45-45-90 Triangle Calculator
Enter any single known value — a leg, the hypotenuse, the area, the perimeter, or even the inradius or circumradius — and instantly solve every other side and property of this special isosceles right triangle.
✓ Five ways to solve it ✓ Exact √2 ratios ✓ Works on mobile
If you've searched "45 45 90 triangle calculator," "how to solve a 45 45 90 triangle," "isosceles right triangle calculator," or "45 45 90 triangle formula," this page is built to answer every version of that question — with a live tool, not just a static ratio.
45-45-90 Triangle Calculator
The Complete Guide to the 45-45-90 Triangle
Everything students, teachers, and builders ask about this special right triangle.
What is a 45-45-90 triangle?
A 45-45-90 triangle is a special right triangle in which the two non-right angles are both exactly 45°, making it the only right triangle that is also isosceles — its two legs (the sides forming the right angle) are always equal in length. This is exactly the shape behind "45 45 90 triangle calculator," "isosceles right triangle calculator," and "how to solve a 45 45 90 triangle" searches, and it's one of only two "special right triangles" (alongside the 30-60-90 triangle) whose side ratios are fixed and memorizable rather than needing a trigonometric table every time.
The 1 : 1 : √2 side ratio
Because both legs are equal and the Pythagorean theorem always holds in a right triangle, the hypotenuse of a 45-45-90 triangle is always exactly √2 times as long as either leg — giving the memorable side ratio 1 : 1 : √2. Knowing just this ratio answers the vast majority of "45 45 90 triangle formula" searches on its own: if a leg is a, the hypotenuse is a√2 (≈1.414a); if the hypotenuse is c, each leg is c ÷ √2 (equivalently, c√2 ÷ 2). The calculator above builds every other property — area, perimeter, inradius, and circumradius — directly from this same core ratio.
Every formula for a 45-45-90 triangle
Once you know the leg length a, every other property follows directly: the hypotenuse is c = a√2; the area (using one leg as the base and the other as the height, since they meet at a right angle) is Area = a² ÷ 2; the perimeter is a(2 + √2); the inradius (radius of the circle inscribed inside the triangle) is r = a(√2 − 1); and the circumradius (radius of the circle passing through all three vertices) is R = a√2 ÷ 2, which — thanks to Thales's theorem for right triangles — is always exactly half the hypotenuse. Interestingly, the altitude from the right angle down to the hypotenuse also works out to a√2 ÷ 2 for this specific triangle, meaning it's numerically identical to the circumradius, a small but satisfying coincidence of this triangle's particular symmetry.
Relationship to a square
A 45-45-90 triangle is exactly what you get when you cut a square in half along its diagonal — this is why "45 45 90 triangle and square" and "diagonal of a square formula" are such closely related searches. If a square has side length s, its diagonal (which becomes the hypotenuse of each resulting triangle) is s√2 — identical in form to the leg-to-hypotenuse relationship of the 45-45-90 triangle, because that's precisely what it is. This connection makes the 45-45-90 triangle calculator above equally useful for "find the diagonal of a square" or "find the side of a square from its diagonal" problems.
Solving from any known value
Because every property of a 45-45-90 triangle derives from a single number (the leg length), the triangle can be fully solved from just one known measurement — a leg, the hypotenuse, the area, the perimeter, the inradius, or the circumradius — by first working backward to find the leg, then applying the standard formulas forward from there. This is exactly why the calculator above offers five separate entry modes rather than a single leg-only input: whichever single measurement you happen to have on hand is enough to solve the entire triangle.
Trigonometric values for 45°
The fixed side ratio of the 45-45-90 triangle is also the geometric reason sin(45°), cos(45°), and tan(45°) take the specific values they do: sin(45°) = cos(45°) = 1 ÷ √2 (≈0.7071), since the opposite and adjacent sides relative to a 45° angle are the two equal legs, divided by the hypotenuse; and tan(45°) = 1, since it's simply leg ÷ leg. This is exactly why the 45-45-90 triangle appears constantly in trigonometry courses as the standard example for evaluating trig functions at 45° without a calculator.
Where 45-45-90 triangles are actually used
Architects and builders use 45-45-90 triangles to design symmetrical structures, miter cuts, and roof pitches that meet at clean 45° angles. Carpenters and framers rely on the fixed 1:1:√2 ratio to cut accurate diagonal braces and corner supports without measuring every single angle. Computer graphics and game development use 45-45-90 geometry extensively in isometric projections and tile-based grid systems. Navigation and compass-rose design use 45° divisions as one of the most common angular references. And of course, geometry and trigonometry students encounter this triangle as one of the very first "special triangles" used to build intuition before moving on to general trigonometric functions.
How the 45-45-90 Triangle Calculator Works
Four simple steps take you from one known value to a fully solved triangle.
Choose What You Know
Select the leg, hypotenuse, area, perimeter, or a radius from the tabs above.
Enter That One Value
You only need a single known measurement — the calculator derives everything else from it.
Automatic Ratio-Based Calculation
The calculator applies the fixed 1 : 1 : √2 side ratio to solve every remaining property.
Review Every Result
See both legs, the hypotenuse, area, perimeter, inradius, and circumradius together in one place.
45-45-90 Triangle Formulas
The core equations behind every calculation mode on this page, given leg a.
Hypotenuse. Always √2 times the length of either equal leg.
Area. Half the product of the two equal legs, since they meet at a right angle.
Perimeter. Both legs plus the hypotenuse, added together.
Inradius. The radius of the circle inscribed inside the triangle.
Circumradius. Always exactly half the hypotenuse, by Thales's theorem.
Altitude to Hypotenuse. Numerically identical to the circumradius for this specific triangle.
Step-by-Step Worked Examples
Eight real calculations, shown with every step.
Example 1 — Solving From a Leg
A leg of 5 units produces a hypotenuse of about 7.07, an area of exactly 12.5, and a perimeter of about 17.07.
Example 2 — Solving From the Hypotenuse
A hypotenuse of 10 corresponds to legs of about 7.07 each, and an area of exactly 25 square units.
Example 3 — Solving From the Area
Working backward from an area of 12.5 recovers the exact leg length of 5 from Example 1.
Example 4 — Solving From the Perimeter
A perimeter of 10 units corresponds to legs of about 2.93 each, with an area of roughly 4.29 square units.
Example 5 — Solving From the Inradius
An inradius of about 2.071 recovers the same leg length of 5 units used in the earlier examples.
Example 6 — Solving From the Circumradius
Since the circumradius is always exactly half the hypotenuse, this reconstructs the same triangle instantly.
Example 7 — Diagonal of a Square
Splitting a square along its diagonal produces exactly this triangle — a side of 7.07 gives a diagonal of about 10.
Example 8 — Ramp or Roof Pitch at 45°
An 8-foot horizontal run at a 45° pitch requires an 8-foot rise and a ramp surface just over 11.3 feet long.
Quick Reference Tables
Common leg lengths, trigonometric values, and special-triangle comparisons at a glance.
Common Leg Lengths & Hypotenuse
| Leg (a) | Hypotenuse (a√2) |
|---|---|
| 1 | 1.414 |
| 5 | 7.071 |
| 10 | 14.142 |
| 12 | 16.971 |
Trigonometric Values at 45°
| Function | Value |
|---|---|
| sin(45°) | √2 / 2 ≈ 0.7071 |
| cos(45°) | √2 / 2 ≈ 0.7071 |
| tan(45°) | 1 |
Special Right Triangles Compared
| Triangle | Side Ratio |
|---|---|
| 45-45-90 | 1 : 1 : √2 |
| 30-60-90 | 1 : √3 : 2 |
Why Use This 45-45-90 Triangle Calculator
Built for anyone who needs a fully solved triangle fast — from just one known value.
Five Ways to Solve It
Start from a leg, hypotenuse, area, perimeter, inradius, or circumradius — whichever you already know.
Instant Calculations
Every mode recalculates live using the exact 1 : 1 : √2 ratio, not a rounded approximation.
Every Property at Once
See both legs, the hypotenuse, area, perimeter, inradius, and circumradius together in one result.
Great for Squares Too
Doubles as a diagonal-of-a-square calculator, since the two shapes share the exact same math.
Built for Learning
Every formula is shown clearly, making it useful for checking geometry homework step by step.
Works Everywhere
A fully responsive layout means the same accurate tool works on desktop, tablet, and mobile.
Practical Applications of the 45-45-90 Triangle
Real fields where this exact ratio comes up constantly.
Architecture & Construction
Designing symmetrical roof pitches, miter cuts, and diagonal bracing at a clean 45°.
Square & Diagonal Problems
Finding a square's diagonal, or a diagonal's implied side length, using the same core ratio.
Computer Graphics & Isometric Design
Building tile-based grids and isometric projections around consistent 45° geometry.
Navigation & Compass Design
Using 45° angular divisions as one of the most common reference directions on a compass rose.
Geometry & Trigonometry Education
Teaching special right triangle ratios and evaluating trig functions at 45° without a calculator.
Carpentry & Framing
Cutting accurate diagonal braces and corner supports using the fixed 1:1:√2 ratio.
Common 45-45-90 Triangle Mistakes
Avoid these classic errors when solving this special right triangle.
Confusing which side is the hypotenuse
The hypotenuse is always opposite the right angle and is always the longest side — never one of the two equal legs.
Forgetting the legs are equal
Both legs are always the same length in a 45-45-90 triangle — solving for only one and forgetting the other is a common oversight.
Mixing up the ratio direction
Multiply a leg by √2 to get the hypotenuse; divide the hypotenuse by √2 to get a leg — reversing this inverts the result.
Rounding √2 too early
Using an overly rounded value like 1.4 instead of 1.41421... introduces avoidable error, especially over larger measurements.
Confusing area with perimeter
Area uses the two legs multiplied together and halved; perimeter is the sum of all three sides — mixing these up gives a nonsensical result.
Misidentifying inradius vs circumradius
The inradius is smaller and sits inside the triangle; the circumradius is always exactly half the hypotenuse — confusing the two leads to a wrong triangle size.
Assuming any right triangle is 45-45-90
Only right triangles with two 45° angles follow this specific 1:1:√2 ratio — other right triangles need different formulas entirely.
Unit mismatches
Mixing inches with centimeters, or feet with meters, without converting first produces an inconsistent result across the triangle's properties.
Frequently Asked Questions
Straight answers to the most common questions about the 45-45-90 triangle.
What is a 45-45-90 triangle?
It's a right triangle with two 45° angles, making it also an isosceles triangle with two equal legs and a fixed 1 : 1 : √2 side ratio.
How do I solve a 45-45-90 triangle?
Enter any one known value — a leg, the hypotenuse, the area, the perimeter, or a radius — into the calculator above, and every other property is derived automatically.
What is the 45-45-90 triangle formula?
If a leg equals a, the hypotenuse is a√2, the area is a²÷2, and the perimeter is a(2+√2).
Are both legs of a 45-45-90 triangle always equal?
Yes — this is exactly what makes it isosceles, and it's true by definition since both non-right angles are equal.
How do I find the hypotenuse of a 45-45-90 triangle?
Multiply either leg by √2 (approximately 1.41421).
How do I find the legs from the hypotenuse?
Divide the hypotenuse by √2, or equivalently multiply it by √2 ÷ 2.
What is the area of a 45-45-90 triangle?
Area = a² ÷ 2, where a is the length of either leg, since the two legs act as base and height.
What is the relationship to a square?
Cutting a square in half along its diagonal produces two 45-45-90 triangles, with the square's side as the leg and its diagonal as the hypotenuse.
What is the inradius of a 45-45-90 triangle?
r = a(√2 − 1), where a is the leg length — the radius of the circle inscribed inside the triangle.
What is the circumradius of a 45-45-90 triangle?
R = a√2 ÷ 2, which is always exactly half the hypotenuse, following from Thales's theorem for right triangles.
Is this calculator accurate?
Yes, it uses the exact 1 : 1 : √2 ratio and standard geometric formulas, not rounded approximations.
Is it free?
Yes, the calculator and every calculation mode on this page are completely free to use, with no signup required.
Does it work on mobile?
Yes, the entire page — including the calculator and diagrams — is fully responsive and works on phones and tablets.
Can students use it?
Yes, it's designed to support geometry and trigonometry coursework by showing every formula used, not just the final answer.
Can I copy my results?
Yes, use the Copy Results button to copy a plain-text summary, or Share to send your results directly.
Solve Any 45-45-90 Triangle Instantly
Find the legs, hypotenuse, area, perimeter, inradius, and circumradius of any 45-45-90 triangle from a single known value using this professional calculator.