Skip to main content

How to Find the Perimeter of Any Polygon

Grade: 6-7 | Topic: Geometry

What You Will Learn​

After this lesson you will be able to calculate the perimeter of any polygon — from simple rectangles and triangles to irregular shapes with many sides. You will know the shortcut formulas for common shapes, understand how to handle regular polygons, and apply perimeter calculations to real-world fencing, framing, and border problems.

Theory​

The universal rule​

A polygon is any closed shape made of straight sides. To find the perimeter of any polygon, add up the lengths of all its sides:

P=s1+s2+s3+⋯+snP = s_1 + s_2 + s_3 + \cdots + s_n

where s1,s2,…,sns_1, s_2, \ldots, s_n are the side lengths and nn is the number of sides. This rule works for every polygon, whether it has 3 sides or 30.

Shortcut formulas for common shapes​

Because certain polygons have equal sides or pairs of equal sides, we can write simpler formulas:

Triangle (3 sides): P=a+b+cP = a + b + c

Equilateral triangle (all 3 sides equal): P=3sP = 3s

Rectangle (opposite sides equal): P=2l+2wP = 2l + 2w

Square (all 4 sides equal): P=4sP = 4s

Regular polygon (all nn sides equal, side length ss): P=n×sP = n \times s

This last formula is very powerful. A regular hexagon with side 5 cm has perimeter P=6×5=30P = 6 \times 5 = 30 cm. A regular octagon with side 3 m has perimeter P=8×3=24P = 8 \times 3 = 24 m.

Irregular polygons​

For polygons where the sides are all different lengths, there is no shortcut — you simply add every side. The key is making sure you have identified all the sides, including any hidden or unlabeled ones. In many problems, you need to figure out a missing side length using information from the rest of the shape.

Finding a missing side​

Sometimes one side is not given directly. Common strategies:

  • Rectangles/parallelograms: Opposite sides are equal. If you know one pair, you know the other.
  • Right-angle shapes: A side can often be found by subtraction. For example, in an L-shaped figure, a hidden edge equals the difference between two labeled edges.
  • Triangles with the perimeter given: If you know the perimeter and two sides, the third side is s3=P−s1−s2s_3 = P - s_1 - s_2.

Worked Examples​

Example 1: Perimeter of a rectangle (easy)​

Problem: A rectangular garden is 18 m long and 11 m wide. How many metres of fencing are needed to go around it?

Step 1: Use the rectangle perimeter formula. P=2l+2w=2(18)+2(11)P = 2l + 2w = 2(18) + 2(11)

Step 2: Calculate. P=36+22=58 mP = 36 + 22 = 58 \text{ m}

Answer: 58 m of fencing

Example 2: Perimeter of a regular hexagon (easy)​

Problem: A regular hexagon has each side measuring 7 cm. Find its perimeter.

Step 1: A regular hexagon has 6 equal sides. P=n×s=6×7=42 cmP = n \times s = 6 \times 7 = 42 \text{ cm}

Answer: 42 cm

Example 3: Perimeter of an irregular quadrilateral (medium)​

Problem: A quadrilateral has sides of 5.2 cm, 8.7 cm, 6.4 cm, and 9.1 cm. Find its perimeter.

Step 1: Add all four sides. P=5.2+8.7+6.4+9.1P = 5.2 + 8.7 + 6.4 + 9.1

Step 2: Calculate. P=29.4 cmP = 29.4 \text{ cm}

Answer: 29.4 cm

Example 4: Finding a missing side from the perimeter (medium)​

Problem: A triangle has a perimeter of 42 cm. Two of its sides are 15 cm and 13 cm. Find the third side.

Step 1: Use the perimeter formula and solve for the unknown side. P=a+b+cP = a + b + c 42=15+13+c42 = 15 + 13 + c

Step 2: Simplify. 42=28+c  ⟹  c=42−28=14 cm42 = 28 + c \implies c = 42 - 28 = 14 \text{ cm}

Answer: 14 cm

Example 5: Perimeter of an L-shaped figure (challenging)​

Problem: An L-shaped room has the following exterior dimensions: the overall length is 10 m and the overall width is 8 m. A rectangular notch is cut from one corner — the notch is 4 m wide and 3 m deep. Find the perimeter of the L-shape.

Step 1: Identify all six sides of the L-shape. Working clockwise from the bottom-left:

  • Bottom: 10 m
  • Right side (full height): 8 m
  • Top (partial): 10−4=610 - 4 = 6 m
  • Drop into the notch: 3 m
  • Across the notch: 4 m
  • Left side (remaining height): 8−3=58 - 3 = 5 m

Step 2: Add all sides. P=10+8+6+3+4+5=36 mP = 10 + 8 + 6 + 3 + 4 + 5 = 36 \text{ m}

Answer: 36 m

Shortcut check: An interesting property of rectilinear L-shapes — the perimeter equals 2×(overall length+overall width)=2(10+8)=362 \times (\text{overall length} + \text{overall width}) = 2(10 + 8) = 36 m. The notch adds extra sides on the inside but removes the same amount from the outside, so the perimeter stays the same as the full rectangle!

Common Mistakes​

Mistake 1: Forgetting a side on irregular shapes

❌ An L-shape has 6 sides, but students often count only 4 (the "outer rectangle") and get a smaller perimeter.

✅ Trace the entire boundary of the shape with your finger. Count each straight segment as a separate side.

Why this matters: Composite and irregular shapes have more sides than a simple rectangle. Skipping even one side means your perimeter is too small. Drawing the shape and labeling every edge prevents this error.

Mistake 2: Confusing perimeter with area

❌ For a rectangle 8 m by 5 m: "Perimeter = 8×5=408 \times 5 = 40 m"

✅ Perimeter =2(8)+2(5)=26= 2(8) + 2(5) = 26 m

Why this matters: Multiplying gives area (in square units), not perimeter (in linear units). Perimeter adds; area multiplies. Check the units — if the answer should be in metres, you need addition, not multiplication.

Mistake 3: Using the wrong number of sides for a regular polygon

❌ "A regular pentagon has 6 sides, so P=6×4=24P = 6 \times 4 = 24"

✅ A pentagon has 5 sides: P=5×4=20P = 5 \times 4 = 20

Why this matters: The prefix tells you the number of sides — "penta" = 5, "hexa" = 6, "octa" = 8, etc. Mixing up the count changes the perimeter.

Practice Problems​

Try these on your own before checking the answers:

  1. A rectangle is 25 cm long and 14 cm wide. Find its perimeter.
  2. A regular octagon has a side length of 6 m. Find its perimeter.
  3. An irregular pentagon has sides of 3 cm, 5 cm, 4 cm, 6 cm, and 7 cm. Find its perimeter.
  4. A triangle has a perimeter of 54 cm. Two sides measure 19 cm and 17 cm. Find the third side.
  5. A swimming pool has an L-shaped outline. The overall dimensions are 15 m by 10 m, with a 5 m by 4 m notch removed from one corner. Find the perimeter.
Click to see answers
  1. P=2(25)+2(14)=50+28=78P = 2(25) + 2(14) = 50 + 28 = 78 cm.
  2. P=8×6=48P = 8 \times 6 = 48 m.
  3. P=3+5+4+6+7=25P = 3 + 5 + 4 + 6 + 7 = 25 cm.
  4. c=54−19−17=18c = 54 - 19 - 17 = 18 cm.
  5. The perimeter of any rectilinear L-shape equals 2(15+10)=502(15 + 10) = 50 m. (Or counting all 6 sides: 15+10+10+4+5+6=5015 + 10 + 10 + 4 + 5 + 6 = 50 m.)

Summary​

  • The perimeter of any polygon is the sum of all its side lengths: P=s1+s2+⋯+snP = s_1 + s_2 + \cdots + s_n.
  • For regular polygons (all sides equal), use the shortcut P=n×sP = n \times s.
  • Common formulas: rectangle P=2l+2wP = 2l + 2w, square P=4sP = 4s, equilateral triangle P=3sP = 3s.
  • For irregular or composite shapes, trace the entire boundary and label every side before adding.
  • Perimeter is measured in linear units (cm, m) — not square units. If you are multiplying dimensions, that is area, not perimeter.

Need help with perimeter problems?

Take a photo of your math problem and MathPal will solve it step by step.

Open MathPal