Mathwords logoMathwords

Rounded Rectangle — Definition, Formula & Examples

A rounded rectangle is a rectangle whose sharp corners have been replaced by quarter-circle arcs, creating smooth, curved edges at each vertex.

A rounded rectangle (also called a rectircle or stadium when maximally rounded) is a plane figure formed by two pairs of parallel line segments connected at their endpoints by four congruent circular arcs of radius rr, where 0<r≤min⁡ ⁣(w2,h2)0 < r \leq \min\!\left(\tfrac{w}{2}, \tfrac{h}{2}\right) and ww and hh are the overall width and height of the shape.

Key Formula

A=w⋅h−(4−π) r2P=2(w+h)−8r+2πrA = w \cdot h - (4 - \pi)\,r^2 \qquad P = 2(w + h) - 8r + 2\pi r
Where:
  • AA = Area of the rounded rectangle
  • PP = Perimeter of the rounded rectangle
  • ww = Overall width of the shape
  • hh = Overall height of the shape
  • rr = Corner radius (radius of each quarter-circle arc)

How It Works

Think of starting with an ordinary rectangle and then filing down each corner into a smooth curve. Each corner becomes a quarter circle with the same radius rr. The straight sides shrink to accommodate the curves, but the overall width and height stay the same. When r=0r = 0, you get a plain rectangle; when rr equals half the shorter side, the two shorter sides disappear entirely and the shape becomes a stadium (an oblong with semicircular ends).

Worked Example

Problem: Find the area and perimeter of a rounded rectangle that is 10 cm wide and 6 cm tall with corner radius 2 cm.
Step 1: Calculate the area using the formula. Start with the full rectangle area, then subtract the material removed from the corners and add back the quarter-circle areas.
A=10×6−(4−π)(2)2=60−(4−3.1416)(4)≈60−3.43=56.57 cm2A = 10 \times 6 - (4 - \pi)(2)^2 = 60 - (4 - 3.1416)(4) \approx 60 - 3.43 = 56.57 \text{ cm}^2
Step 2: Calculate the perimeter. Replace the 8r of straight edge removed from corners with the curved arcs totaling one full circle of radius r.
P=2(10+6)−8(2)+2π(2)=32−16+12.566≈28.57 cmP = 2(10 + 6) - 8(2) + 2\pi(2) = 32 - 16 + 12.566 \approx 28.57 \text{ cm}
Answer: The area is approximately 56.57 cm² and the perimeter is approximately 28.57 cm.

Why It Matters

Rounded rectangles appear everywhere in design — phone screens, app icons, road signs, and swimming pools. Understanding their geometry helps in fields like graphic design, architecture, and manufacturing where calculating material area or border length for curved shapes is essential.

Common Mistakes

Mistake: Using the plain rectangle perimeter formula without adjusting for the rounded corners.
Correction: Rounding the corners removes straight edge length and replaces it with curved arc length. You must subtract 8r8r and add 2πr2\pi r to get the correct perimeter.