Perimeter is the total continuous distance around the outermost edge of a flat, two-dimensional shape.
If you imagine walking all the way around the very edge of a sports field or a room and returning to your exact starting spot, the total distance you walked is the perimeter.
The official SI unit (International System of Units) for measuring perimeter is the meter (m).
Because perimeter measures a direct line path, it is always expressed in regular linear units.
Common units include millimeters (mm), centimeters (cm), meters (m), and kilometers (km).
You must never use square units like cm2 for perimeter, because square units are strictly reserved for measuring flat area space, not boundary lengths.
Perimeter is a tool used to measure spaces in daily life.
Daily Examples:
To find the perimeter of a shape, add the lengths of all its outside sides.
Problem:
A garden has four sides. The sides measure 6m, 8m, 9m, and 5m. Find the total perimeter of the garden.
Explanation:
Step 1: Write down all the side lengths.
Perimeter = 6m + 8m + 5m + 9m
Step 2: Add the numbers together.
6 + 8 = 14
14 + 5 = 19
19 + 9 = 28
Final Answer = 28m
This section covers flat shapes made with straight lines. To find the perimeter of these shapes, add up the lengths of all their outside sides. Below are the steps, diagrams, and examples for each straight shape.
A triangle has three straight outer sides. To find its perimeter, simply add the lengths of all three sides together.
Formula :
For a triangle with sides a, b and c ;
Perimeter = a + b + c
Example: Find the perimeter of the triangle shown above with sides 5 cm, 6 cm, and 7 cm.
Formula:
Perimeter = a + b + c
a = 5 cm
b = 6 cm
c = 7 cm
Perimeter = 5 + 6 + 7 = 18 cm
A square is a four-sided shape with four identical, equal sides. You can shortcut the addition process by multiplying one side length by 4.
Formula :
For a square with side S ;
Perimeter = S × 4
Example: Find the perimeter of a square floor where one side (s) is 6 meters.
Formula:
Perimeter = 4 × S
S = 6 m
Perimeter = 4 × 6 = 24 m
A rectangle has two equal opposite lengths and two equal opposite widths. You add the length and width together, then double that total value.
Formula :
For a rectangle with length L and width W ;
Perimeter = 2 ( L + W )
Example: Find the perimeter of a rectangle window with length (L) 8 cm and width (W) 3 cm.
Formula:
Perimeter = 2 (L + W)
L = 8 cm
W = 3 cm
Calculation:
Perimeter = 2 × (8 + 3)
Perimeter = 2 × 11 = 22 cm
A parallelogram has opposite slanted sides that are equal in length. Its perimeter is found by doubling the sum of its two adjacent outer sides (a and b).
Formula :
For a parallelogram with adjacent sides a and b ;
Perimeter = 2 ( a + b )
Example: Find the perimeter of a parallelogram tile with matching outer sides of 9 mm and 5 mm.
Formula:
Perimeter = 2 (a + b)
a = 9 mm
b = 5 mm
Calculation:
Perimeter = 2 × (9 + 5)
Perimeter = 2 × 14 = 28 mm
A trapezium is a four-sided shape with one pair of parallel lines. All four outer edges can be completely different lengths, meaning you must add all four sides independently to find the total outer edge path.
Formula :
For a trapezium with sides a, b, c and d ;
Perimeter = a + b + c + d
Example: Find the perimeter of the trapezium shown above with sides measuring 4 cm, 6 cm, 10 cm, and 5 cm.
Formula:
Perimeter = a + b + c + d
a = 4 cm
b = 6 cm
c = 10 cm
d = 5 cm
Calculation:
Perimeter = 4 + 6 + 10 + 5 = 25 cm
A rhombus is a slanted four-sided shape where all four sides are exactly equal in length. Because every side is identical, you can find the perimeter easily by multiplying one side length by 4.
Formula :
For a rhombus with side S ;
Perimeter = S × 4
Example: Find the perimeter of a rhombus sign where each identical side (s) measures 6 meters.
Formula:
Perimeter = 4 × S
S = 6 m
Calculation:
Perimeter = 4 × 6 = 24 m
Multi-sided flat shapes built purely from straight lines joined corner-to-corner are called polygons. These boundaries follow strict addition or multiplication properties depending on their unique side patterns.
A kite is a four-sided shape with two pairs of equal-length adjacent sides. To calculate its perimeter, add the two different side lengths (a and b) together and double the total value.
Formula:
For a kite with adjacent side pairs a and b ;
Perimeter = 2 ( a + b )
Example: Find the perimeter of a kite frame where the shorter upper edge (a) is 5 cm and the longer lower edge (b) is 9 cm.
a = 5 cm
b = 9 cm
Calculation:
Perimeter = 2 × (5 + 9)
Perimeter = 2 × 14 = 28 cm
A polygon is regular if every single one of its outer side lines is exactly the same length, and all of its interior corners match perfectly. Because of this perfect uniformity, you do not need to add the sides one by one. You can use a direct multiplication shortcut.
Formula:
For a regular shape with n number of sides and side length s ;
Perimeter = n × s
Example: Find the perimeter of a regular 6-sided hexagon where each individual outer side line (s) measures 8 meters.
n = 6 sides
s = 8 m
Calculation:
Perimeter = 6 × 8 = 48 m
A polygon is irregular if its outer border lines are completely different lengths, or its interior corners have different measurements. Because there is no consistency, no shortcut multiplication layout exists. You must trace along the outside boundary path and add every individual side independently.
Formula:
Perimeter = Sum of all individual outer boundary sides
Example: Find the perimeter of an irregular 5-sided building lot with straight side lines measuring 3m, 5m, 4m, 2m, and 3m.
Sides = 3m, 5m, 4m, 2m, 3m
Calculation:
Perimeter = 3 + 5 + 4 + 2 + 3 = 17 m
This section covers flat shapes made with curved lines. To find the perimeter of a curved shape, you measure the length of its curved outer boundary instead of adding straight sides. Below are the steps, diagrams, and examples for each curved shape.
The perimeter of a perfect circle is called the circumference. It measures the continuous line path around the outside of the circle. You can find this value using either the radius (the distance from the center to the edge) or the diameter (the straight line across the entire circle through the center).
Formulas:
Using Diameter (d): Circumference = π × d
Using Radius (r): Circumference = 2 × π × r
*(Note: The diameter is always exactly twice the length of the radius, so d = 2r)*
Example: Find the outer circumference of a round dinner plate with a radius (r) of 7 cm.
Formula:
Circumference = 2 × π × r
Calculation:
Circumference = 2 × 22 7 × 7
Circumference = 2 × 22 × 1
Circumference = 44 cm
A semicircle is exactly half of a perfect circle. Its full perimeter path is made of two separate parts that you must calculate and add together: the curved outer arc plus the straight flat diameter line that seals the bottom base.
Formula:
Perimeter = π × d 2 + d
Example: Find the complete perimeter of a solid semicircle window with a straight base diameter (d) of 14 meters.
Step 1: Find the length of the curved outer arc by dividing a full circumference by 2.
Curved Arc = 22 7 × 14 ÷ 2 = 44 ÷ 2 = 22 m
Step 2: Add the straight base diameter length to the curved arc length.
Total Perimeter = Curved Arc + Diameter
Total Perimeter = 22 m + 14 m = 36 m
A quadrant is a quarter section of a circle (one-fourth of a full circle). Its boundary layout contains a curved quarter-arc path plus two straight matching radius edges that run out from the center pivot point.
Formula:
Perimeter = 2 × π × r 4 + 2r
Example: Find the total perimeter of a quadrant plate layout that has a radius (r) of 7 cm.
Step 1: Find the length of the curved quarter arc by dividing a full circumference by 4.
Curved Arc = ( 2 × 22 7 × 7 ) ÷ 4 = 44 ÷ 4 = 11 cm
Step 2: Add both straight radius boundary borders to your curved arc total.
Total Perimeter = Curved Arc + Radius + Radius
Total Perimeter = 11 cm + 7 cm + 7 cm = 25 cm
Composite arcs are custom boundaries created by mixing individual curved sections along with regular flat line tracks. These combined tracks form unusual perimeters, such as running tracks or stadium layouts.
Formula:
Perimeter = Sum of all straight segments + Sum of all curve arc lengths
Example: A running track lane has two matching straight parallel sides of 10 meters each, capped on both ends by two matching curved semicircle loops with an inner diameter of 14 meters.
Step 1: Combine the two curved semicircle caps.The two semicircular arcs together form one complete circle..
Combined Curve = 22 7 × 14 = 44 m
Step 2: Add the straight track paths to the combined curved total.
Total Track Perimeter = Combined Curve + Top Straight + Bottom Straight
Total Track Perimeter = 44 m + 10 m + 10 m = 64 m
Review these first three step-by-step worked examples to see how perimeter formulas and properties are applied to solve regular and compound geometric problems.
Example 1 (Rectangle Fence Layout)
A rectangular chicken coop has an outer length of 12 meters and an outer width of 5.5 meters. Calculate the total length of wire mesh fencing required to completely seal its outer boundary.
Solution:
Step 1: Identify the shape parameters and write down the matching perimeter formula.
Length (L) = 12 m
Width (W) = 5.5 m
Formula: Perimeter = 2 ( L + W )
Step 2: Substitute the values into the formula and solve inside the brackets first.
Perimeter = 2 × (12 + 5.5)
Perimeter = 2 × 17.5
Perimeter = 35 meters
Final Answer = 35 m
Example 2 (L-Shaped Missing Sides)
Calculate the total outer perimeter of the compound L-shaped rectilinear floor layout shown below.
Solution:
Step 1: Find the missing horizontal cut-out step line along the top inner indentation.
Total bottom width = 15 cm. Leftmost top horizontal side = 8 cm.
Missing Horizontal Side = 15 cm − 8 cm = 7 cm
Step 2: Find the missing vertical cut-out step line along the right inner indentation.
Total left vertical height = 16 cm. Rightmost vertical wall side = 9 cm.
Missing Vertical Side = 16 cm − 9 cm = 7 cm
Step 3: Sum all six outer boundary edge segments together to find the total outer path.
Perimeter = 16cm + 8cm + 7cm + 7cm + 9cm + 15cm
Perimeter = 24 + 14 + 24
Perimeter = 62 cm
Final Answer = 62 cm
Example 3 (Semicircle Protractor Edge)
A geometric protractor layout is shaped like a perfect closed semicircle. If its straight base diameter line measures 28 centimeters, calculate its total boundary perimeter path length.
Solution:
Step 1: Note the perimeter formula for a closed semicircle layout combining curved and flat segments.
Perimeter = π × d 2 + d
Diameter (d) = 28 cm, Pi (π) = 22/7
Step 2: Calculate the length of the curved top arc segment.
Curved Arc = ( 22 7 × 28 ) ÷ 2
Curved Arc = (22 × 4) ÷ 2 = 88 ÷ 2 = 44 cm
Step 3: Add the straight flat base diameter line to finish the outer boundary loop.
Total Perimeter = 44 cm (Curved) + 28 cm (Flat Base)
Total Perimeter = 72 cm
Final Answer = 72 cm
Example 4 (Algebraic Square Sheet)
The side length of a regular square metal sheet is defined by the algebraic term (3x − 2) cm. Write a simplified expression for its perimeter, and calculate its actual value when x = 5.
Solution:
Step 1: Write out the perimeter property equation for a regular square template.
Perimeter = 4 × Side Length (S)
S = (3x − 2)
Step 2: Expand the multiplication expression by applying the distributive property.
Perimeter = 4 × (3x − 2)
Perimeter = (4 × 3x) − (4 × 2)
Simplified Expression = (12x − 8) cm
Step 3: Substitute the variable condition parameter x = 5 into the expression.
Perimeter = 12(5) − 8
Perimeter = 60 − 8
Perimeter = 52 cm
Final Answer = 52 cm
Example 5 (Regular Octagon Sign)
A stop sign is manufactured in the shape of a perfect regular 8-sided octagon. If its total continuous outer perimeter measures 240 centimeters, calculate the length of one single boundary edge wall.
Solution:
Step 1: Set up the regular polygon formula for a shape with matching side lengths.
Perimeter = n × s
Total number of sides (n) = 8
Total continuous perimeter = 240 cm
Step 2: Isolate the single side variable (s) by modifying the equation layout.
240 = 8 × s
s = 240 ÷ 8
s = 30 cm
Final Answer = 30 cm
Attempt the following questions to test your understanding of perimeter concepts:
(1.) A regular rhombus-shaped tile has an individual outer edge wall measuring exactly 15 centimeters. Calculate its total perimeter.
(2.) Find the complete boundary track length of a closed semicircle protractor layout that has a straight base diameter of 7 meters.
(3.) An L-shaped rectilinear floor plan has a total vertical height of 18 meters on the far left and a total horizontal width of 14 meters across the absolute bottom. The inner horizontal indentation wall measures 6 meters, and the inner vertical step wall measures 8 meters. Determine the missing wall lengths and calculate the total outer perimeter.
(4.) A running track lane layout is built by combining two matching horizontal straight parallel paths of 40 meters each, capped on both ends by two matching curved semicircle loops that have an inner diameter line of 21 meters. Find the total outer boundary path length of the track.
(5.) A regular 8-sided octagon layout has a single straight outer side length measuring 12 centimeters. Calculate its total continuous perimeter.
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