Similar Triangles Calculator

Q: What is the ratio of areas of similar triangles?

If the scale factor between two similar triangles is k, the ratio of their areas is k². Example: if Triangle 1 has sides twice as long as Triangle 2 (k = 2), then Triangle 1 has 4× the area of Triangle 2. If Triangle 1 has area 36 cm² and Triangle 2 has area 9 cm², then the scale factor is √(36/9) = √4 = 2. This quadratic relationship between linear measurements and area is a fundamental geometric principle.

Q: What are the three criteria for triangle similarity?

AA (Angle-Angle): Two pairs of corresponding angles are equal. Since angles sum to 180°, the third pair is automatically equal. SAS (Side-Angle-Side): Two pairs of corresponding sides are proportional AND the included angles are equal. SSS (Side-Side-Side): All three pairs of corresponding sides are proportional. The AA criterion is the most commonly used in proofs because it requires the least information.

Q: How is the ratio of perimeters related to the scale factor?

The ratio of perimeters of two similar triangles equals the scale factor k. If Triangle 1 has sides a, b, c and Triangle 2 has sides ka, kb, kc, then: Perimeter₁ = a+b+c, Perimeter₂ = ka+kb+kc = k(a+b+c). So Perimeter₁/Perimeter₂ = 1/k. This linear relationship - unlike the quadratic relationship for areas - means a triangle with 3× longer sides has exactly 3× the perimeter but 9× the area.

Q: What is the AA similarity theorem?

The AA (Angle-Angle) similarity theorem states: if two angles of one triangle equal two angles of another triangle, the triangles are similar. This works because the three angles of a triangle always sum to 180°. If A₁ = A₂ and B₁ = B₂, then C₁ = 180°−A₁−B₁ = 180°−A₂−B₂ = C₂. AA is the most efficient criterion - you only need to verify two angles, and similarity is guaranteed.

Q: Can triangles be similar if they have the same area but different shapes?

No. Similar triangles must have the same shape (proportional sides and equal angles). Two triangles can have the same area but completely different shapes - for example, a 1×10 right triangle and a 2×5 right triangle both have area 10 but are not similar (their angles differ). Conversely, similar triangles with scale factor ≠ 1 always have different areas.

Q: How do similar triangles appear in real life?

Shadow problems: a person's shadow and the person form a triangle similar to the lamppost and its shadow - use proportions to find heights. Map scales: map distances to real distances use a fixed scale factor. Optical instruments: lenses create similar projections. Architecture: scaled blueprints and models. Indirect measurement: find the height of a building or tree by measuring its shadow and using similar triangles with a known reference object.

Q: What is the SAS similarity theorem?

SAS (Side-Angle-Side) similarity states: if two sides of one triangle are proportional to two sides of another triangle AND the included angles are equal, the triangles are similar. Unlike SAS congruence (which requires equal sides), SAS similarity requires proportional (not equal) sides. Example: Triangle 1 with sides 3, 5 and included angle 60°; Triangle 2 with sides 6, 10 and the same 60° angle - both SAS similar with scale factor 2.

Calculate missing sides, scale factor, and area ratio for two similar triangles using proportional side ratios.

🔺 What are Similar Triangles?

Two triangles are similar if they have the same shape but not necessarily the same size. This means all three pairs of corresponding angles are equal and all three pairs of corresponding sides are in the same ratio. That constant ratio is called the scale factor k. When two triangles are similar, every length in Triangle 2 is exactly k times the corresponding length in Triangle 1.

The concept of similarity is one of the most powerful ideas in geometry. It allows mathematicians and engineers to work with scaled models, maps, and blueprints because proportional relationships between sides are preserved regardless of overall size. A blueprint drawn at 1:50 scale uses similar triangles implicitly every time a triangular truss or gable is represented.

There are three standard criteria for proving that two triangles are similar. The AA (Angle-Angle) criterion states that if two pairs of corresponding angles are equal, the third pair must also be equal - since the angles in any triangle sum to 180°. The SAS (Side-Angle-Side) criterion applies when two pairs of sides are proportional and the included angles are equal. The SSS (Side-Side-Side) criterion applies when all three pairs of corresponding sides share the same ratio, which is the relationship this calculator uses to find all missing sides of Triangle 2 from Triangle 1 and one known side.

The scale factor k encodes the relationship between the two triangles for every linear measurement: individual sides, altitudes, medians, and the inradius all scale by k. Areas scale by k². If k = 2, Triangle 2 has twice the side lengths, twice the perimeter, and four times the area of Triangle 1. If k = 1, the triangles are congruent - identical in every measurement. Understanding this relationship is essential for shadow problems, map reading, optical instruments, architectural design, and any field involving scaled representations of the physical world.

Similar Triangles Formula

The fundamental proportionality relationship for similar triangles is:

a₁ / a₂ = b₁ / b₂ = c₁ / c₂ = k

a₁, b₁, c₁ — Side lengths of Triangle 1 (reference triangle)

a₂, b₂, c₂ — Corresponding side lengths of Triangle 2

k — Scale factor: the constant ratio of all corresponding sides

Finding the scale factor from one known side of Triangle 2:

k = a₂ / a₁ (or b₂ / b₁ or c₂ / c₁)

Use whichever pair of corresponding sides you know

Finding all sides of Triangle 2:

a₂ = k × a₁ b₂ = k × b₁ c₂ = k × c₁

Ratio of perimeters:

P₂ / P₁ = k

Perimeters scale linearly - the same ratio as individual sides

Ratio of areas:

A₂ / A₁ = k²

Areas scale as the square of k, because area involves two linear dimensions

If k = 3, the area ratio is 9. If k = 0.5, the area ratio is 0.25.

The area scales as k² because the area of a triangle is ½ × base × height; both base and height scale by k individually, giving k × k = k² overall. The perimeter, being a sum of linear dimensions each scaling by k, scales linearly as k.

How to Use This Calculator

Enter Triangle 1 sides — Fill in all three side lengths (a₁, b₁, c₁) of the reference triangle. These must form a valid triangle: the sum of any two sides must exceed the third.
Enter the known side of Triangle 2 — Type the value of the one side of Triangle 2 that you know into the “Known side value” field.
Select which side is known — Use the dropdown to indicate whether the known value corresponds to side a, b, or c of Triangle 2. This determines the scale factor.
Click Calculate — The calculator instantly computes the scale factor k, all three sides of Triangle 2, the perimeter ratio, and the area ratio.

Example Calculations

Example 1 — Classic 3-4-5 Scaled Up

Triangle 1 has sides 3, 4, 5. Triangle 2 has side a = 6. Find all sides of Triangle 2.

Scale factor k = a₂ / a₁ = 6 / 3 = 2.0000

T2 side b = 4 × 2 = 8.0000 T2 side c = 5 × 2 = 10.0000

Perimeter ratio = 2.0000 · Area ratio = k² = 4.0000

Triangle 2 sides: 6, 8, 10 · Scale factor k = 2.0000 · Area ratio = 4.0000

Try this example →

Example 2 — Scaling Down (6-8-10 to 3-4-5)

Triangle 1 has sides 6, 8, 10. Triangle 2 has side c = 5. Find all sides of Triangle 2.

Scale factor k = c₂ / c₁ = 5 / 10 = 0.5000

T2 side a = 6 × 0.5 = 3.0000 T2 side b = 8 × 0.5 = 4.0000

Perimeter ratio = 0.5000 · Area ratio = 0.25

Triangle 2 sides: 3, 4, 5 · Scale factor k = 0.5000 · Area ratio = 0.2500

Try this example →

Example 3 — 5-12-13 Pythagorean Triple Doubled

Triangle 1 has sides 5, 12, 13. Triangle 2 has side a = 10. Find all sides of Triangle 2.

Scale factor k = a₂ / a₁ = 10 / 5 = 2.0000

T2 side b = 12 × 2 = 24.0000 T2 side c = 13 × 2 = 26.0000

Perimeter ratio = 2.0000 · Area ratio = 4.0000

Triangle 2 sides: 10, 24, 26 · Scale factor k = 2.0000 · Area ratio = 4.0000

Try this example →

Example 4 — Shadow Problem: Find a Tree Height

A person 1.8 m tall casts a shadow 2.4 m long (hypotenuse 3 m). A nearby tree casts a shadow 8 m long. How tall is the tree?

Triangle 1 (person): height = 1.8, shadow = 2.4, hypotenuse = 3

Scale factor k = tree shadow / person shadow = 8 / 2.4 = 3.3333

Tree height = 1.8 × 3.3333 = 6.0000 m Tree hypotenuse = 3 × 3.3333 = 10.0000 m

Tree height = 6.0000 m · Scale factor k = 3.3333 · Area ratio = 11.1111

Try this example →

Frequently Asked Questions

What are similar triangles?+

Two triangles are similar if they have the same shape but not necessarily the same size. This means: (1) all corresponding angles are equal, and (2) all corresponding sides are in the same ratio (proportion). Similar triangles satisfy one of: AA (two pairs of equal angles), SAS similarity (two sides proportional with the included angle equal), or SSS similarity (all three sides proportional). The constant ratio of any pair of corresponding sides is called the scale factor k. Similarity is denoted ▵ABC ∼ ▵DEF.

How do you find the missing side of similar triangles?+

Set up a proportion using corresponding sides. If triangles ABC and DEF are similar, then a/d = b/e = c/f = k (the scale factor). If you know all sides of triangle 1 and one side of triangle 2, divide to find k, then multiply all sides of triangle 1 by k to get all sides of triangle 2. Example: Triangle 1 has sides 3, 4, 5. Triangle 2 has one known side of 6 corresponding to side a. Since k = 6/3 = 2, the other sides of Triangle 2 are b = 4×2 = 8 and c = 5×2 = 10.

What is the scale factor of similar triangles?+

The scale factor k is the common ratio of corresponding sides: k = a₂/a₁ = b₂/b₁ = c₂/c₁. If k > 1, Triangle 2 is larger than Triangle 1. If k < 1, Triangle 2 is smaller. If k = 1, the triangles are congruent. The scale factor relates all linear measurements: individual sides, altitudes, medians, and the inradius all scale by k, while areas scale by k² and volumes of prisms built on the triangles would scale by k³.

What is the ratio of areas of similar triangles?+

If the scale factor between two similar triangles is k, the ratio of their areas is k². Example: if Triangle 1 has sides twice as long as Triangle 2 (k = 2), then Triangle 1 has 4× the area of Triangle 2. If Triangle 1 has area 36 cm² and Triangle 2 has area 9 cm², then the scale factor is √(36/9) = √4 = 2. This quadratic relationship between linear measurements and area is a fundamental geometric principle - doubling all sides quadruples the area.

What are the three criteria for triangle similarity?+

AA (Angle-Angle): Two pairs of corresponding angles are equal. Since angles sum to 180°, the third pair is automatically equal. This is the most commonly used criterion. SAS (Side-Angle-Side): Two pairs of corresponding sides are proportional AND the included angles are equal. SSS (Side-Side-Side): All three pairs of corresponding sides are proportional - the ratio a₁/a₂ = b₁/b₂ = c₁/c₂ must hold. This calculator uses the SSS approach to compute all missing sides from a known scale factor.

How is the ratio of perimeters related to the scale factor?+

The ratio of perimeters of two similar triangles equals the scale factor k. If Triangle 1 has sides a, b, c and Triangle 2 has sides ka, kb, kc, then: Perimeter₁ = a+b+c, Perimeter₂ = ka+kb+kc = k(a+b+c). So Perimeter₂/Perimeter₁ = k. This linear relationship contrasts with the quadratic relationship for areas. A triangle with 3× longer sides has exactly 3× the perimeter but 9× the area - important in cost estimation for materials.

What is the AA similarity theorem?+

The AA (Angle-Angle) similarity theorem states: if two angles of one triangle equal two angles of another triangle, the triangles are similar. This works because the three angles of a triangle always sum to 180°. If A₁ = A₂ and B₁ = B₂, then C₁ = 180°−A₁−B₁ = 180°−A₂−B₂ = C₂. AA is the most efficient criterion - you only need to verify two angles, and similarity is guaranteed. It is especially useful in parallel-line and shadow problems where angles can be deduced from geometry rather than measurement.

Can triangles be similar if they have the same area but different shapes?+

No. Similar triangles must have the same shape (proportional sides and equal angles). Two triangles can have the same area but completely different shapes - for example, a 1×10 right triangle and a 2×5 right triangle both have area 10 but are not similar (their angles differ). Conversely, similar triangles with scale factor ≠ 1 always have different areas because A₂/A₁ = k² ≠ 1 when k ≠ 1. Same area with different shape means neither similar nor congruent.

How do similar triangles appear in real life?+

Shadow problems: a person's shadow and the person form a right triangle similar to the lamppost and its shadow - use proportions to find heights. Map scales: map distances to real distances use a fixed scale factor. Optical instruments: lenses and cameras create similar projections - the image size relates to the object size through similar triangles at the focal point. Architecture and engineering: scaled blueprints and physical models use similarity so every dimension corresponds to the real structure by the same ratio. Indirect measurement: find the height of a building or tree by measuring its shadow alongside a known reference object.

What is the SAS similarity theorem?+

SAS (Side-Angle-Side) similarity states: if two sides of one triangle are proportional to two sides of another triangle AND the included angles are equal, the triangles are similar. Unlike SAS congruence (which requires equal sides), SAS similarity requires proportional - not equal - sides. Example: Triangle 1 with sides 3, 5 and included angle 60°; Triangle 2 with sides 6, 10 and the same 60° angle. The ratios 6/3 = 10/5 = 2 are equal and the included angles match, so the triangles are SAS similar with scale factor k = 2.

🔗 Related Calculators

📌 Quick Tips

💡Two triangles are similar if all three pairs of corresponding angles are equal (AA, AAA) - the sides are then proportional.

💡The scale factor k = a₁/a₂ = b₁/b₂ = c₁/c₂. All three ratios must be equal for true similarity.

💡Ratio of perimeters = scale factor k. Ratio of areas = k². So if k = 2, the larger triangle has 4× the area.

💡Similar triangles appear in shadow problems, map scales, optical lenses, and architectural drawings.

💡If the scale factor is 1, the triangles are congruent (identical in size and shape).