Similar Triangles and Polygons
Definition: Triangles and other polygons are similar if all their corresponding sides are in the same proportion.
Try this
Drag any orange dot at P,Q or R. The triangle will remain similar to the triangle LMN even though it changes size.
(If there is no image below, see support page.)
Polygons that are similar have all corresponding sides in the same proportion. This means they have the same shape, but can be different sizes.
In the figure above, drag one of the points P,Q or R. The yellow triangle PQR changes size, but is always the same shape as LMN.
Think of it as "zooming" in or out.
Rotation and reflection
Polygons can still be similar even if one of them is rotated, and/or mirror image of the other.
In the figure below, all three polygons are similar.
Starting with the original polygon on the left, the center polygon is rotated clockwise 90°, the right one is flipped vertically.
Mark the angles in the two right polygons that correspond with the angle P on the left.
This is illustrated in more depth for triangles in
"Similar Triangles",
but is true for all similar polygons, not just triangles.
Formal notation
Given two similar polygons ABCD and JKLM, we can write
ABCD ~ JKLM
which is read as "polygon ABCD is similar to polygon JKLM". The wavy line symbol means 'similar to'.
Properties of Similar Polygons
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Corresponding angles are the same
So in the figure above,
∠P =∠L, ∠Q =∠M, and ∠R =∠N.
From this, it follows that the corresponding exterior angles will also be the same.
-
Corresponding sides are all in the same proportion
By definition each pair of corresponding sides are in the same proportion, or ratio.
Formally, in two similar triangles PQR and LMN :
So, for example, if in two similar polygons one side is twice the length of the corresponding side in the other,
Then all the other sides will be twice the length of their corresponding side also.
-
Corresponding diagonals are in the same proportion
In each polygon the corresponding diagonals are in the same proportion. Their ratio is the same as the ratio of the sides.
-
Area ratio
The ratio of the areas of the two polygons is the square of the ratio of the sides.
So if the sides are in the ratio 3:1 then the areas will be in the ratio 9:1. This is
illustrated in more depth for triangles in
"Similar Triangles - ratio of areas",
but is true for all similar polygons, not just triangles.
Regular Polygons
For any two regular polygons with the same number of sides:
- They are always similar, regardless of size. Since they have the sides all the same length, they must always be in the same proportions, and so are always similar.
- The apothems and radii are in the same proportions as each other and the sides.
Related topics
Similar Polygons
(C) 2007 Copyright John Page
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