How to Find Side Lengths in a 45 45 90 Triangle (With Examples)
A 45 45 90 triangle is a special case in geometry, commonly seen in school maths, competitive exams, and real-life applications. Its unique structure makes calculating side lengths and areas much easier, benefiting students in topics like trigonometry, algebra, and coordinate geometry. Understanding this triangle is essential for board exams, JEE, NEET, and beyond.
Understanding the 45 45 90 Triangle
A 45 45 90 triangle is a right triangle with two 45-degree angles and one 90-degree angle. This means the two legs (sides opposite 45° angles) are always equal. Since both legs are the same, it's also called an isosceles right triangle. The unique property of this triangle is that you can easily determine all sides and area if you know just one side.
Properties of a 45 45 90 Triangle
- Two equal angles: 45° each
- One right angle: 90°
- Two equal sides (legs)
- Hypotenuse is longer than either leg
- Sides are in the ratio 1 : 1 : √2 (leg : leg : hypotenuse)
- It’s half of a square, split along the diagonal
Formulae and Side Lengths in 45 45 90 Triangles
The special triangle ratio helps you quickly find missing sides:
| Given | Leg (a, b) | Hypotenuse (c) |
|---|---|---|
| Leg = x | x | x√2 |
| Hypotenuse = h | h/√2 | h |
Key formulas:
- Hypotenuse c = x√2 (if leg = x)
- Each Leg x = c / √2 (if hypotenuse = c)
- Area = (x × x) / 2 = x² / 2
- Perimeter = x + x + x√2 = x(2 + √2)
Step-by-Step Example: Finding Sides and Area
Let’s solve a problem.
Example: If one leg of a 45 45 90 triangle is 6 cm, find the hypotenuse and area.
- Find the hypotenuse:
Hypotenuse = leg × √2 = 6 × 1.414 = 8.49 cm (rounded to 2 decimals) - Find the area:
Area = (6 × 6) / 2 = 36 / 2 = 18 cm²
Practice Problems
- If the hypotenuse of a 45 45 90 triangle is 10 units, what are the lengths of the legs?
- Find the area of a 45 45 90 triangle with legs of 7 cm.
- The perimeter of a 45 45 90 triangle is 17.1 cm. Find the length of each leg (Round your answer to 2 decimal places).
- True/False: All isosceles right triangles are 45 45 90 triangles.
- If the area of a 45 45 90 triangle is 32 cm², what is the length of each leg?
Common Mistakes to Avoid
- Not applying the √2 factor to the hypotenuse (hypotenuse is not just double the leg).
- Confusing the side ratio with that of a 30 60 90 triangle.
- Forgetting to use the correct formula for area (should be leg × leg / 2).
- Rounding √2 too early; keep at least 2 decimals for accurate results.
Real-World Applications
45 45 90 triangles are everywhere – in carpentry, art, and architecture. Cutting a square diagonally creates two such triangles, useful for framing, tiling, and even bridge design. In maths, they’re essential for quick trigonometric calculations, and they appear often in coordinate geometry problems.
At Vedantu, we guide students through these applications and help make sense of geometry in exams and practical life. You can also explore related triangles like the isosceles triangle and apply these skills in trigonometry problems.
In summary, mastering the 45 45 90 triangle unlocks faster problem solving in maths exams and real-world design. This triangle’s consistent properties make side calculations, area finding, and geometry tasks much simpler. With regular practice and proper understanding, you’ll find geometry questions easier and more enjoyable.
FAQs on 45 45 90 Triangle: Properties, Formula, and Practice
1. What is the formula for a 45 45 90 triangle?
In a 45 45 90 triangle, the two legs are equal in length. If we let the length of each leg be 'x', then the hypotenuse will always be x√2. This is because of the Pythagorean theorem. This simple formula makes calculations easier. This is a special case of the Pythagorean theorem, where it helps to find the missing side lengths quickly.
2. What is the easy way to remember the 45 45 90 triangle?
Remember the side ratios: 1:1:√2. The two shorter sides (legs) are equal, and the hypotenuse is √2 times the length of a leg. Visualize an isosceles right triangle, which is another name for a 45-45-90 triangle. This visual representation will assist with memorization.
3. What are special right triangles 45 45 90 and 30 60 90?
Special right triangles are right-angled triangles with specific angle measures that lead to easy-to-remember side ratios. A 45 45 90 triangle has angles of 45°, 45°, and 90°, with sides in the ratio 1:1:√2. A 30 60 90 triangle has angles of 30°, 60°, and 90°, with sides in the ratio 1:√3:2. Knowing these ratios speeds up calculations.
4. What are the properties of a 45 45 90 triangle?
A 45 45 90 triangle is a special type of isosceles right triangle. Its key properties include: two angles measuring 45 degrees each and one right angle (90 degrees); two sides of equal length (legs); and the hypotenuse that is √2 times the length of either leg.
5. How do I solve for a missing side in a 45 45 90 triangle?
Use the 45 45 90 triangle formula. If you know the length of one leg (x), the other leg is also 'x', and the hypotenuse is x√2. If you know the hypotenuse (h), then each leg's length is h/√2. Remember to rationalize the denominator if needed, simplifying to h√2/2.
6. What is the formula for sides in a 45 45 90 triangle?
If one leg is 'x', then the other leg is also 'x', and the hypotenuse is x√2. This relationship simplifies solving problems involving these triangles significantly. The formula directly derives from the Pythagorean theorem applied to an isosceles right triangle.
7. How does the 45 45 90 formula compare to the Pythagorean theorem?
The 45 45 90 triangle formula is a specific application of the Pythagorean theorem (a² + b² = c²). Because the legs are equal (a=b), the formula simplifies to x² + x² = (x√2)². It's a shortcut for isosceles right triangles.
8. Can you derive trigonometric values using a 45 45 90 triangle?
Yes! In a 45 45 90 triangle, since both legs are equal, the trigonometric ratios for sine, cosine, and tangent at 45° are easily derived: sin 45° = cos 45° = 1/√2 = √2/2, and tan 45° = 1. These values are fundamental in trigonometry.
9. What is the proof of the side ratio for a 45 45 90 triangle?
The 1:1:√2 ratio is proven using the Pythagorean theorem. Start with an isosceles right triangle with legs of length 'x'. By the Pythagorean theorem, the hypotenuse (h) is √(x² + x²) = √(2x²) = x√2. This gives the ratio x : x : x√2, which simplifies to 1:1:√2.
10. What real-life shapes use 45 45 90 triangles?
45 45 90 triangles are found in many symmetrical structures. Examples include: the diagonals of a square which create two 45 45 90 triangles; various architectural designs with symmetrical elements; and in some engineering and construction applications. They are used frequently because of their simplicity and symmetry.
11. Is a 45 45 90 triangle always isosceles?
Yes, a 45 45 90 triangle is always an isosceles triangle because, by definition, it has two equal angles (45° each), which means the sides opposite these angles are also equal in length. This is a defining characteristic.
12. What’s the best way to remember a 45 45 90 triangle?
Visualize an isosceles right triangle. Remember the simple ratio of sides: 1:1:√2. If one leg is 'x', the other leg is 'x', and the hypotenuse is x√2. Practice a few problems to solidify your understanding of this special triangle.
