15 75 90 Triangle Side Ratio

Is There a Perfect Triangle That Shows Up Everywhere From Roofs to Radar?

You've probably seen it somewhere—a triangle with sides in the ratio 15:75:90. Consider this: maybe it was on a chalkboard, a textbook, or tucked into a geometry problem. At first glance, it looks like any other set of numbers. But here's what makes it interesting: this isn't just some random triplet. It's a scaled version of a Pythagorean triple, and understanding its proportions reveals something elegant about how right triangles work in the real world.

What Is a 15:75:90 Triangle?

A 15:75:90 triangle is a right triangle whose three side lengths are in the exact ratio of 15 to 75 to 90. Consider this: that means if the shortest side measures 15 units, the middle side is 75 units, and the longest side (the hypotenuse) stretches to 90 units. But—and this is key—those numbers can be scaled up or down proportionally. So you might also encounter a triangle with sides 5:25:30 or 30:150:180. They're all the same shape, just different sizes.

To understand why this ratio matters, let’s break down what kind of triangle we’re really talking about.

Understanding the 15:75:90 Ratio Through Pythagorean Triples

The numbers 15, 75, and 90 might seem arbitrary until you realize they’re all multiples of a simpler set: 1, 5, and 6. Multiply each by 15, and you get 15, 75, and 90. But wait—that’s not quite right either. Let’s do the math properly.

Actually, the full story is that 15:75:90 is a multiple of the well-known Pythagorean triple 3:4:5. That said, multiply each part of 3:4:5 by 5, and you get 15:20:25—not 15:75:90. So where does 15:75:90 come from?

It turns out that 15:75:90 is actually a multiple of another Pythagorean triple: 5:25:25√2? On the flip side, no, that doesn’t work either. Let’s dig deeper. No workaround needed.

The correct breakdown is that 15:75:90 simplifies to 1:5:6 when divided by 15. And indeed, 1² + 5² = 1 + 25 = 26, while 6² = 36. So no, it’s not a Pythagorean triple in its simplest form.

Hold on.

Wait.

Actually, let’s check again.

If the sides are in the ratio 15:75:90, then dividing each by 15 gives us 1:5:6. That means 15:75:90 does not satisfy the Pythagorean theorem. But 1² + 5² = 26 ≠ 36 = 6². It’s not a right triangle at all.

Hmm.

That’s a big deal.

Because if it’s not a right triangle, then calling it a “15:75:90 triangle” is misleading. So what’s really going on here?

Clarifying the Misconception: Is 15:75:90 Even a Right Triangle?

Let’s run the numbers carefully.

Given three sides in the ratio 15:75:90, we can assign actual lengths:

• Side A = 15
• Side B = 75
• Side C = 90

Now apply the Pythagorean theorem:
a² + b² = c²?

Compute:

• 15² = 225
• 75² = 5625
• 90² = 8100

Check: 225 + 5625 = 5850
Compare to 8100

Nope. 5850 ≠ 8100.

So this triangle is not a right triangle. That’s a critical point.

But then why do people keep mentioning a “15:75:90 triangle”?

Because they’re thinking of a different ratio—one that is valid.

The Real Pythagorean Triple Behind the Confusion

What most people mean when they say “15:75:90 triangle” is actually referring to a triangle with sides in the ratio 15:36:39, which is a right triangle.

Why?

Because 15:36:39 is simply 3:4:5 scaled up by a factor of 3.

Check it:

• 15² = 225
• 36² = 1296
• 39² = 1521

And 225 + 1296 = 1521 ✅

So 15² + 36² = 39² — it works!

So where did 15:75:90 come from?

It might be a typo, a misremembered formula, or confusion with another ratio. But it’s definitely not a right triangle.

That said, there is a real triangle with sides in the ratio 15:75:90—and it’s a perfectly valid triangle, just not a right one.

Continue exploring with our guides on how many feet is in a quarter mile and what does at least mean in math.

What Kind of Triangle Is 15:75:90?

Even though 15:75:90 isn’t a right triangle, it’s still a legitimate triangle. Let’s figure out what type it is.

We already know:

• 15² + 75² = 225 + 5625 = 5850
• 90² = 8100

Since 5850 < 8100, the largest angle is greater than 90 degrees.

Because of this, a triangle with sides in the ratio 15:75:90 is an obtuse triangle—specifically, one where the angle opposite the longest side (90) is obtuse.

So if someone refers to a “15:75:90 triangle,” they’re likely either:

1. Mistaking it for 15:36:39 (a right triangle), or
2. Talking about an obtuse triangle with those exact proportions.

Either way, it’s worth clearing up the confusion.

But let’s go back to something more useful: what is the correct 15:36:39 triangle all about?

The 15:36:39 Triangle — A Valid Right Triangle

Now that we’ve clarified the confusion, let’s focus on the real deal: the 15:36:39 triangle.

This is a right triangle because it's a multiple of the classic 3:4:5 Pythagorean triple. Multiply each side by 3:

• 3 × 3 = 9
• 4 × 3 = 12
• 5 × 3 = 15

Wait—that gives 9:12:15, not 15:36:39.

Let’s try again.

Multiply 3:4:5 by 5:

• 3 × 5 = 15
• 4 × 5 = 20
• 5 × 5 = 25

So 15:20:25 is a right triangle.

But we said earlier that 15:36:39 is also a right triangle.

Let’s verify that.

• 15² = 225
• 36² = 1296
• 39² = 1521

225 + 1296 = 1521 ✅

Yes, 15:36:39 is a right triangle.

Actually, let's correct that. Practically speaking, multiply each term by 3:
5 × 3 = 15, 12 × 3 = 36, 13 × 3 = 39. Instead, it is a scaled version of the 5:12:13 Pythagorean triple. This works because 5² + 12² = 25 + 144 = 169 = 13². Now, the 15:36:39 triangle is not derived from the 3:4:5 triple. Scaling by 3 preserves the right angle, so 15² + 36² = 225 + 1296 = 1521 = 39².

This triangle has some interesting properties. Think about it: the smallest angle, opposite the side of length 15, is approximately 22. 6°, and the other acute angle is about 67.4°. Its area is (15 × 36)/2 = 270 square units. Such triangles appear in construction and design where specific angle measures are needed, like in roof trusses or staircase stringers.

Now, returning to the 15:75:90 triangle: since it's obtuse, we can find its angles using the Law of Cosines. But the largest angle, opposite the side 90, satisfies:
cos(C) = (15² + 75² - 90²) / (2 × 15 × 75) = (225 + 5625 - 8100) / 2250 = (-2250) / 2250 = -1. Wait, that gives cos(C) = -1, which would mean angle C is 180°—but that's impossible for a triangle. Let's recalculate carefully:
15² + 75² = 225 + 5625 = 5850.90² = 8100.
So cos(C) = (5850 - 8100) / (2 × 15 × 75) = (-2250) / 2250 = -1.
Indeed, cos(C) = -1 implies C = 180°, which suggests the points are collinear.

is exactly the triangle inequality condition for degeneracy. A triangle with sides 15, 75, and 90 does not exist because the sum of the two shorter sides (15 + 75) is equal to the longest side (90). This violates the fundamental rule that the sum of any two sides must be greater* than the third.

This explains the earlier confusion perfectly. The "15:75:90 triangle" is a geometric impossibility. In real terms, it's not just obtuse; it's a degenerate case—a straight line segment of length 90, theoretically divided into parts of 15 and 75. This is why the Law of Cosines calculation pointed to a 180° angle.

So, the initial premise was flawed from the start. There is no triangle with those proportions to classify as right, acute, or obtuse.

Conclusion

The journey to untangle the "15:75:90 triangle" reveals a valuable lesson in geometry: always check the triangle inequality first. While the ratio sparked an interesting discussion about obtuse and right triangles, it ultimately points to a non-existent shape.

The real focus, and a perfectly valid example, is the 15:36:39 triangle. As a scaled version of the 5:12:13 Pythagorean triple, it is a genuine right triangle with practical applications. The next time you encounter such a ratio, you'll know exactly how to separate geometric fact from fiction.