Solve a 45 45 90 Triangle
Use a single positive value. Select what that value represents. The calculator finds all major dimensions.
Formula Used
Let each equal leg be x. A 45 45 90 triangle has two matching legs and one longer hypotenuse.
Hypotenuse = x√2
Area = x² ÷ 2
Perimeter = x(2 + √2)
Altitude to hypotenuse = x ÷ √2
Inradius = x(2 − √2) ÷ 2
Circumradius = x√2 ÷ 2
The calculator first converts your selected measurement into x. It then derives every requested value using these fixed relationships.
How to Use This Calculator
- Select the measurement you already know.
- Enter its positive numerical value.
- Add a unit label, such as cm or inches.
- Choose your preferred decimal precision and result detail.
- Press Calculate Triangle to view results above the form.
- Download CSV or PDF after reviewing the values.
Example Data Table
| Known input | Input value | Equal legs | Hypotenuse | Area |
|---|---|---|---|---|
| One leg | 6 cm | 6 cm each | 8.485 cm | 18 cm² |
| Hypotenuse | 10 m | 7.071 m each | 10 m | 25 m² |
| Area | 50 in² | 10 in each | 14.142 in | 50 in² |
| Altitude | 5 ft | 7.071 ft each | 10 ft | 25 ft² |
Working With 45 45 90 Triangles
A 45 45 90 triangle is a special right triangle. Its two acute angles are equal. Each acute angle measures forty-five degrees. The third angle measures ninety degrees. Because the angles match, the two legs also match. This symmetry makes calculations fast and reliable.
Why One Value Is Enough
The side relationship follows a fixed pattern. Let one leg equal x. The other leg is also x. The hypotenuse equals x times square root of two. This relationship replaces repeated trigonometric calculations. One trusted measurement can reveal every other main dimension.
Choose the measurement you know before starting. A leg is the most direct choice. The calculator multiplies it by square root of two. A known hypotenuse works equally well. The calculator divides it by square root of two. Area, perimeter, and altitude can also determine the missing sides.
Useful Measurements
Area is especially useful for floor plans and design sketches. The area equals one half times leg squared. Therefore, the leg equals the square root of twice the area. Perimeter is another helpful starting point. Divide the perimeter by two plus square root of two. The resulting value is the length of either leg.
The altitude here reaches from the right angle to the hypotenuse. It is shorter than either leg. Its length equals the leg divided by square root of two. When the altitude is known, multiply it by square root of two. That gives each leg. The hypotenuse then equals twice the altitude.
Accuracy and Learning
Units matter in every geometry problem. Keep all inputs in one unit system. Use centimetres, metres, inches, or another consistent choice. Square units apply only to area. Linear units apply to lengths and perimeter. The calculator preserves your typed unit label in the results.
Rounded answers are convenient for construction estimates and homework checks. Exact radical expressions are better for proofs and symbolic work. Compare both forms when possible. This page displays formulas, decimal values, and relationships. It can also export the results for notes or reports.
Check that your input is positive before solving. A triangle cannot have zero or negative side lengths. Review the selected measurement type carefully. Then compare the diagram with your answer. The equal legs should remain equal. The hypotenuse should always be longest. These quick checks prevent common mistakes. Support everyday geometry practice and revision.
Frequently Asked Questions
1. What makes a 45 45 90 triangle special?
It has angles of 45°, 45°, and 90°. The two equal angles create two equal legs. Its side ratio is always 1 : 1 : √2.
2. How do I find the hypotenuse from one leg?
Multiply the leg length by √2. For example, a leg of 8 gives a hypotenuse of 8√2, or about 11.314.
3. How do I find a leg from the hypotenuse?
Divide the hypotenuse by √2. You may also multiply it by √2 ÷ 2. Both methods give the same leg length.
4. Can I enter the area instead of a side?
Yes. The calculator uses leg = √(2 × area). It then finds the hypotenuse, perimeter, altitude, and circle radii.
5. What is the area formula?
The area is x² ÷ 2, where x is either equal leg. This comes from one half times base times height.
6. What is the perimeter formula?
The perimeter equals x(2 + √2). Add the two equal legs, then add the hypotenuse x√2.
7. What does the altitude option mean?
It is the perpendicular distance from the right-angle vertex to the hypotenuse. In this triangle, altitude = leg ÷ √2.
8. Why is the hypotenuse always longest?
It lies opposite the 90° angle, the largest angle. Since √2 is greater than one, x√2 is longer than either leg.
9. Can I use centimetres, feet, or inches?
Yes. Use any consistent length unit. Area results use the matching square unit, such as cm², ft², or in².
10. Are rounded values exact?
No. Rounded values are approximations. The radical form, such as x√2, keeps the exact relationship for symbolic work.
11. Can I export my calculation?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a printable results summary.