You’re staring at a calculator or a piece of code, and things just aren't adding up. Maybe you're building a game engine in JavaScript, or perhaps you're just trying to pass a trig quiz without losing your mind. Either way, you need to convert 90 deg in radians, and you need it now.
It’s $\pi/2$.
Roughly 1.57079 if you’re doing the decimal thing. That’s the "cheat sheet" answer. But honestly, if you just stop there, you're missing the entire reason why your computer refuses to use degrees in the first place. Computers don't care about the 360-degree circle we invented because of the Babylonians and their obsession with the number 60. They care about the relationship between a circle’s radius and its arc.
The weird history of why we use degrees anyway
We use degrees because of ancient history. It’s kinda wild when you think about it. The Persians and Babylonians liked the number 360 because it’s close to the number of days in a year and it’s divisible by almost everything. It’s "human-friendly." But it’s totally arbitrary. There is no physical reason a circle has 360 parts. We just decided it did.
Radians are different. They aren't arbitrary. A radian is a "natural" unit.
When you talk about 90 deg in radians, you are essentially saying, "How many radius-lengths do I need to wrap around a quarter of this circle?"
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The answer involves $\pi$. Since a full circle is $2\pi$ radians, a quarter of that circle (90 degrees) is $2\pi / 4$. Simplify that fraction, and you get $\pi / 2$. This isn't just a math trick; it's a fundamental property of the universe's geometry. If you take a string the length of a circle's radius and lay it along the edge (the circumference), it will fit exactly $2\pi$ times.
Converting 90 deg in radians without a calculator
You don't need a fancy Texas Instruments brick to do this. You just need one simple ratio.
$\text{Radians} = \text{Degrees} \times (\pi / 180)$
So, for 90 degrees:
$90 \times (\pi / 180) = 90\pi / 180$
Divide both by 90. You get $1/2$. Toss the $\pi$ back on top. Boom. $\pi/2$.
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If you’re working in a programming environment like Python or C++, you’ll usually see this expressed as M_PI_2 or math.pi / 2. Using the decimal 1.57 is usually a bad idea because of floating-point errors. You want that precision. Precision is the difference between a character in your game looking at a door and that same character staring at the wall three inches to the left.
Why does Calculus hate degrees?
If you ever venture into higher-level math, you'll notice degrees completely disappear. Why? Because the derivatives of trigonometric functions only work simply if you use radians.
If you try to find the derivative of $\sin(x)$ where $x$ is in degrees, you end up with this messy constant $(\pi / 180)$ floating around your equation. It’s gross. It’s clunky. But in radians, the derivative of $\sin(x)$ is just $\cos(x)$. It’s elegant. Math experts like Leonhard Euler pushed for this because it makes the underlying patterns of nature visible without the "noise" of human-invented degree scales.
Real-world 90-degree applications
Think about your phone’s accelerometer. When you flip your phone from portrait to landscape, the hardware detects a 90-degree shift. But the low-level firmware processing that motion is likely calculating that rotation in radians.
1.5708 radians per second.
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That’s the speed of a sharp turn. In robotics, engineers use "state-space representation." If a robotic arm needs to move to a right angle to pick up a semiconductor, the command sent to the servo motor is almost certainly a radian value. Degrees are basically just a "translation layer" for us humans because our brains find "90" easier to visualize than "1.57."
Common mistakes to watch out for
The biggest pitfall? Forgetting to set your calculator mode. We've all been there. You're doing a physics problem, you type in sin(90), and you expect to get 1. Instead, you get 0.8939.
That’s because your calculator was in radian mode. It thought you meant 90 radians—which is about 14 full trips around a circle plus a little extra. Always, always check the top of the screen for that tiny "DEG" or "RAD" icon.
Another one: rounding $\pi$ too early. If you use 3.14, your 90-degree conversion becomes 1.57. But if you're doing high-precision engineering or GPS calculations, those missing decimals mean you're off by miles when you project that angle over long distances.
Moving forward with radians
If you’re coding, stop using degrees in your logic. Write a wrapper function if you must, but keep the internal math in radians.
Actionable Steps:
- Memorize the Big Four: 90° is $\pi/2$, 180° is $\pi$, 270° is $3\pi/2$, and 360° is $2\pi$.
- Code Safely: Always use built-in constants like
math.PIinstead of typing3.14159. - Visualize the Arc: When you see 90 deg in radians, don't just see a number. See a quarter-circle arc that is roughly one-and-a-half times the length of the radius.
Understanding this conversion isn't just about passing a test. It’s about speaking the language that the physical world—and the digital world—actually uses.