Calculator for Right Angles and Hypotenuse

A right triangle is fully defined by just two of its measurements. Enter any two known values, such as sides or angles, into the calculator below and it will instantly calculate everything else: the hypotenuse, both acute angles, the altitude from the right angle to the hypotenuse, area, perimeter, and the radii of both the inscribed and circumscribed circles.

Right triangle calculator — enter any two side or angle values to compute all triangle properties

Right triangle calculator

Enter any two known values (sides or acute angles) to solve all properties.

a Side a (opposite)

b Side b (adjacent)

c Hypotenuse c

α Angle α (opposite a) °

β Angle β (opposite b) °

Results

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Diagram

For information on other tools and topics:

• A list of many types of Angles from Geometry: So many varieties! [Do you know of angle types that didn't make the list?.]
• Collatz Conjecture Visualizer: Explore one of math's great unsolved problems interactively. [Explore the Collatz paths]
• Earth's Looming Expiration Date: Why Earth faces an early deadline as the Sun's energy output increases. [Idea on how to preserve our planet]
• Number Rounding Tool: Multiple methods of rounding supported. [Type your number and round it up or down!]

Escape Velocity Calculator

Escape velocity is the minimum speed an object needs to break free from a body's gravitational pull without any further propulsion. The formula is simple: v = √(2GM/r), where G is the gravitational constant, M is the mass of the body, and r is the distance from its center. The result tells you how fast something must be launched to escape that gravity well entirely with no engines required after the initial push.

It can be surprising as to how much this number varies.  Earth's escape velocity is about 11.2 km/s. Jupiter's is over 59 km/s. The Sun's surface escape velocity is around 617 km/s. And for a neutron star, it can be a significant fraction of the speed of light. Use the calculator below to explore escape velocities for planets, stars, galaxies, and any custom body you wish.

Escape Velocity Calculator

Quick-select a known body: — Custom / enter manually — Moon Mars Earth Venus Jupiter Saturn Sun Sirius A Neutron Star (typical) Sgr A* (Milky Way black hole) Milky Way Galaxy Mass of body (kg): Radius / distance from center (meters):

For information on other tools and topics:

• Collatz Conjecture Visualizer: Explore one of math's great unsolved problems interactively. [Explore the Collatz paths]
• Futhorc Runic Bi-Directional Converter: Turn your message into First Millennium runes and back! [Try out this rune conversion tool on your own messages.]
• Earth's Looming Expiration Date: Why Earth faces an early deadline as the Sun's energy output increases. [Idea on how to preserve our planet]
• Number Rounding Tool: Multiple methods of rounding supported. [Type your number and round it up or down!]

Many Paths of the Collatz Conjecture

The Collatz Conjecture is one of mathematics' strange unsolved problems. The rule is deceptively simple: take any positive integer, and if it's even divide it by 2, if it's odd multiply by 3 and add 1. Repeat this process and the sequence seems to always eventually reach 1. Always. This is true for every number ever tested! Even still, no one has ever been able to prove it, though some attempts have got close.

The sequences themselves are practically unpredictable. For example, the number 27 takes 111 steps while rocketing up to 9,232 before finally collapsing to 1. Nearby numbers can reach 1 in just a handful of steps, while others take hundreds of chaotic steps before converging. Use the interactive math tool below to explore and compare up to 5 numbers at once.

Collatz Conjecture Visualizer

Pick any positive integer. If it's even, divide by 2. If it's odd, multiply by 3 and add 1. Repeat. No matter what number you start with, the sequence always seems to reach 1, but nobody has ever proved why. Enter up to 5 numbers to compare their paths.

Enter Numbers to Compare

About the Collatz Conjecture: Mathematician Paul Erdős said: “Mathematics is not yet ready for such problems.”

For information on other tools and topics:

• Futhorc Runic Bi-Directional Converter: Turn your message in to First Millennium runes and back! [Try out this rune conversion tool on your own messages.]

• Earth's Looming Expiration Date: Why Earth faces an early deadline as Sun's energy output increases. [Idea on how to preserve our planet] 

• Number Rounding Tool: This the rounding tool that you didn't know you need! Multiple methods of rounding are supported. [Type Your Number and Round it up or down!]

Tool that Rounds to the Nearest Fraction

When working in trades like carpentry, machining or cooking, you often run into decimal measurements that are difficult or impossible to translate to a standard ruler, tape measure, or measuring cup. Sometimes, you just need to work with fractions.

While a CAD program might output a required component hole as 0.6875 inches, a person on the shop floor needs to select a tool or check a dimension using the common fraction 11/16 inches. Similarly, scaling a recipe can result in awkward numbers like 0.833 cups, which is much easier to manage when converted to a practical fraction like 5/6 or the nearest standard measuring cup size. Below is the tool that is designed to bridge that gap by converting any decimal into its closest usable fraction.

The Fractional Rounding Tool (below) takes any decimal number and, based on your chosen level of granularity (the maximum denominator, such as 1/8 or 1/16), it determines the nearest possible fraction. This is essential because it allows you to standardize your precision and use common measuring instruments effectively. You also have full control over the rounding method, which dictates how the tool handles numbers that fall exactly halfway between two fractions. This is a great feature when working with tolerances, negative numbers or specific industry standards like rounding half up or half even. Use the tool below to instantly convert your decimal plans into measurable, actionable fractions.

Fractional Rounding Tool 📏

Decimal Number to Round (e.g., 4.375)

Round to the Nearest Fraction (Max Denominator)

Nearest 1/2 Nearest 1/4 Nearest 1/8 Nearest 1/16 (Standard Measure) Nearest 1/32 Nearest 1/64 Nearest 1/10 (Metric) Nearest 1/100

Rounding Mode (Determines how to choose the nearest numerator)

0: Round Up 1: Round Down 2: Round Ceil 3: Round Floor 4: Round Half Up (Standard) 5: Round Half Down 6: Round Half Even (Banker's) 7: Round Half Ceil

Rounded Fractional Result (Simplified)

For information on other tools and topics:

• Futhorc Runic Bi-Directional Converter: Turn your message in to First Millennium runes and back! [Try out this rune conversion tool on your own messages.]

• Earth's Looming Expiration Date: Why Earth faces an early deadline as Sun's energy output increases. [Idea on how to preserve our planet] 

• Number Rounding Tool: This the rounding tool that you didn't know you need! Multiple methods of rounding are supported. [Type Your Number and Round it up or down!]

Number Rounding Tool You Might Need

Everyone learns the "Schoolhouse Rule" of rounding. This is where you look at the next digit and if it's 5 or greater, round up. This method (Round Half Up, or 5 always rounds up) works for everyday math, but it introduces a hidden and cumulative problem that is often not considered: upward bias.

In financial, scientific, or engineering calculations involving hundreds of figures, the "5 always rounds up" rule causes you to round up more often than you round down. This subtle bias can compound into a significant error in the final result. Our tool provides professional rounding systems designed specifically to eliminate this problem.

Reducing Bias

These methods are used when the total sum of all figures must be as accurate as possible, minimizing accumulated error.

Mode	What It Does	Why You Use It
Round Half Even	When a number is exactly halfway (e.g., 5.5 or 6.5), it rounds to the nearest even digit.	This is Banker's Rounding. By rounding equally to even numbers, it eliminates the upward bias of the schoolhouse method. It's the standard for professional financial and scientific calculations.
Stochastic Rounding	Uses random chance to decide whether to round up or down when exactly halfway.	Used in high-precision scientific simulation and modeling to introduce statistical fairness and prevent bias in complex, non-linear calculations.

Strict Control Over Direction

These modes are used when your calculation must never exceed (or never fall short of) the true value.

Mode	Rule	Example Use Case
Round Floor	Always rounds down (towards negative infinity).	Resource Allocation: Calculating how many full containers, shipments, or packages you can create from a given amount, ensuring you never over-count.
Round Ceil	Always rounds up (towards positive infinity).	Safety Margins: Calculating how much material to order or capacity you need, ensuring you always have at least the required amount.

By using this tool, you move beyond simple arithmetic to achieve the precise, mandate-required accuracy necessary for serious data analysis and computation.

Multiple Methods Rounding Tool 🎯

Number to Round (e.g., 123.456789)

Rounding Place (Exponent 10^N, N from 9 to -9)

Rounding Mode (decimal.js options)

0: Round Up (Rounds away from zero) 1: Round Down (Rounds towards zero / Truncates) 2: Round Ceil (Rounds towards +Infinity) 3: Round Floor (Rounds towards -Infinity) 4: Round Half Up (Nearest, 0.5 away from zero) 5: Round Half Down (Nearest, 0.5 towards zero) 6: Round Half Even (Nearest, 0.5 to even neighbor) 7: Round Half Ceil (Nearest, 0.5 to +Infinity) 8: Stochastic Rounding (Random, zero-bias)

Rounded Result

Note: The rounding place is specified by its 10^N exponent, covering every single place value from 10^9 down to 10^-9.

For information on other topics and tools:

• Futhorc Runic Bi-Directional Converter: Turn your message in to First Millennium runes and back! [Try out this rune conversion tool on your own messages.]

• Earth's Looming Expiration Date: Why Earth faces an early deadline as Sun's energy output increases. [Idea on how to preserve our planet] 

• The Number Converter Utility: Convert numerals to words and words to numerals! [Type Your Number and Convert it!]

Weird math - 10in plus 6mm equals 26cm

Instagram: https://www.instagram.com/p/CH_Mv_Dlzjh

Fighting the Future, One Square Root at a time

Find the square root for Y
Stage One:
1. Determine the largest squared whole number less than Y.
2. Use the square root of the largest squared whole number to be the first part of the answer. Place this number to the left of the decimal place within the answer.
3. Find the difference of Y and the squared whole number.

Stage Two:
1. Multiple the difference by 100, designated as A.
2. Multiple the answer so far by 2 (without the decimal point), designated as B.
3. Multiple B by 10.
4. Give C one of the following values: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9
5. Find D where (B+C)*C equals the greatest value less than A.
6. C becomes the next digit right of the decimal in the answer.
7. Find the difference between of A and D.
8. Repeat Stage Two until the answer reaches the desired number of digits after the decimal.

Example:

PDF File: Example to find the square root of 3

(Sorry, I had to make it a PDF file because html isn't good at showing math equations and I didn't want to scan in my chicken scratch writing. Free Acrobat Reader is a must, but if you don't already have it, go here to get it: http://www.adobe.com/products/acrobat/readstep2.html)

Now, is anyone ever going to use this? Hey, if anyone has seen this method in print, please let me know.

Preface to Square Root

Back when I was in high school, I learned something that isn’t known by very many people. I learned the method to manually find a square root in a way that is similar to long division. This method allows you to find each decimal place with certainty. You can solve to as many places after the decimal point as you want.
I've never found this long method in print anywhere. I’ve found other simpler methods to finding a square root, but they usually involve closing in on the square root by continuously rerunning the same method. You are never left with a perfect answer because you can never be sure if the successive decimal places are correct. I’m also not sure which method is used by calculators (on which we all depend for square roots these days, which is the beginning of Asimov’s vision for our world coming true, but that’s a future blog entry).
I have no clue why this long method works. But, in a very small effort to fight the future, I’m going to show the method here, soon.

UPDATE: Here's the link to the long method of finding square roots: http://fcsuper.blogspot.com/2005/05/fighting-future-one-square-root-at.html#comments