Rounded rectangles are rectangles in which the corners have been replaced by quarter circular arcs. This planar figure is a popular shape for interior design, landscaping, architecture, and web design. The area of a rounded rectangle is slightly less than that of a rectangle with square corners and depends on the radius of curvature at the corners. If you know the radius and the overall width and length of the rectangle, you can compute the total area using the geometry equation below, or using a specialized geometry calculator.
For this explanation, assume that the radus of the corners is R, the longer dimension of the rectangle L, and the shorter dimension W. First, divide the rectangle into four quarter circles and 5 smaller rectangles as shown in the thumbnail above. The four quarter circles form a complete circle, so the total area of that portion is pi*R^2.
Next, find the areas of the longer rectangles around the border. The longer ones each have a width of R and length of L – 2R. Thus, their total area is 2R(L – 2R).
Now find the areas of the shorter rectangles around the border. The longer ones each have a width of R and length of W – 2R. Thus, their total area is 2R(W – 2R).
Finally, find the area of the rectangle in the center. Its length is L – 2R and width W – 2R, thus its area is (L – 2R)(W – 2R).
Therefore, the total area of the rounded rectangle is pi*R^2 + 2R(L – 2R) + 2R(W – 2R) + (L – 2R)(W – 2R). You can use algebra to simplify this expression by explanding the factors and combining like terms. Worked out, this is LW – (4 – pi)R^2.
As an example of how to use this formula, imagine a concrete patio in the shape of a rounded rectangle, where R = 3 feet, L = 18 feet, and W = 12 feet. Using the equation above, its area is 18*12 – (4 – pi)3^2 = 208.274 square feet. This knowledge may be useful if you need the cover the surface with some material, and the amount of material needed depends on the area.