相似多边形计算器

Two polygons are similar when they have exactly the same shape but possibly different sizes. The Similar Polygons Calculator finds the scale factor between two similar shapes from a pair of corresponding sides, and computes how that scale factor affects perimeter and area. This tutorial defines similarity precisely, derives the area scales-as-k² rule, walks through worked examples, and contrasts similarity with the stronger condition of congruence.

What "similar" means

Two polygons are similar if BOTH of the following hold:

1. Corresponding angles are equal. If you label the vertices in matching order, every angle in one polygon matches the corresponding angle in the other.
2. Corresponding sides are proportional. The ratio of any side of Polygon 2 to its corresponding side in Polygon 1 is the same for every pair. This common ratio is the scale factor k.

Both conditions matter. A square and a non-square rhombus have proportional sides (all equal) but unequal angles — not similar. A square and a rectangle have all right angles but disproportionate sides — also not similar.

The scale factor

The scale factor from Polygon 1 to Polygon 2 is:

k = (side of Polygon 2) / (corresponding side of Polygon 1)

• k > 1: Polygon 2 is larger (an enlargement).
• 0 < k < 1: Polygon 2 is smaller (a reduction).
• k = 1: the polygons are congruent (same shape AND same size).

The calculator returns k from a single pair of corresponding sides. From k, you can derive every other side of Polygon 2 by multiplying the matching side of Polygon 1.

How perimeter and area scale

This is the key insight that students often miss:

• Perimeter ratio = k (linear scaling)
• Area ratio = k² (quadratic scaling)

If Polygon 2 has twice the side lengths of Polygon 1 (k = 2), its perimeter is 2× larger but its area is 4× larger. A triangle with sides 3-4-5 has area 6; a similar triangle with sides 6-8-10 has area 24.

Why area scales as k²: area depends on the product of two length measurements (base × height, for example, or side² for a square). Multiplying both lengths by k multiplies the area by k × k = k².

The same logic extends to 3D: volume ratio = k³ for similar solids. Doubling all dimensions of a box multiplies the volume by 8.

Worked example

Polygon 1: a rectangle with sides 4 and 6 (perimeter 20, area 24).
Polygon 2: a similar rectangle with one corresponding side of length 8.

1. Scale factor: k = 8 / 4 = 2.
2. Other side of Polygon 2: 6 × 2 = 12.
3. Perimeter of Polygon 2: 20 × 2 = 40. (Or compute directly: 2(8 + 12) = 40.)
4. Area of Polygon 2: 24 × k² = 24 × 4 = 96. (Or compute directly: 8 × 12 = 96.)

How to test if two polygons are similar

Three standard tests work for triangles specifically:

• AA (Angle-Angle): if two pairs of angles are equal, the triangles are similar. (The third pair must also be equal because angle sums are 180°.)
• SSS-similarity: if all three pairs of corresponding sides are proportional with the same ratio, the triangles are similar.
• SAS-similarity: if two pairs of sides are proportional with the same ratio AND the included angles are equal, the triangles are similar.

For general polygons (not just triangles), you must verify both equal angles AND proportional sides — there is no shortcut. Even four-sided figures require checking both because of the rhombus / rectangle counterexamples above.

Similarity vs congruence

Every pair of congruent polygons is similar (with k = 1), but most similar pairs are not congruent. Congruence is the strict subset.

Real-world applications

• Maps: a map's scale (e.g., 1 : 50,000) is a similarity scale factor. Every distance on the map is 1/50,000 of the real-world distance.
• Blueprints and architectural drawings: same idea — a drawing scaled to 1/96 or 1/48 of the real building.
• Scale models: physical models of buildings, cars, airplanes are similar to their full-size counterparts. The model car at 1:24 scale has 1/24 the length, 1/576 the surface area, 1/13824 the volume (and proportional mass for the same material).
• Photo enlargements: every digital photo enlargement is a similarity transformation. Doubling the print size quadruples the paper area.
• Indirect measurement: similar triangles measure inaccessible heights (a tree's height from its shadow vs a known stick's shadow at the same time of day).

Common mistakes

• Scaling area by k instead of k². A common student error. If you double linear dimensions, the area increases by a factor of 4, not 2.
• Reading the scale factor backwards. k from 1 to 2 means "Polygon 2 is k times Polygon 1". To go from 2 to 1, the reciprocal scale factor 1/k applies.
• Assuming proportional sides without equal angles. A general rhombus and a square have all equal sides but only the square has equal angles — they are not similar.
• Using non-corresponding sides for the ratio. The scale factor is computed from corresponding sides. If you pair a 3-side of Polygon 1 with a 12-side of Polygon 2 but they actually correspond to different positions, your ratio is meaningless.