Perimeter Calculator
Calculate perimeter for common 2D shapes with clear formulas, step-by-step work, labeled sketches, and unit support.
Background
Students often know the shape but are not sure which boundary lengths count. This calculator helps by showing the correct perimeter formula first, then substituting the values, and explaining shape-specific rules such as semicircle perimeter includes the diameter, sector perimeter includes two radii plus the arc, and ellipse perimeter is an approximation.
How to use this calculator
- Choose the shape you want.
- Enter the required side lengths, radii, or angle values.
- Select a unit such as cm, m, in, ft, or yd.
- Click Calculate to get the perimeter, formula, steps, and sketch.
- Use Custom sides to add any list of side lengths.
How this calculator works
- Basic polygons: perimeter is the sum of side lengths.
- Circle-family shapes: perimeter may include curved arc length plus one or more straight edges.
- Regular polygons: perimeter is P = ns.
- Ellipse: perimeter is shown using a high-quality approximation.
- Custom sides: enter a comma-separated list like 3, 4, 5, 6 or an expression like 3 + 4 + 5 + 6.
Formula & Equations Used
Annulus: P = 2πR + 2πr = 2π(R + r)
Circle: P = 2πr = πd
Ellipse (approx.): P ≈ π[3(a+b) − √((3a+b)(a+3b))]
Kite: P = 2a + 2b = 2(a+b)
Parallelogram: P = 2(a+b)
Quadrilateral: P = a+b+c+d
Rectangle: P = 2(l+w)
Regular Polygon: P = ns
Rhombus: P = 4s
Sector: P = 2r + (θ/360)(2πr)
Semicircle: P = πr + 2r
Square: P = 4s
Trapezoid: P = a+b+c+d
Triangle: P = a+b+c
Custom sides: P = s₁ + s₂ + s₃ + ...
Example Problems & Step-by-Step Solutions
Example 1 — Rectangle
Find the perimeter of a rectangle with length 8 and width 5.
- Use P = 2(l+w).
- Substitute: P = 2(8+5).
- Simplify: P = 2(13) = 26.
Example 2 — Semicircle
Find the perimeter of a semicircle with radius 7.
- Use P = πr + 2r.
- Substitute: P = 7π + 14.
- Approximate: P ≈ 35.99.
Example 3 — Sector
Find the perimeter of a sector with radius 7 and central angle 60°.
- Arc length = (60/360)(2π·7) = 7π/3.
- Add the two radii: P = 14 + 7π/3.
- Approximate: P ≈ 21.33.
Example 4 — Custom sides
Find the perimeter of a shape with side lengths 3, 4, 5, 6.
- Add all side lengths.
- P = 3+4+5+6.
- P = 18.
Frequently Asked Questions
Q: Is perimeter the same as circumference?
For a circle, the boundary length is usually called circumference. In this calculator, it is shown as the perimeter result for circle-family shapes.
Q: Does a semicircle perimeter include the straight edge?
Yes. The perimeter of a semicircle includes the curved half-circle plus the diameter.
Q: Does a sector perimeter include the arc only?
No. A sector perimeter includes the arc length plus the two radii.
Q: Why is ellipse perimeter approximate?
Unlike many school shapes, an ellipse does not have a simple elementary exact perimeter formula, so the calculator uses a strong approximation.
Q: What can I enter in Custom sides?
Use a comma-separated list like 3, 4, 5, 6 or a plus-separated expression like 3 + 4 + 5 + 6.
