Perimeter Formulas

You need the perimeter formula for a specific shape and you need it now. Here is the formula, a one-line example, and the one thing that goes wrong with each. All formulas assume you measure every side in the same unit. Mix inches and centimetres and your perimeter will be wrong before you start.

Perimeter is a linear measurement: feet, metres, inches. If your answer comes out in square units you have calculated area, not perimeter. The most common mistake is confusing the two. A fence (perimeter) is not the same as paint (area).

Use the chart below to find your shape. Each entry lists the formula, the variables, and one worked example so you can check your arithmetic.

Every polygon follows the same rule: add the lengths of all its sides. For a triangle, P = a + b + c. For a quadrilateral such as a trapezoid, P = a + b + c + d, where a and b are the parallel bases and c and d are the legs. For an irregular pentagon or hexagon, you still sum each side individually, there is no shortcut.

The exception is a regular polygon. All sides equal, all angles equal. The equation is P = n × s, where n is the number of sides and s is the side length. A regular hexagon (n = 6) with side 4 m has perimeter 24 m. A regular octagon (n = 8) with side 3 ft has perimeter 24 ft.

Common Polygon Equations

• Rectangle: P = 2(l + w). Length 10 m, width 4 m → P = 2(10 + 4) = 28 m.
• Square: P = 4s. Side 6 cm → P = 24 cm.
• Parallelogram: P = 2(b + s). Base 8 m, side 5 m → P = 26 m.
• Trapezoid: P = a + b + c + d. Bases 10 cm and 6 cm, legs 5 cm each → P = 26 cm.
• Triangle: P = a + b + c. Sides 3 m, 4 m, 5 m → P = 12 m.

For a composite shape such as an L-shaped room, do not sum the perimeters of the sub-shapes. Their shared interior edge would be counted twice. Instead, add every exterior side length once. Missing a side? Subtract the known sides from the total perimeter. That is how the Common Core standard 3.MD.D.8 frames missing-side problems.

The perimeter of a circle is its circumference. Two equivalent equations: C = 2πr or C = πd. Radius 7 cm → C = 2 × 3.14 × 7 ≈ 43.96 cm. That is exact for a perfect circle; real-world circles (pipes, wheels) are close enough.

A semicircle is half the circumference plus the diameter: P = πr + 2r. Radius 5 cm → P = (3.14 × 5) + 10 = 25.7 cm. Do not forget the straight edge. Students commonly report only 15.7 cm and miss the diameter.

For a sector (a pie slice), the perimeter is the arc length plus two radii. Arc length = (θ/360) × 2πr, where θ is the central angle in degrees. Radius 10 cm, angle 60° → arc length = (60/360) × 62.8 ≈ 10.47 cm, so P = 10.47 + 20 = 30.47 cm.

The Common Core standard 7.G.B.4 expects you to know the circumference equation and to derive the relationship between circumference and area. This covers the equation; for a full walkthrough of manual calculation see the circumference of a circle page.

The ellipse has no exact elementary perimeter equation. Its true perimeter requires a complete elliptic integral of the second kind, which only a mathematician would calculate by hand. For everyone else, the best practical equation is Ramanujan's first approximation (1914):

P ≈ π [ 3(a + b) - √( (3a + b)(a + 3b) ) ]

where a is the semi-major axis (half the longest diameter) and b is the semi-minor axis (half the shortest diameter).87] ≈ 25.53 m. This approximation is accurate to within 0.04% for most ellipses.

The simpler average equation π(a+b) underestimates the real ellipse perimeter by 5 to 15 percent, depending on the ellipse's eccentricity. Do not use it unless you are estimating for a rough sketch and the shape is nearly circular. For an oval garden bed or elliptical window, use Ramanujan's equation or a dedicated ellipse perimeter calculator.

That is the only case where the exact answer is simple. For deeper reading on ellipse perimeter approximations, see the perimeter of an ellipse page.

Each equation below is paired with one worked example. Copy this chart to stick on a wall or keep in a notebook.

• Rectangle: P = 2(l + w). l = 10 m, w = 4 m → 28 m.
• Square: P = 4s. s = 6 cm → 24 cm.
• Triangle: P = a + b + c. Sides 3, 4, 5 m → 12 m.
• Circle (circumference): C = 2πr or πd. r = 7 cm → 43.98 cm.
• Parallelogram: P = 2(b + s). b = 8 m, s = 5 m → 26 m.
• Trapezoid: P = a + b + c + d. Bases 10, 6 cm, legs 5 cm → 26 cm.
• Regular polygon: P = n × s. n = 6, s = 4 m → 24 m.
• Semicircle: P = πr + 2r. r = 5 cm → 25.71 cm.
• Sector: P = (θ/360)×2πr + 2r. r = 10 cm, θ = 60° → 30.47 cm.
• Ellipse (Ramanujan approx.): P ≈ π[3(a+b) - √((3a+b)(a+3b))]. a = 5 m, b = 3 m → 25.53 m.