How this calculator works
A balloon arch is usually built as a half ellipse: it rises from one side of the base, curves over the top and comes down the other side. The length of that curve decides how many balloons are needed. The page approximates the curve with the perimeter of an ellipse, taken over its top half. Each single balloon or each cluster then takes up about one balloon diameter along the arch. A cluster of four or five balloons is the standard way to make an arch look full, while a single row uses far fewer balloons and leaves gaps between them.
Formula
arch length = half the perimeter of an ellipse with semi-axes width/2 and height, using Ramanujan's approximation pi(a+b)(1 + 3h/(10 + sqrt(4 - 3h))) with h = ((a-b)/(a+b))^2. clusters = arch length / balloon diameter, rounded up. balloons = clusters x balloons per cluster.
Worked example
An arch 2.4 m high and 3 m wide has an arch length of about 6.21 m. With 25 cm balloons in clusters of four, that gives 25 clusters and 100 balloons. At 35 cents each the balloons cost $35.00. The same arch in a single row needs 25 balloons.
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Frequently asked questions
Why does the arch need to be at least half as tall as it is wide?
A half ellipse whose height is less than half its width cannot form a complete arch between the two base points. If your design is a low, wide ribbon of balloons, measure a shorter span across the base and treat the rest as extra length.
How much does the arch length change with the shape?
A semicircular arch of the same height and width is a little shorter than a tall half ellipse. The ellipse formula responds to both numbers, so enter the measurements of the finished shape rather than the frame.
Should I buy extra balloons?
Yes, add about 10 percent for popped and mis-sized balloons. The count here is the number placed on the arch only, and it assumes the balloons touch along the length.
Does balloon size change the count?
Yes, strongly. Clusters take about one balloon diameter each, so doubling the diameter halves the number of clusters and therefore halves the number of balloons.