Similar Triangles Calculator

How the similar triangles calculator works

Two triangles are similar when they have the same shape, even if one is larger. Their matching angles are equal, and the lengths of their matching sides are all in the same proportion, called the scale factor. This calculator checks that proportion for you. Enter two sides of the first triangle (a and b) and the two corresponding sides of the second triangle (c and d). It compares the ratio a / b with the ratio c / d and tells you whether the triangles are similar on those sides, then shows both ratios and the scale factor. Similar triangles are used to find heights from shadows, scale maps and models, and solve geometry problems where a missing side can be found by proportion.

Required Inputs & Typical Use Cases

• Side a and Side b

  Two sides of the first triangle, as positive numbers in the same unit.

• Side c and Side d

  The corresponding sides of the second triangle. c matches a, and d matches b.

The Formula

a / b = c / d      (equivalently a × d = b × c)
Scale factor k = c / a = d / b
Missing side: d = b × c / a

• a, b

  Two sides of triangle 1.

• c, d

  The matching sides of triangle 2, where c corresponds to a and d to b.

• k

  The scale factor, the number you multiply triangle 1's sides by to get triangle 2's sides.

Worked Example

1. Example 1: 3-4-5 and 6-8-10

   Take a = 3, b = 4, c = 6, d = 8. Then a / b = 0.75 and c / d = 0.75, so the ratios match. The triangles are similar with scale factor k = 6 / 3 = 2. The third sides agree too, since 5 × 2 = 10.

2. Example 2: sides that do not match

   With a = 3, b = 4, c = 6, d = 9, a / b = 0.75 but c / d ≈ 0.6667. Cross-multiplying gives 3 × 9 = 27 and 4 × 6 = 24, which are not equal, so the triangles are not similar.

3. Example 3: finding a missing side by proportion

   Triangle 1 has sides 6 and 9. In a similar triangle the side matching 6 is 10. The side matching 9 is d, so 6 / 9 = 10 / d. Solving gives d = 9 × 10 / 6 = 15. Check with the calculator: a = 6, b = 9, c = 10, d = 15 returns similar.

Tips, Assumptions & Limitations

• Common mistake: pairing the wrong sides. Match shortest with shortest, longest with longest, or use equal angles to decide which sides correspond. A mismatched pair gives a false 'not similar'.
• Common mistake: mixing units. Convert all lengths to the same unit before entering them, since inches and centimeters in one ratio will break the proportion.
• Common mistake: confusing similar with congruent. Congruent triangles are identical in size and shape, so their scale factor is exactly 1. Similar triangles can have any positive scale factor.
• Common mistake: checking only one pair of sides. This tool tests the ratio of two sides, so for a full verification repeat it with the third side as well, or confirm that two angles are equal (AA similarity).
• Scale factors apply to lengths only. Areas scale by k squared and volumes of similar solids by k cubed.
• Lengths must be positive numbers. Tiny rounding differences in decimal inputs are tolerated.