An isosceles right triangle (also known as a 45°-45°-90° triangle) is a special right triangle where the two perpendicular legs are equal in length (a = b). Using the Pythagorean relationship a² + a² = c², the hypotenuse is simplified directly to c = a√2.
Calculate equal leg lengths, hypotenuse, area, and perimeter in seconds with exact radical expressions, decimal approximations, step-by-step formula proofs, and visual SVG diagrams.
Select your calculation mode below, input known dimensions, and view instant results with dynamic SVG diagrams.
The isosceles right triangle is defined by five core geometric relationships:
Find the hypotenuse c when equal leg a is known.
Find equal legs a and b when hypotenuse c is known.
Calculate area and total perimeter around boundaries.
An isosceles right triangle is a unique geometric figure that combines the defining characteristics of both an isosceles triangle and a right-angled triangle. It contains exactly one 90° right angle formed by two perpendicular sides of identical length (a = b). Because the two legs are equal in length, the two remaining acute angles must also be equal to one another, making each of them exactly 45°.
In classical Euclidean geometry, an isosceles triangle is defined as any triangle with at least two equal sides, while a right triangle is defined as any triangle containing a 90° angle. The isosceles right triangle is the single specific triangle that satisfies both conditions simultaneously.
Not every isosceles triangle is a right triangle! General isosceles triangles can be acute (all angles under 90°) or obtuse (one angle over 90°). Standard Pythagorean calculations (a² + b² = c²) apply strictly to right-angled triangles. For general non-right isosceles triangles, you must draw a central vertical altitude to split the shape into two smaller right-angled triangles before calculating missing dimensions.
An isosceles right triangle is widely referred to in high school geometry and trigonometry as a 45-45-90 triangle because of its internal angle measures:
The 45-45-90 triangle is classified as one of two primary special right triangles (the other being the 30-60-90 triangle). Special right triangles possess fixed angle measures and predictable, exact side ratios, allowing mathematicians, engineers, and students to solve for unknown side lengths instantly without evaluating complex trigonometric functions like sine or cosine.
The defining mathematical feature of every 45-45-90 isosceles right triangle is its invariant side length ratio:
Leg A : Leg B : Hypotenuse C
Regardless of the actual physical scale of the triangle—whether measuring a fraction of a millimeter on a microchip or hundreds of meters across an architectural site—the relationship between the sides remains identical:
This 1 : 1 : √2 ratio is a primary semantic differentiator in geometry and serves as the foundation for instant mental calculations.
Because the two perpendicular legs are equal (a = b), the general Pythagorean theorem equation a² + b² = c² simplifies into dedicated formulas organized by known parameters:
Both legs forming the 90° right angle have identical length.
Hypotenuse is always leg length multiplied by √2.
Dividing hypotenuse by √2 yields both equal leg lengths.
Simple single-variable formulas for area and perimeter.
Our specialized online calculator executes instant algebraic evaluations using exact 45-45-90 relationships. Follow these quick steps:
1 : 1 : √2 ratio to calculate all remaining sides.½a² and total perimeter via 2a + c.To find the hypotenuse c of an isosceles right triangle when an equal leg length a is given, you do not need to perform complex multiplication or long addition under a square root. You can derive the simplified formula directly from the Pythagorean theorem:
Worked Example: Find the hypotenuse of an isosceles right triangle with equal leg lengths of a = 10 cm.
a = 10 cmc = 10 × √2c = 10√2 cm10 × 1.41421356 = 14.1421 cmFor general right-angled triangles where the two perpendicular legs are unequal (a ≠ b), use our primary Pythagorean Theorem Calculator.
When the hypotenuse c is known, both equal perpendicular legs (a and b) can be determined by rearranging the 45-45-90 formula:
To eliminate the irrational square root from the denominator, multiply the numerator and denominator by √2 to produce the rationalized form:
Worked Example: Given a hypotenuse c = 20 inches, calculate the length of equal legs a and b.
c = 20 ina = (20√2) / 2 = 10√2 in10 × 1.41421356 = 14.1421 ina = b = 14.1421 inTo solve general triangles with unequal sides, see our Pythagorean Theorem Calculator.
The general formula for the area of any triangle is Area = ½ × base × height. In a right triangle, the two legs perpendicular to each other serve as the base and height. Because an isosceles right triangle has equal legs (base = height = a), the area formula simplifies directly to:
If you only know the hypotenuse c, you can calculate area directly using:
Example: An isosceles right triangle has leg length a = 8 cm. Its area is A = 8² / 2 = 64 / 2 = 32 cm².
The perimeter P of any triangle is the total distance around its three outer edges:
Substituting b = a and c = a√2 yields:
Because 2 + √2 ≈ 3.41421356, the perimeter of an isosceles right triangle is always approximately 3.4142 times the length of one equal leg.
Example: For leg a = 10 ft, P = 2(10) + 10√2 = 20 + 14.1421 = 34.1421 ft.
Review these five practical, worked mathematical examples demonstrating all calculation modes:
Given: Leg a = 5 cm
Hypotenuse: c = 5√2 cm ≈ 7.071 cm
Area: ½(5²) = 12.5 cm²
Perimeter: 2(5) + 5√2 = 17.071 cm
Given: Hypotenuse c = 10 in
Legs: a = b = 10 / √2 = 5√2 in ≈ 7.071 in
Area: 10² / 4 = 25 in²
Perimeter: 2(7.071) + 10 = 24.071 in
Given: Area A = 32 m²
Legs: a = √(2 × 32) = √64 = 8 m
Hypotenuse: c = 8√2 m ≈ 11.314 m
Perimeter: 2(8) + 11.314 = 27.314 m
Given: Leg a = 25 ft
Hypotenuse: c = 25 × 1.41421356 = 35.355 ft
Area: ½(25²) = 312.5 sq ft
Perimeter: 50 + 35.355 = 85.355 ft
Given: Hypotenuse c = 14 ft
Exact Form: Leg a = 14 / √2 = 7√2 ft
Decimal Form: Leg a ≈ 9.8995 ft
When computing side lengths in 45-45-90 isosceles right triangles, results can be written in two distinct ways:
√2 explicitly intact (such as 10√2 or 7√2). Because √2 is an irrational number with non-repeating infinite decimals, writing expressions in exact radical form avoids all rounding errors in pure mathematics.√2 ≈ 1.41421356 to produce a rounded floating-point number (such as 14.142 cm). Decimal forms are essential for real-world physical measurements, engineering construction tape pulls, and CAD software inputs.Our calculator displays both representations side-by-side to serve classroom mathematics and practical field applications equally.
The mathematical proof that the hypotenuse of an isosceles right triangle equals leg times √2 is straightforward and elegant:
a² + b² = c²b = a): a² + a² = c²2a² = c²c = √(2a²)√(x · y) = √x · √y: c = √(a²) · √2 = a√2Thus, the factor √2 is a direct mathematical consequence of having two equal perpendicular legs forming a right angle.
The proof that the two non-right angles in an isosceles right triangle are exactly 45° stems from fundamental geometric axioms:
180°.90° right angle.180° - 90° = 90°.90° / 2 = 45°.An isosceles triangle is broadly defined as any triangle with two equal sides. However, an isosceles right triangle is a highly restricted subcategory:
1 : 1 : √2 side ratio.The terms Isosceles Right Triangle and 45-45-90 Triangle are completely synonymous and describe the exact same geometric object from two complementary perspectives:
In practice, geometry educators and engineers use both terms interchangeably.
Compare the distinct mathematical characteristics of general right triangles against isosceles right triangles:
| Property | General Right Triangle | Isosceles Right Triangle |
|---|---|---|
| Right Angle (90°) | Yes (1 right angle) | Yes (1 right angle) |
| Equal Legs (a = b) | Rare (Usually unequal) | Always strictly equal (a = b) |
| Acute Interior Angles | Variable (e.g. 30° & 60°, 36.87° & 53.13°) | Always exactly 45° & 45° |
| Side Ratio | Variable (e.g. 3:4:5, 5:12:13) | Always fixed 1 : 1 : √2 |
| Hypotenuse Formula | c = √(a² + b²) | c = a√2 |
Do not confuse isosceles right triangles with equilateral triangles:
The hypotenuse of a 45-45-90 isosceles right triangle directly represents the diagonal of a square. If a square has side length a, drawing a straight line from one corner to the opposite corner splits the square into two congruent isosceles right triangles.
Example: A square floor tile measures 12 inches on each side. The diagonal distance across the tile is 12√2 ≈ 16.97 inches.
Because every square is composed of two identical 45-45-90 triangles joined along their hypotenuse, all geometric properties of squares convert directly into isosceles right triangle formulas:
a).c = a√2).a² (exactly double the isosceles triangle area ½a²).Our calculator supports standard metric and imperial units. When calculating isosceles right triangles, unit dimensional consistency must be maintained:
Our calculator fully supports decimal input values for legs, hypotenuses, areas, and perimeters. For example, if a leg length is entered as a = 7.5 cm:
Decimal precision allows builders, designers, and students to calculate accurate physical dimensions without being restricted to integer side lengths.
Because √2 is an irrational number containing infinite non-repeating decimal digits, intermediate calculations should retain maximum precision:
√2 to 1.41 in multi-step calculations. Keep exact radical forms or high-precision floating-point values until the final step.Avoid these ten frequent pitfalls when solving 45-45-90 triangle problems:
Forgetting that non-right isosceles triangles require an altitude line to split them before using Pythagorean formulas.
Mistakenly setting the hypotenuse equal to a instead of a√2.
Using a + √2 instead of a × √2 for hypotenuse calculation.
Leaving c / √2 un-simplified instead of multiplying by √2 / √2 to get (c√2) / 2.
Using a² (square area) instead of ½a² (triangle area).
Rounding √2 to 1.4 early in calculations, introducing cumulative rounding errors.
Applying the 1 : √3 : 2 ratio of 30-60-90 triangles to an isosceles 45-45-90 triangle.
Combining feet and inches directly without converting to uniform units first.
Assuming 14.142 is an exact integer solution rather than an approximation of 10√2.
Forgetting that perimeter requires adding all three sides: a + a + c = 2a + c.
To compute valid isosceles right triangle dimensions, inputs must meet specific mathematical criteria:
value > 0).Isosceles right triangles appear frequently across architecture, engineering, computer graphics, and construction design:
Calculating diagonal dimensions across square concrete slabs, floor tiles, decks, and tabletop frames.
Determining cut lengths for 45-degree architectural trim, picture frame miters, and metal corner gussets.
Designing 12:12 pitch roof structures where horizontal run equals vertical rise, forming a 45° angle.
Sizing diagonal 45-degree cross-braces for timber frames, steel trusses, and scaffolding towers.
For specialized job site layout tools, visit our dedicated Construction Pythagorean Theorem Calculator.
The 45-45-90 isosceles right triangle represents a simplified special case in trigonometry of the general Pythagorean theorem a² + b² = c². Substituting b = a simplifies the formula from two independent variables down to one single variable:
To explore general right triangle calculations where legs may have unequal lengths, visit our main Pythagorean Theorem Calculator.
It is important to understand when to use this specialized calculator versus a general right triangle tool:
c = √(a² + b²) for any arbitrary hypotenuse right triangle.a = b, allowing you to find the hypotenuse from just one leg input using c = a√2, or find equal legs from a single hypotenuse input using a = c / √2.If your triangle has unequal legs, use our main Pythagorean Theorem Calculator.
While a general right triangle solver calculates a = √(c² - b²) given a hypotenuse and one specific leg, the Isosceles Right Triangle Calculator solves for both equal legs simultaneously given only the hypotenuse (a = b = c / √2), without requiring any prior leg measurements.
An isosceles right triangle is a special right-angled triangle containing one 90-degree right angle and two equal perpendicular legs (a = b). The remaining two acute interior angles are each exactly 45 degrees.
A 45-45-90 triangle is another name for an isosceles right triangle, named after its three internal angle measurements: 45°, 45°, and 90°.
The fundamental side ratio of a 45-45-90 isosceles right triangle is 1 : 1 : √2, representing Leg A : Leg B : Hypotenuse C.
Multiply the equal leg length by √2 (c = a√2 ≈ a × 1.41421356). For example, a leg of 10 gives a hypotenuse of 10√2 ≈ 14.142.
Divide the hypotenuse by √2 (a = c / √2 = c√2 / 2). For example, a hypotenuse of 10 yields equal legs of 10 / √2 = 5√2 ≈ 7.071.
The area formula for an isosceles right triangle is Area = ½ × a², where a is the length of either equal leg. Alternatively, in terms of hypotenuse c, Area = c² / 4.
The perimeter formula is P = 2a + c = 2a + a√2 = a(2 + √2) ≈ 3.4142 × a.
By substituting equal legs a = b into the Pythagorean theorem: a² + a² = c² => 2a² = c² => c = √(2a²) = a√2.
All triangles contain 180 degrees. Subtracting the 90-degree right angle leaves 90 degrees. Because the legs are equal in length, the opposite base angles must be equal: 90° / 2 = 45°.
No. An isosceles triangle only requires two equal side lengths. It can be acute, right, or obtuse depending on its vertex angle.
Yes. Any triangle with two equal interior angles (45° and 45°) automatically has equal opposite leg lengths, making all 45-45-90 triangles isosceles right triangles.
Drawing a single straight diagonal across any square divides it into two identical, congruent 45-45-90 isosceles right triangles.
Yes. The calculator accepts any positive real number or decimal value (such as a = 7.5 or c = 14.25) and outputs accurate exact and rounded decimal results.
A 45-45-90 triangle has equal legs and a 1 : 1 : √2 side ratio. A 30-60-90 triangle has unequal legs with a 1 : √3 : 2 side ratio formed by bisecting an equilateral triangle.
Explore specialized right triangle tools tailored to your exact mathematical or engineering requirements: