Scale Factor Calculator

How it works

A scale factor k is the number every length is multiplied by when a figure is enlarged or reduced without changing its shape. Because area is length × length, it grows by k × k = k²; volume is length × length × length, so it grows by k³. Double every edge of a box and it holds eight times as much.

A drawing ratio says the same thing as a factor: 1:48 means one unit on the drawing is 48 of the same unit in reality, so the factor from drawing to real is 48. The architect's ¼-inch scale is 1:48 because a foot is 48 quarter inches; a 1:50,000 map puts 50,000 cm, half a kilometre, into every centimetre. To go the other way, from real to drawing, divide instead, which is the inverse factor 1/k.

When you have an original and a scaled copy of the same length, the factor is simply scaled ÷ original, and the calculator reports it as a 1:k ratio along with the area and volume factors it implies.

Formula

k = scale factor     or     k = ratio right ÷ ratio left     or     k = scaled length ÷ original length
scaled length = k  × original length
scaled area   = k² × original area
scaled volume = k³ × original volume
ratio = 1:k          inverse factor = 1 ÷ k

Example

Scaling by a factor of 2: a 10 in length becomes 20 in, a 50 sq in area becomes 2² × 50 = 200 sq in, and a 100 cu in volume becomes 2³ × 100 = 800 cu in. The ratio is 1:2 and the inverse factor is 0.5.

A ¼ in = 1 ft floor plan is 1:48. A wall that measures 2.5 in on the plan is 48 × 2.5 = 120 in, or 10 ft, in the building. A room drawn as 50 sq in covers 48² × 50 = 115,200 sq in, which is 800 sq ft.

If a photo 4 in wide is printed 10 in wide, the factor is 10 ÷ 4 = 2.5, the ratio is 1:2.5, and the print has 2.5² = 6.25 times the area.

Assumptions and limitations

• The scaling is uniform, the same factor in every direction. A drawing stretched differently in width and height has no single scale factor.
• The ratio is drawing : real, so 1:48 enlarges a plan measurement to full size. Use the inverse factor to go from real to drawing.
• The original and scaled lengths are in the same unit. The unit you choose labels the results and converts nothing: a ¼ in = 1 ft plan measured in inches gives real inches.
• The area and volume you enter are in the chosen unit squared and cubed.

Frequently asked questions