Golden cameo
The golden rectangle uses the ratio 1:φ; slice off a square and a smaller golden rectangle remains—self-similarity that can repeat forever.
Self-similarity
Rectangle Calculator — Width, Height, Diagonal, Perimeter, Area
Diagram & Inputs
Tip: Provide any two independent values (e.g., w & h, w & A, P & A, d & A, P & d, etc.). The solver infers the rest and flags inconsistencies if your extra inputs don’t match.
Results
How the Rectangle Calculator Works (with Visuals)
All rectangle properties connect through a handful of formulas. With width w and height h, we have perimeter P = 2(w + h), area A = w·h, diagonal d = √(w² + h²) (Pythagoras), and circumradius R = d/2. Because a rectangle has two defining lengths, **any two independent values** determine the entire shape. The calculator recovers w and h first, then computes the rest. If you enter more than two values, we check for consistency (within a small numerical tolerance) and still display a helpful “best-fit” set for learning and quick estimates.
Area from Width & Height
Given w and h, the tool immediately computes A = w·h, P = 2(w+h), and the diagonal.
Diagonal (Pythagoras)
With w and d, solve h = √(d² − w²). With h and d, solve w = √(d² − h²).
Perimeter
Walking the boundary adds two widths and two heights, so P = 2(w + h).
Solving from P & A
Let S = w + h = P/2. Then w,h are roots of t² − S·t + A = 0 ⟶ w,h = (S ± √(S² − 4A))/2.
Solving from d & A
Compute s = w + h = √(d² + 2A), then solve t² − s·t + A = 0 to get w and h.
Aspect Ratio & Scaling
Holding the **ratio** w:h fixed preserves shape while scaling size. Useful for print/layout fits.
Typical input pairs include: (w, h); (w, A) ⟶ h = A/w; (h, P) ⟶ w = P/2 − h; (w, d) ⟶ h = √(d² − w²); (P, A) via the quadratic above; (P, d) ⟶ A = ((P/2)² − d²)/2 then use (P, A); and (d, A) using the sum/product approach. Everything runs privately in your browser, with outputs labeled in your chosen unit (areas use squared units automatically).
5 Fun Facts about Rectangles
Diagonals spill secrets
Each diagonal is √(w² + h²) and they always bisect each other. Two matching diagonals are a reliable proof of right angles, even if the shape looks skewed.
Pythagoras inside
Square = rectangle on easy mode
When w = h, the rectangle maximizes area for a fixed perimeter and its circumradius collapses to R = d/2 = s/√2—a tidy bundle of equalities.
Special case
Aspect ratio fingerprints
4:3, 16:9, and √2:1 (A-series paper) are all rectangle ratios. Hold area constant but stretch the ratio and the perimeter balloons—a quick layout sanity check.
Everyday geometry
Perimeter vs area
Among rectangles with the same perimeter, the square encloses the largest area. Push the ratio toward a skinny strip and area collapses while perimeter holds steady.
Optimization clue
Rectangle Calculator: FAQs
Which inputs are valid?
Any two independent values among \(w, h, d, P, A, R\). Extra values are checked for consistency.
What about units?
Lengths use your chosen unit; area shows squared units (e.g., cm²).
Is this private?
Yes—100% client-side.