How to Find the Perimeter of Polygons

How to Find the Perimeter of Polygons

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The perimeter of a polygon is the total distance around it. Add every side length. For a regular polygon, all side lengths are equal, so multiply one side by the number of sides.

Core Formulas

  • Any polygon: \(P=s_1+s_2+s_3+\cdots\).
  • Regular polygon: \(P=ns\), where \(n\) is the number of sides and \(s\) is one side length.
  • Missing side: subtract the known sides from the total perimeter.

Worked Example

A regular hexagon with side length \(8\) has perimeter \(6(8)=48\).

Video Lesson

Original Practice Figures

These saved figures are kept with the lesson for continuity. The exercise text below gives all needed measurements.

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Perimeter of Polygons

Think of this lesson as more than a rule to memorize. Perimeter of Polygons is about shape relationships, formulas, and units. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Geometry formulas work because they measure a specific feature: length around, space inside, or space enclosed by a solid. Match the question to the measurement first.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

  • Sketch or label the shape.
  • Decide whether the question asks for length, area, volume, or surface area.
  • Substitute values into the matching formula.
  • Keep units squared for area and cubed for volume.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Exercises

Solve each ACT-style practice problem. The questions increase in difficulty.

1) Find the perimeter of a triangle with side lengths 5, 7, and 9 cm.

2) Find the perimeter of a rectangle with length 12 ft and width 5 ft.

3) Find the perimeter of a regular pentagon with side length 6 in.

4) Find the perimeter of a quadrilateral with sides 8, 10, 13, and 15 yd.

5) A regular octagon has side length 4.5 cm. Find its perimeter.

6) A triangle has perimeter 36 m and two sides 10 m and 11 m. Find the third side.

7) A regular decagon has perimeter 120 ft. Find one side length.

8) A rectangle has perimeter 50 in and length 16 in. Find its width.

9) Find the perimeter of a regular 12-gon with side length 7.

10) A garden has sides 18, 24, 18, 24, and 30 ft. Find the perimeter.

11) A hexagon has side lengths 9, 10, 11, 12, 13, and 14 cm. Find the perimeter.

12) A regular polygon has 15 sides of length 3.2 m. Find its perimeter.

13) A fence surrounds a rectangular field 80 yd by 45 yd. Find the perimeter.

14) A polygon has perimeter 94 and known sides 12, 17, 20, and 25. Find the missing side.

15) A regular hexagon has the same perimeter as a square with side 18. Find the hexagon side length.

16) The sides of a triangle are \(x\), \(x+3\), and \(2x+1\). If the perimeter is 40, find \(x\).

17) A rectangle has width \(w\) and length \(3w\). Its perimeter is 96. Find the length.

18) A regular polygon has perimeter 84 and side length 7. How many sides does it have?

19) An irregular octagon has side lengths 4, 6, 6, 7, 8, 8, 10, and 11. Find the perimeter.

20) A walking path goes around a regular 18-sided garden with side length 2.5 m. Two laps are walked. How far is that?

 
1) Perimeter means add all side lengths.
\(P=5+7+9=21\).
The perimeter is \(21\) cm.
2) A rectangle has two lengths and two widths.
\(P=12+5+12+5=34\).
The perimeter is \(34\) ft.
3) A regular pentagon has \(5\) equal sides.
\(P=5(6)=30\).
The perimeter is \(30\) in.
4) Add the four side lengths.
\(P=8+10+13+15=46\).
The perimeter is \(46\) yd.
5) A regular octagon has \(8\) equal sides.
\(P=8(4.5)=36\).
The perimeter is \(36\) cm.
6) Add known sides: \(10+11=21\).
Subtract from total: \(36-21=15\).
The third side is \(15\) m.
7) For a regular polygon, \(P=ns\).
\(120=10s\).
So \(s=12\) ft.
8) Use \(P=2L+2w\).
\(50=2(16)+2w=32+2w\).
\(w=9\) in.
9) A regular 12-gon has \(12\) equal sides.
\(P=12(7)=84\).
The perimeter is \(84\) units.
10) Add all sides.
\(P=18+24+18+24+30=114\).
The perimeter is \(114\) ft.
11) Add the six sides.
\(P=9+10+11+12+13+14=69\).
The perimeter is \(69\) cm.
12) Use \(P=ns\).
\(P=15(3.2)=48\).
The perimeter is \(48\) m.
13) Use \(P=2L+2w\).
\(P=2(80)+2(45)=160+90=250\).
The perimeter is \(250\) yd.
14) Add known sides: \(12+17+20+25=74\).
Missing side \(=94-74=20\).
The missing side is \(20\) units.
15) The square perimeter is \(4(18)=72\).
A regular hexagon has \(6\) equal sides, so each side is \(72/6=12\).
16) Add the sides: \(x+(x+3)+(2x+1)=40\).
\(4x+4=40\), so \(4x=36\).
\(x=9\).
17) Use \(P=2L+2w\) and \(L=3w\).
\(96=2(3w)+2w=8w\), so \(w=12\).
The length is \(3w=36\).
18) Use \(P=ns\).
\(84=n(7)\).
Thus \(n=12\) sides.
19) Add all eight sides.
\(P=4+6+6+7+8+8+10+11=60\).
The perimeter is \(60\) units.
20) One lap is the perimeter: \(18(2.5)=45\) m.
Two laps are \(2(45)=90\) m.
The total distance is \(90\) m.

More Practice

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