Understanding Perimeter

The perimeter of a shape is the total distance around its boundary. It is the sum of all the sides of a shape. Perimeter is measured in linear units such as meters, centimeters, feet, inches, etc.

Regular Polygons

1. Square

Formula: P = 4s

where s = length of one side

Example: If a square has sides of 5 cm, the perimeter is:

P = 4 × 5 cm = 20 cm

2. Rectangle

Formula: P = 2(l + w)

where l = length and w = width

Example: If a rectangle has length 8 cm and width 5 cm, the perimeter is:

P = 2 × (8 cm + 5 cm) = 2 × 13 cm = 26 cm

3. Regular Triangle (Equilateral)

Formula: P = 3s

where s = length of one side

Example: If an equilateral triangle has sides of 7 cm, the perimeter is:

P = 3 × 7 cm = 21 cm

4. Regular Pentagon

Formula: P = 5s

where s = length of one side

Example: If a regular pentagon has sides of 6 cm, the perimeter is:

P = 5 × 6 cm = 30 cm

5. Regular n-sided Polygon

Formula: P = n × s

where n = number of sides and s = length of one side

Example: If a regular octagon (8 sides) has sides of 4 cm, the perimeter is:

P = 8 × 4 cm = 32 cm

Circles

Circle (Circumference)

Formula: C = 2πr = πd

where r = radius, d = diameter, and π ≈ 3.14159

Example: If a circle has radius 7 cm, the circumference is:

C = 2 × π × 7 cm = 14π cm ≈ 43.98 cm

Alternative calculation: If the diameter is 14 cm:

C = π × 14 cm = 14π cm ≈ 43.98 cm

Irregular Shapes

1. Irregular Polygon

Formula: P = sum of all sides

P = a + b + c + d + e + f + ...

Example: If an irregular hexagon has sides of 8 cm, 6 cm, 7 cm, 9 cm, 5 cm, and 7 cm, the perimeter is:

P = 8 cm + 6 cm + 7 cm + 9 cm + 5 cm + 7 cm = 42 cm

2. Composite Shapes

Method: Identify all outer edges of the shape

P = sum of all outer edges

Example: For a shape composed of a rectangle with a semicircle:

P = 2 × length + width + semicircle arc

If the rectangle is 8 cm × 5 cm and the semicircle has diameter 5 cm:

P = 8 cm + 5 cm + 8 cm + (π × 5 cm)/2 = 21 cm + 7.85 cm = 28.85 cm

Special Cases and Applications

1. Perimeter of a Triangle Using Coordinates

When you have the vertices (coordinates) of a triangle:

Given points A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃):

Side AB = √[(x₂ - x₁)² + (y₂ - y₁)²]

Side BC = √[(x₃ - x₂)² + (y₃ - y₂)²]

Side CA = √[(x₁ - x₃)² + (y₁ - y₃)²]

Perimeter = AB + BC + CA

Example: For a triangle with vertices at (0,0), (4,0), and (2,3):

AB = √[(4 - 0)² + (0 - 0)²] = 4

BC = √[(2 - 4)² + (3 - 0)²] = √(4 + 9) = √13 ≈ 3.61

CA = √[(0 - 2)² + (0 - 3)²] = √(4 + 9) = √13 ≈ 3.61

Perimeter = 4 + 3.61 + 3.61 = 11.22 units

2. Perimeter of a Sector

Formula: P = 2r + arc length

Where arc length = r × θ (θ in radians)

Example: For a sector with radius 10 cm and central angle 60° (π/3 radians):

P = 2 × 10 cm + 10 cm × (π/3) = 20 cm + 10.47 cm = 30.47 cm

3. Perimeter with Missing Information

When some measurements are unknown, you can use geometric properties:

Pythagorean Theorem: In a right triangle, if legs are a and b, and hypotenuse is c, then c² = a² + b²
Properties of Special Triangles:
- Isosceles triangle: two sides are equal
- 30-60-90 triangle: sides are in the ratio 1 : √3 : 2
- 45-45-90 triangle: sides are in the ratio 1 : 1 : √2
Properties of Parallelograms: Opposite sides are equal