Some shapes are exact copies. Some are bigger or smaller copies. Stretch shapes on a grid, find the scale factor and see how length and area change.
Start here. Two shapes can be related in two special ways. They can be congruent: exactly the same shape and exactly the same size. You can slide, flip or turn one on top of the other and it fits perfectly. Or they can be similar: the same shape, but one is a bigger or smaller copy of the other. Think of a photo and its enlargement, or a map and the land it shows.
Scale factor. If a shape is made 2 times as big, every side becomes 2 times as long. The number 2 is the scale factor. If the factor is 3 you triple every side. If it is ½ every side is cut in half. A scale factor of 1 changes nothing, so the two shapes are congruent.
Angles stay the same. When you enlarge a shape evenly, its corners do not change. A square corner (90°) is still a square corner in the big copy. That is why similar shapes look the same: same angles, and sides in the same proportion.
Uneven stretching breaks it. If you stretch only the width, the shape gets squashed. The sides no longer grow by the same number, so the new shape is not similar. Try it in the Stretch Lab with "Stretch width only".
Missing sides. If two shapes are similar, you can find a missing side by multiplying by the scale factor. A 2 by 3 rectangle made 3 times as big is 6 by 9. Do not add the same amount to every side: that is a common mistake and it changes the shape.
Perimeter and area. Make a shape k times as big. The perimeter (the distance around) becomes k times as big. But the area (the space inside) becomes k × k times as big. Double every side and the area is 4 times as big. Triple every side and the area is 9 times as big.
Similar triangles. Two triangles are similar if two of their angles match. The three angles of any triangle add up to 180°, so if two angles match, the third one matches too. This is useful for shadows: at the same time of day, a tall thing and a short thing make shadows with matching angles, so heights and shadows are in the same proportion.
Try this above. In the Stretch Lab, pick a shape and press ×2, then ×3, then ×½. Watch every side. Then switch to "Stretch width only" and see what goes wrong.
Congruent shapes are exact copies: the same shape and the same size. A flipped or turned copy is still congruent. Only the position and the direction it faces are different.
Similar shapes have the same shape but not always the same size. All the angles are equal, and every side of one is the same number of times as long as the matching side of the other.
Yes, in the strict sense: congruent shapes are the special case where the scale factor is 1. In this game, to keep things simple, "similar" means the same shape but a different size.
Area measures two directions at once: across and up. Both grow by k, so the area grows by k × k. A 2 by 2 square has area 4. Make it 3 times as big and it is 6 by 6 with area 36, which is 9 times as big.
Pick the level, not the age. "Just starting" sorts shapes into congruent, similar or neither. "I know the basics" uses scale factors and missing sides. "Challenge me" uses perimeter, area, similar triangles, shadows and maps.
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