Perimeter of Polygons - Sixth Grade Mathematics

Perimeter is the total continuous distance around the outer boundary of a closed two-dimensional polygon, measured in linear units.

The Concept of Linear Boundary Measurement

To determine the perimeter of any polygon, imagine unrolling its outer edges into a single straight line or walking directly along the perimeter fence of an enclosed property.

Walking the Perimeter of a Polygon:
                 Side b (5 m)
             +-----------------+
            /                   \
    Side a /                     \ Side c
    (4 m) /                       \ (4 m)
         +-------------------------+
                 Side d (7 m)

    Perimeter = a + b + c + d = 4 + 5 + 4 + 7 = 20 linear meters (m).

Linear units measure one-dimensional length. Common units include centimeters (cm), meters (m), kilometers (km), inches (in.), feet (ft), and yards (yd). Perimeter is never expressed in square units.

Perimeter Formulas for Common Polygons

While the fundamental method is always summing all individual side lengths, geometric symmetries yield simplified algebraic formulas.

+----------------------+-----------------------------+-----------------------------+
| Polygon              | Diagram / Characteristic    | Perimeter Formula           |
+----------------------+-----------------------------+-----------------------------+
| Any General Polygon  | Sides s1, s2, ..., sn       | P = s1 + s2 + ... + sn      |
+----------------------+-----------------------------+-----------------------------+
| Rectangle            | Length l, Width w           | P = 2l + 2w  or  2(l + w)   |
+----------------------+-----------------------------+-----------------------------+
| Square               | 4 equal sides of length s   | P = 4s                      |
+----------------------+-----------------------------+-----------------------------+
| Parallelogram        | Adjacent sides a and b      | P = 2a + 2b  or  2(a + b)   |
+----------------------+-----------------------------+-----------------------------+
| Equilateral Triangle | 3 equal sides of length s   | P = 3s                      |
+----------------------+-----------------------------+-----------------------------+
| Regular n-gon        | n equal sides of length s   | P = n * s                   |
+----------------------+-----------------------------+-----------------------------+

Example 1: Rectangle with length 14 cm and width 9 cm
           P = 2l + 2w = 2(14) + 2(9) = 28 + 18 = 46 cm.

Example 2: Regular Octagon with side 6.5 cm
           P = n * s = 8 * 6.5 = 52 cm.

Finding Missing Side Lengths from Perimeter

When the total perimeter is known along with some side lengths, algebraic equations allow us to solve for the unknown side.

Problem: A rectangle has a perimeter of 54 inches and a length of 18 inches. Find its width w.

Step 1: Write the formula:
        2l + 2w = P

Step 2: Substitute known values:
        2(18) + 2w = 54
        36 + 2w    = 54

Step 3: Solve for w:
        2w = 54 - 36
        2w = 18
        w  = 9 inches.
Check: 2(18) + 2(9) = 36 + 18 = 54 inches. (Confirmed!)

Perimeter with Algebraic Expressions

Geometric word problems frequently express side lengths in algebraic terms.

A triangle has sides measuring (x + 3), (2x - 1), and (3x + 4).
If the perimeter is 42 cm, find x and the length of each side.
Equation: (x + 3) + (2x - 1) + (3x + 4) = 42
Combine like terms: (x + 2x + 3x) + (3 - 1 + 4) = 42
                    6x + 6 = 42
                    6x     = 36
                    x      = 6
Sides:
Side 1 = 6 + 3 = 9 cm
Side 2 = 2(6) - 1 = 11 cm
Side 3 = 3(6) + 4 = 22 cm
Check: 9 + 11 + 22 = 42 cm.

Exercises

Exercise 1. Calculate the perimeter of each figure: (a) A rectangle measuring 15.5 meters by 8.2 meters; (b) A square with side length 7.4 inches; (c) A regular hexagon with side length 12.3 cm; (d) An isosceles triangle with legs of 11 cm and a base of 6 cm.

Exercise 2. A rectangular garden has a perimeter of 78 feet. If its width is 14 feet, what is its length?

Exercise 3. A farmer wants to fence a pentagonal pasture. Four of the sides measure 45 m, 60 m, 52 m, and 38 m. If the total perimeter is 240 m, find the length of the fifth side.

Exercise 4. The sides of a quadrilateral are x, 2x, (x + 5), and (3x - 2). If the perimeter is 66 inches, find the value of x and the length of the longest side.

Solutions

Solution to Exercise 1. For (a), P = 2(15.5) + 2(8.2) = 31 + 16.4 = 47.4 meters. For (b), P = 4(7.4) = 29.6 inches. For (c), P = 6(12.3) = 73.8 cm. For (d), P = 11 + 11 + 6 = 28 cm.

Solution to Exercise 2. Using P = 2l + 2w: 78 = 2l + 2(14). 78 = 2l + 28. Subtract 28 from both sides: 50 = 2l. Divide by 2: l = 25 feet. The length of the garden is 25 feet.

Solution to Exercise 3. Sum the four known sides: 45 + 60 + 52 + 38 = 195 m. Subtract this sum from the total perimeter: 240 - 195 = 45 m. The fifth side is 45 meters long.

Solution to Exercise 4. Add the side expressions to equal the perimeter: x + 2x + (x + 5) + (3x - 2) = 66. Combining like terms gives 7x + 3 = 66. Subtract 3 from both sides: 7x = 63. Divide by 7: x = 9. The four sides are: x = 9 inches, 2x = 18 inches, x + 5 = 14 inches, and 3x - 2 = 3(9) - 2 = 25 inches. The longest side measures 25 inches.