5-12-13 Triangle Calculator – Scaled Multiples, Angles & Trig Values

What Is the 5-12-13 Triangle?

The 5-12-13 triangle is a right triangle whose three sides are integers — making it one of the classic Pythagorean triples in mathematics. A Pythagorean triple is any set of three positive integers (a, b, c) that satisfy the Pythagorean theorem: a² + b² = c². For the 5-12-13 triangle, the verification is clean and exact:

The side of length 13 is the hypotenuse — the longest side, always opposite the right angle. The sides of length 5 and 12 are the two legs of the right triangle, and they form the 90° angle between them.

The 5-12-13 triangle is considered a primitive Pythagorean triple because the three sides (5, 12, 13) share no common factor other than 1 — you can't simplify them to a smaller triple. This places it alongside the 3-4-5 triangle, the 8-15-17 triangle, and the 7-24-25 triangle as one of the foundational Pythagorean triples that every math student should know.

5-12-13 Triangle Angles (Degrees and Radians)

The three angles of the 5-12-13 triangle are determined by the side ratios. One angle is always exactly 90° (since it's a right triangle). The other two are found using the inverse sine function:

Note that the two non-right angles (22.62° and 67.38°) are complementary — they add up to exactly 90°. This is always true in any right triangle, because all three angles must sum to 180°, and one of them is already 90°.

5-12-13 Triangle Angles in Radians

Converting to radians is straightforward using the formula: radians = degrees × (π/180).

5-12-13 Triangle Formula: Area, Perimeter & Key Measurements

All the key measurements of the 5-12-13 triangle follow directly from the three side lengths. Here is the complete formula summary:

Property	Formula	Result
Area	½ × leg₁ × leg₂ = ½ × 5 × 12	30 sq units
Perimeter	5 + 12 + 13	30 units
Semi-perimeter (s)	(5 + 12 + 13) / 2	15
Inradius (r)	Area / s = 30 / 15	2 units
Circumradius (R)	hypotenuse / 2 = 13 / 2	6.5 units
Altitude to hypotenuse (h)	(5 × 12) / 13	≈ 4.615 units
Angle opposite 5	arcsin(5/13)	22.62° (0.3948 rad)
Angle opposite 12	arcsin(12/13)	67.38° (1.1760 rad)

5-12-13 Triangle Sin, Cos, Tan – Exact Values

One of the most valuable properties of the 5-12-13 triangle is that all six trigonometric ratios produce exact fractions — no irrational numbers or decimals. This makes it extremely useful in trigonometry problems, especially on standardized tests where exact values are preferred over approximations.

Using SOH-CAH-TOA (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent):

For angle A (≈22.62°, opposite the side of length 5):

For angle B (≈67.38°, opposite the side of length 12):

Notice the symmetry: sin(A) = cos(B) and cos(A) = sin(B). This is always true for the two non-right angles in any right triangle — they are complementary, so each is the co-function of the other.

5-12-13 Triangle Example Problems

Here are four common types of problems involving the 5-12-13 triangle, each with a step-by-step solution:

Example 1: Verify it's a right triangle

Example 2: Find the area

Example 3: Find angle A using trigonometry

Example 4: Scaled version (10-24-26 triangle)

5-12-13 vs 3-4-5 Triangle: Key Differences

Both are Pythagorean triples, but they are fundamentally different shapes — not similar to each other:

Property	3-4-5 Triangle	5-12-13 Triangle
Sides	3, 4, 5	5, 12, 13
Small angle	≈ 36.87° (0.6435 rad)	≈ 22.62° (0.3948 rad)
Large angle	≈ 53.13° (0.9273 rad)	≈ 67.38° (1.1760 rad)
Area	6 sq units	30 sq units
Perimeter	12 units	30 units
Inradius	1 unit	2 units
Shape	More square-like	More elongated/thin
Similar?	No — different angles, not similar triangles

The 3-4-5 triangle is the smallest primitive Pythagorean triple — every side is smaller. The 5-12-13 triangle is "thinner" because its small angle (22.62°) is much smaller than the 3-4-5 triangle's small angle (36.87°). Both are used constantly in geometry, trigonometry, and engineering because their integer sides make calculations clean.

Pythagorean Triple Family: Multiples of 5-12-13

Every integer multiple of the 5-12-13 triple is also a Pythagorean triple — the angles stay identical while the triangle scales up. These scaled versions are not primitive triples (since all sides share a common factor k), but they appear frequently in practical problems:

Scale (k)	Side a	Side b	Hypotenuse c	Area	Perimeter
k = 1	5	12	13	30	30
k = 2	10	24	26	120	60
k = 3	15	36	39	270	90
k = 4	20	48	52	480	120
k = 5	25	60	65	750	150
k = 10	50	120	130	3,000	300

Notice that area scales by k² (doubling the triangle doubles the linear dimensions but quadruples the area), while perimeter scales linearly by k.

Why Is the 5-12-13 Triangle Important?

The 5-12-13 triangle comes up in multiple contexts across mathematics and applied fields:

• Trigonometry education: Because all six trig ratios produce exact fractions (5/13, 12/13, 5/12, etc.), the 5-12-13 triangle is used to teach SOH-CAH-TOA without a calculator. Students can verify their work exactly rather than relying on decimal approximations.
• Standardized tests (SAT, ACT, GRE): Right triangles with integer sides appear frequently. Recognizing the 5-12-13 (and 3-4-5) pattern lets you skip Pythagorean theorem calculations entirely — a significant time-saver.
• Construction and surveying: The 3-4-5 and 5-12-13 patterns have been used for centuries to check right angles in construction. If you can measure three lengths and they satisfy the triple, you know you have a perfect right angle.
• Competition mathematics: The equal area-perimeter property (both equal 30) makes the 5-12-13 triangle a recurring figure in math olympiad problems.
• Physics and engineering: Right triangles with known exact ratios simplify vector component calculations in mechanics, electrical engineering, and navigation.

Frequently Asked Questions

Q: Is 5-12-13 a right triangle?
A: Yes. 5² + 12² = 25 + 144 = 169 = 13². It satisfies the Pythagorean theorem exactly, making it a Pythagorean triple with a right angle opposite the side of length 13.

Q: What are the angles of a 5-12-13 triangle?
A: 90° (right angle), approximately 22.62° (opposite side 5), and approximately 67.38° (opposite side 12). In radians: π/2, 0.3948, and 1.1760.

Q: What is the area of a 5-12-13 triangle?
A: 30 square units. Area = ½ × 5 × 12 = 30. Notably, the perimeter is also 30 — a rare property of this particular triangle.

Q: What is sin, cos, tan of the 5-12-13 triangle?
A: For angle A (22.62°): sin = 5/13, cos = 12/13, tan = 5/12. For angle B (67.38°): sin = 12/13, cos = 5/13, tan = 12/5. All values are exact fractions.

Q: How is 5-12-13 different from 3-4-5?
A: Both are Pythagorean triples but they are not similar — they have completely different angles (22.62°/67.38° for 5-12-13 vs 36.87°/53.13° for 3-4-5). The 5-12-13 triangle is thinner and more elongated.