Perimeter

Perimeter is the total distance around the outside of a two-dimensional shape. It is essentially the length of the boundary of the shape. Perimeter is measured in linear units, such as inches, feet, meters, or kilometers, because it represents a length.

Formulas for Calculating Perimeter

The formula for calculating perimeter also varies depending on the shape:

• Square: Perimeter = 4 × side = 4s
• Rectangle: Perimeter = 2 × (length + width) = 2(l + w)
• Triangle: Perimeter = side1 + side2 + side3 = a + b + c
• Circle: Perimeter = 2 × π × radius = 2πr (also known as the circumference)
• Parallelogram: Perimeter = 2 × (side1 + side2) = 2(a + b)
• Trapezoid: Perimeter = side1 + side2 + side3 + side4 = a + b + c + d

Examples of Perimeter Calculations

1. Rectangle: Using the same rectangle from before, with a length of 10 inches and a width of 5 inches, the perimeter would be:

   Perimeter = 2 × (length + width) Perimeter = 2 × (10 inches + 5 inches) Perimeter = 2 × 15 inches Perimeter = 30 inches

   The total distance around the rectangle is 30 inches.

2. Circle: For the circle with a radius of 7 meters, the perimeter (circumference) is:

   Perimeter = 2 × π × radius Perimeter = 2 × π × 7 meters Perimeter = 14π meters Perimeter ≈ 43.98 meters

   The distance around the circle is approximately 43.98 meters.

3. Triangle: Let's calculate the perimeter of a triangle with sides of 5 cm, 7 cm, and 9 cm.

   Perimeter = side1 + side2 + side3 Perimeter = 5 cm + 7 cm + 9 cm Perimeter = 21 cm

   The total length around the triangle is 21 cm.

Practical Applications of Perimeter

Perimeter calculations are useful in different practical scenarios:

• Fencing: Homeowners use perimeter to determine the amount of fencing needed to enclose a yard or garden.
• Framing: Artists and carpenters use perimeter to calculate the amount of framing material needed for a picture or a window.
• Sewing: Seamstresses use perimeter to calculate the amount of trim needed for a piece of fabric.
• Landscaping: Landscapers use perimeter to determine the amount of edging needed for a flower bed.

Key Differences Between Area and Perimeter

Visualizing Area and Perimeter

To further illustrate the difference between area and perimeter, consider a square with sides of 4 units each.

• Area: The area of the square is calculated as side × side = 4 units × 4 units = 16 square units. This means that 16 squares, each one unit by one unit, would be needed to cover the entire surface of the larger square.
• Perimeter: The perimeter of the square is calculated as 4 × side = 4 × 4 units = 16 units. This is the total length of all the sides added together, representing the distance around the square.

Notice that in this particular case, the numerical values of the area and perimeter are the same (16). However, they represent different attributes and are measured in different units. The area is 16 square units, while the perimeter is 16 linear units.

How Changing Dimensions Affects Area and Perimeter

Understanding how changes in dimensions affect area and perimeter is important in many practical applications. Let's examine a rectangle to illustrate this concept.

Increasing Dimensions

Consider a rectangle with an initial length of 5 units and a width of 3 units.

• Initial Area: Area = length × width = 5 units × 3 units = 15 square units
• Initial Perimeter: Perimeter = 2 × (length + width) = 2 × (5 units + 3 units) = 16 units

Now, let's double both the length and the width: new length = 10 units, new width = 6 units.

• New Area: Area = length × width = 10 units × 6 units = 60 square units
• New Perimeter: Perimeter = 2 × (length + width) = 2 × (10 units + 6 units) = 32 units

Notice that when the dimensions are doubled, the area is quadrupled (increased by a factor of 4), while the perimeter is only doubled.

Decreasing Dimensions

Now, let's consider the same initial rectangle (length = 5 units, width = 3 units) and halve both dimensions: new length = 2.5 units, new width = 1.5 units.

• New Area: Area = length × width = 2.5 units × 1.5 units = 3.75 square units
• New Perimeter: Perimeter = 2 × (length + width) = 2 × (2.5 units + 1.5 units) = 8 units

When the dimensions are halved, the area is reduced to one-quarter of its original value, while the perimeter is halved.

General Observations

• Changing the dimensions of a shape affects its area more significantly than its perimeter because area is a two-dimensional measurement.
• When the dimensions of a shape are multiplied by a factor of k, the area is multiplied by k², while the perimeter is multiplied by k.
• Understanding this relationship is crucial in various fields, such as scaling designs in architecture or resizing objects in manufacturing.

Complex Shapes and Composite Figures

Calculating the area and perimeter of complex shapes, also known as composite figures, involves breaking them down into simpler shapes for which the area and perimeter formulas are known.

Strategy for Complex Shapes

1. Decomposition: Divide the complex shape into simpler shapes such as rectangles, triangles, circles, and semicircles.
2. Individual Calculations: Calculate the area and perimeter of each of the simpler shapes.
3. Summation: Add the areas of all the simpler shapes to find the total area of the complex shape.
4. Perimeter Calculation: For the perimeter, sum the lengths of the outer boundaries of the complex shape. Be careful not to include the lengths of the internal lines where the simpler shapes meet.

Example: L-Shaped Figure

Consider an L-shaped figure composed of two rectangles. One rectangle has dimensions of 8 units by 3 units, and the other has dimensions of 5 units by 3 units. The two rectangles are joined to form the L-shape.

1. Area Calculation:

   • Area of Rectangle 1 = 8 units × 3 units = 24 square units
   • Area of Rectangle 2 = 5 units × 3 units = 15 square units
   • Total Area = 24 square units + 15 square units = 39 square units

2. Perimeter Calculation: The perimeter of the L-shape consists of the outer boundaries. Adding the lengths:

   • Perimeter = 8 units + 3 units + 5 units + 3 units + 3 units + 6 units = 28 units

Example: Shape Composed of a Rectangle and a Semicircle

Consider a shape composed of a rectangle (length = 10 units, width = 4 units) with a semicircle attached to one of its sides (diameter = 4 units, radius = 2 units).

1. Area Calculation:

   • Area of Rectangle = 10 units × 4 units = 40 square units
   • Area of Semicircle = ½ × π × radius² = ½ × π × (2 units)² = 2π square units ≈ 6.28 square units
   • Total Area = 40 square units + 6.28 square units = 46.28 square units

2. Perimeter Calculation: The perimeter consists of three sides of the rectangle and the arc length of the semicircle.

   • Perimeter of Rectangle sides = 10 units + 4 units + 10 units = 24 units
   • Perimeter of Semicircle (arc length) = ½ × 2π × radius = π × 2 units = 2π units ≈ 6.28 units
   • Total Perimeter = 24 units + 6.28 units = 30.28 units

Practical Examples

Landscaping Project

Suppose you are planning a garden in your backyard. The garden will be rectangular, with a length of 15 feet and a width of 10 feet. You need to buy soil to fill the garden bed and fencing to enclose it.

1. Area Calculation:

   • Area = length × width = 15 feet × 10 feet = 150 square feet
   • This tells you how much soil you need to buy to cover the garden bed.

2. Perimeter Calculation:

   • Perimeter = 2 × (length + width) = 2 × (15 feet + 10 feet) = 50 feet
   • This tells you how much fencing you need to enclose the garden.

Interior Design Project

You are redecorating a room in your house. The room is rectangular, with a length of 12 feet and a width of 10 feet. You want to paint the walls and put a decorative border around the top of the walls.

1. Area Calculation: First, calculate the area of the walls. Assuming the walls are 8 feet high:

   • Area of one long wall = 12 feet × 8 feet = 96 square feet
   • Area of the other long wall = 12 feet × 8 feet = 96 square feet
   • Area of one short wall = 10 feet × 8 feet = 80 square feet
   • Area of the other short wall = 10 feet × 8 feet = 80 square feet
   • Total wall area = 96 + 96 + 80 + 80 = 352 square feet
   • This tells you how much paint you need to buy to cover the walls.

2. Perimeter Calculation:

   • Perimeter = 2 × (length + width) = 2 × (12 feet + 10 feet) = 44 feet
   • This tells you how much decorative border you need to go around the top of the walls.

Conclusion

Area and perimeter are distinct yet related concepts in geometry, each providing valuable information about the properties of a shape. Area measures the space inside a shape, while perimeter measures the distance around it. Understanding the difference between area and perimeter is crucial for solving a wide range of practical problems in fields such as construction, gardening, interior design, and more. By knowing the appropriate formulas and how to apply them, one can accurately calculate these measurements for various shapes, from simple squares and rectangles to complex composite figures. As demonstrated, changes in dimensions affect area and perimeter differently, with area being more sensitive to dimensional changes due to its two-dimensional nature.