A scale copy is a figure that has the exact same shape as the original but a different size. Every length in the original is multiplied by the same number, and every angle stays exactly the same. If you’ve ever seen a smaller version of a triangle that looks identical to a bigger one, just shrunk down, that’s a scale copy. This concept shows up in 7th-grade math and forms the foundation for understanding similarity in geometry.
How Scale Copies Work
Two rules define a scale copy. First, every side length in the original figure gets multiplied by the same number. Second, all the angles remain unchanged. A scale copy of a rectangle with angles of 90 degrees still has angles of 90 degrees. A scale copy of a triangle with a 45-degree corner still has that 45-degree corner. The shape is preserved perfectly, only the size changes.
Think of it like zooming in or out on a photo. The proportions stay consistent. Nothing gets stretched wider or squished taller. If the original triangle has sides of 3, 4, and 5, and you multiply each by 1.5, the scale copy has sides of 4.5, 6, and 7.5. That ratio of 1.5 holds for every single side.
What the Scale Factor Tells You
The number you multiply every length by is called the scale factor. To find it, divide any side length in the copy by the corresponding side length in the original. If the original triangle has a side of 3 and the copy has a corresponding side of 9, the scale factor is 3.
A scale factor greater than 1 makes the copy larger than the original. A scale factor less than 1 (like 0.5 or 2/3) makes the copy smaller. A scale factor of exactly 1 produces a copy that’s the same size, which is still technically a scale copy.
Scale factors work in both directions. If triangle DEF is a larger scale copy of triangle ABC with a scale factor of 3/2, then triangle ABC is a smaller scale copy of triangle DEF with a scale factor of 2/3. The two numbers are always reciprocals of each other.
How Perimeter and Area Change
The scale factor affects perimeter and area differently, and this trips up a lot of students. Perimeter scales by the same factor as the side lengths. If your scale factor is 3, the perimeter of the copy is 3 times the perimeter of the original. That makes sense because perimeter is just the sum of the sides, and each side tripled.
Area, however, scales by the square of the scale factor. With a scale factor of 3, the area of the copy is 9 times (3 squared) the area of the original. This happens because area depends on two dimensions, length and width, and both get multiplied by the scale factor. So the effect compounds. A scale factor of 2 quadruples the area. A scale factor of 0.5 reduces the area to one-quarter.
How to Spot a Figure That Is Not a Scale Copy
The most reliable test is checking whether every pair of corresponding sides shares the same ratio. If one side doubles but another side stays the same, the figure is not a scale copy. For example, imagine Figure A has sides of 3 and 5. Figure B has sides of 5 and 5. To go from 3 to 5 you’d multiply by 5/3, but to go from 5 to 5 you’d multiply by 1. Those are different multipliers, so Figure B is not a scale copy of Figure A.
Another common non-example: adding a constant instead of multiplying. If someone adds 2 to every side of a triangle (turning sides of 3, 4, 5 into 5, 6, 7), the result is not a scale copy. The ratios between corresponding sides are 5/3, 6/4, and 7/5, which are all different. Scale copies require multiplication by the same factor, not addition of the same number.
Stretching in only one direction also disqualifies a figure. If you take a square and make it twice as wide but keep the height the same, you get a rectangle. That rectangle is not a scale copy of the square because the horizontal sides doubled while the vertical sides didn’t change, and the angles (while still 90 degrees in this case) don’t save it from having non-proportional sides.
Scale Copies in Everyday Life
Maps are one of the most familiar examples. A map with a scale of 1:50,000 means every centimeter on the map represents 50,000 centimeters (or 500 meters) in real life. The map is a scale copy of the actual geography, shrunk down so it fits on paper. Blueprints and architectural floor plans work the same way, letting builders represent a full-sized building at a fraction of the size while keeping all proportions accurate.
Model cars, miniature figurines, and dollhouse furniture are all physical scale copies. A 1:24 scale model car means every measurement on the model is 1/24th of the real car’s measurement. The angles of the windshield, the proportions of the wheels relative to the body, all stay the same. Only the size changes.
Quick Way to Check Your Work
When you’re asked whether one figure is a scale copy of another, follow these steps:
- Pick any two corresponding sides and divide the copy’s length by the original’s length to get a ratio.
- Repeat for every other pair of corresponding sides.
- Compare all the ratios. If they’re all equal, it’s a scale copy. If even one ratio differs, it’s not.
You can also check angles. If any corresponding angle changes between the original and the copy, it’s not a scale copy. In practice, checking side ratios is usually faster and more definitive, especially when you’re working with coordinates or measured lengths on a grid.