The perimeter of a shape is the total distance around its boundary; add all sides or use the shape’s rule.
Perimeter shows how far it is around a shape. You use it to buy fence, frame a picture, size trim, or mark a track. The method is simple: identify the shape, grab the right rule, keep units straight, and add with care. This guide gives quick steps, crisp rules, and worked examples so you can solve any perimeter task with confidence.
What Perimeter Means
Perimeter is the length of the closed path that outlines a two-dimensional figure. For a polygon, you add the side lengths. For a circle, you use the circumference rule. Real-life uses pop up all the time—planning turf edging, laying a wire, or measuring a yard for a pet run.
Common Shapes And Perimeter Rules
Use these core rules. Keep variables and units straight and you will move fast.
| Shape | Perimeter Formula | Variables / Notes |
|---|---|---|
| Any polygon | Sum of all sides | Add each listed edge length. |
| Rectangle | 2(l + w) | l = length, w = width. |
| Square | 4s | s = side length. |
| Triangle | a + b + c | a, b, c = side lengths. |
| Regular n-gon | n × s | n = number of sides, s = side length. |
| Circle | C = 2πr = πd | r = radius, d = diameter; “perimeter” is called circumference. |
| Semicircle | πr + 2r | Curved edge plus the diameter. |
| Composite figure | Add each outer edge | Trace only the outside boundary. |
Step-By-Step Method
1) Identify The Shape And Known Data
Sketch the figure. Label every given side. Note whether the figure is a standard shape (rectangle, square, triangle, regular polygon, circle) or a mix. If a side is missing but parallel to a known side, mark that match. On a grid, count unit segments along the outer path.
2) Pick The Rule
Choose from the table above. For a regular polygon, one side and the side count is enough. For a circle, use circumference. For a composite shape, split it into parts only to reveal missing sides, then add the full outer rim only once.
3) Keep Units Consistent
Convert before adding. If some sides are in meters and others in centimeters, convert to a single unit first. If you mix feet and inches, switch all values to inches, then change back at the end if you like.
4) Add With Care
Write the expression with parentheses where needed. Evaluate cleanly. Round only after you finish, and only when a context requests it.
Keyword Variant: Finding Perimeter For Any Shape—Practical Cases
This section shows the method in action for a range of figures. You will see how the same core idea—sum around the rim—handles straight edges and curves.
Rectangles And Squares
Rectangle example. A photo frame is 12 in by 8 in. Perimeter P = 2(l + w) = 2(12 + 8) = 40 in. You need 40 inches of trim.
Square example. A patio tile has side 0.6 m. P = 4s = 4 × 0.6 = 2.4 m.
Triangles
Plain sum. Sides are 7 cm, 9 cm, and 5 cm. P = 7 + 9 + 5 = 21 cm.
Right triangle with grid legs. Legs 6 ft and 8 ft with a fence along the two legs and the hypotenuse. If the hypotenuse is installed, add it too. Hypotenuse h = √(6² + 8²) = 10 ft, so total P = 6 + 8 + 10 = 24 ft.
Regular Polygons
A stop-sign shape (regular octagon) has side 14 inches. P = n × s = 8 × 14 = 112 inches.
Circles And Arcs
For a circle edge, use C = 2πr or C = πd. If you only know radius 3.5 m, C = 2πr ≈ 2 × 3.14159 × 3.5 ≈ 21.99 m. For quarter-circle borders attached to straight edges, take the curved part as (¼ of the circle’s circumference) and add the two straight legs.
Composite Figures
Look at the outline as one path. Avoid counting shared interior edges. If an “L” shape is made of two rectangles that meet, the inside seam is not part of the outside rim. If a side length is missing, use matches from parallel segments or subtract from totals.
Measurement Tips That Save Time
- Trace the rim with a finger or pencil. Say the lengths out loud as you pass them to avoid skips.
- Sort units first. Make every length use the same unit before you add.
- Round at the end. Keep full precision during the sum.
- Label your answer. Always include units.
Worked Problems With Explanations
1) Garden Bed Border (Mixed Units)
A rectangular bed is 2.4 m by 180 cm. Convert 180 cm → 1.8 m. P = 2(l + w) = 2(2.4 + 1.8) = 2 × 4.2 = 8.4 m of edging.
2) Fencing An L-Shaped Yard
The outer sides run 30 m, 10 m, 6 m, 14 m, 20 m, and 8 m. Add them: P = 30 + 10 + 6 + 14 + 20 + 8 = 88 m. Do not add the interior corner seam.
3) Quarter-Circle Flower Bed
The curved edge is a quarter of a circle with radius 5 ft, attached to two straight legs of 5 ft each. Curved part = (¼) × 2πr = (½)πr = 0.5 × π × 5 ≈ 7.854 ft. Total P ≈ 7.854 + 5 + 5 ≈ 17.854 ft. Round to 17.9 ft if a store lists tenths.
4) Regular Hexagon Patio
Side s = 0.9 m. P = 6s = 5.4 m. If pavers come in 30 cm pieces, that is 18 pieces around the rim.
5) Irregular Polygon From A Grid
On a grid where each square is 1 unit, count the outer edge segments. Add every straight run. Diagonal runs need length given; if all edges align to the grid, you only add whole units.
When A Side Is Missing
Use structure. For rectangles, opposite sides match. In a composite figure, write totals along a long side and subtract a known part to get the missing segment. If you know coordinates, distance between points (x1, y1) and (x2, y2) is √[(x2 − x1)² + (y2 − y1)²]. Use it to fill in slanted edges.
Accuracy, Units, And Rounding
Match the problem’s unit and precision. If lengths are given to the nearest millimeter, keep that through your sum. If a circle uses π, carry π in the expression until the last step for best accuracy.
Use clear unit symbols (cm, m, in, ft) and leave a space when writing. For circles, use π ≈ 3.14159 or 22/7; keep one choice in the same problem.
Reference Links For Deeper Study
For a crisp math definition and examples, see the perimeter entry. For teaching videos and interactive practice, see Khan Academy’s introduction to perimeter.
Unit Conversions You Often Need
| From | To | Conversion |
|---|---|---|
| 1 m | cm | 1 m = 100 cm |
| 1 cm | mm | 1 cm = 10 mm |
| 1 km | m | 1 km = 1000 m |
| 1 ft | in | 1 ft = 12 in |
| 1 yd | ft | 1 yd = 3 ft |
| 1 mi | ft | 1 mi = 5280 ft |
Frequent Mistakes And Easy Fixes
Counting Interior Edges
Only the outside counts. If a shape is stitched from smaller blocks, the seams in the middle are not part of the rim.
Mixing Units Mid-Sum
Pick one unit and convert everything before you add. A quick line that shows the conversion will save a rework later.
Dropping A Side
Trace the rim in one sweep while listing lengths. Many misses happen at corners. A light tick mark helps hold your place.
Using π Too Soon
Keep π symbolic while you add other parts. Plug the rounded value at the last step to keep small errors from stacking up.
Practice Set
- A rectangle is 18 cm by 11 cm. Find the rim length.
- A triangle has sides 5 m, 13 m, and 12 m. What is the total?
- A regular pentagon has side 2.2 m. How far around is it?
- A path forms a semicircle of radius 2 m attached to a diameter walkway. What length of edging is needed?
- An “L” shape has outer sides 9, 7, 3, 4, 6, and 2 meters in order. What is the total length around?
Method Notes
Criteria Used In This Guide
The steps favor speed, clear units, and accuracy. Shapes are grouped by shared rules. Examples show both neat numbers and mixed units, since real jobs often blend them.
Why These Sources
Britannica gives a clean language definition and usage examples. Khan Academy offers short videos and practice that help you cement the skill. Both are trusted, well-maintained references.