45 45 90 triangle calculator

What is a 45 45 90 triangle?

A 45 45 90 triangle, also known as an isosceles right triangle, possesses unique properties that make it a particular interest in geometry. This is one type of special triangle where the angles measure 45°, 45°, and 90°. Such a triangle is symmetric, hence its two legs are equal in length.

Characteristics

This geometric figure is appealing due to its simple yet elegant structure. The key characteristics include:

• Equality of legs: In a 45 45 90 triangle, the legs are equal, simplifying the process of studying and calculating its dimensions.

• Side ratios: The length of the hypotenuse is equal to the length of a leg times the square root of two ($c = a\sqrt{2}$, where $a$ is the length of a leg, and $c$ is the length of the hypotenuse).

• Right angle: The hypotenuse always faces the 90° angle, important for calculations using trigonometry.

Properties of a 45 45 90 triangle

• Symmetry: Due to the equality of angles and legs, this triangle is symmetric, which simplifies its analysis. The triangle is symmetrical about the bisector of the 90° angle, allowing the use of properties of mirror reflection.

• Trigonometric functions: The sine and cosine of 45° angles are both $\frac{\sqrt{2}}{2}$ (or approximately 0.7071).

• Area and perimeter: The area and perimeter are also easily calculated due to simple ratios and formulas.

Formulas

Formulas with a known leg

If a leg $a$ is known, we can find the hypotenuse, area, and perimeter using:

1. Hypotenuse: $c = a\sqrt{2}$
2. Area: $\text{A} = \frac{a^2}{2}$
3. Perimeter: $\text{P} = 2a + a\sqrt{2}$

Formulas with a known hypotenuse

If the hypotenuse $c$ is known, we can find the leg, area, and perimeter using:

1. Leg: $a = \frac{c}{\sqrt{2}}$
2. Area: $\text{A} = \frac{c^2}{4}$
3. Perimeter: $\text{P} = 2 \left(\frac{c}{\sqrt{2}}\right) + c = c\left(1 + \frac{2}{\sqrt{2}}\right) = c(1 + \sqrt{2})$

Formulas with a known area

If the area $A$ is known, the leg, hypotenuse, and perimeter can be found using:

1. Leg: $a = \sqrt{2 \times \text{A}}$
2. Hypotenuse: $c = \sqrt{4 \times \text{A}}$
3. Perimeter: $\text{P} = 2a + c = 2\sqrt{2} \times \text{A} + \sqrt{4 \times \text{A}}$

Formulas with a known perimeter

If the perimeter $P$ is known, the leg, hypotenuse, and area can be found using:

1. Leg: $a = \frac{\text{P}}{2 + \sqrt{2}}$
2. Hypotenuse: $c = \sqrt{2} \times a$
3. Area: $\text{A} = \frac{a^2}{2}$

Calculation examples

Example 1: Known leg

Suppose a leg of the triangle is 5 cm. Find the hypotenuse, area, and perimeter:

1. Hypotenuse: $c = 5\sqrt{2} \approx 7.07$ cm
2. Area: $\text{A} = \frac{5^2}{2} = 12.5$ sq cm
3. Perimeter: $\text{P} = 2 \times 5 + 5\sqrt{2} \approx 17.07$ cm

Example 2: Known hypotenuse

If the hypotenuse of the triangle is 10 cm, find the leg, area, and perimeter:

1. Leg: $a = \frac{10}{\sqrt{2}} \approx 7.07$ cm
2. Area: $\text{A} = \frac{10^2}{4} = 25$ sq cm
3. Perimeter: $\text{P} = 10 + 2 \times 7.07 \approx 24.14$ cm

Example 3: Known area

Assume the area of a 45 45 90 triangle is 18 sq cm. Find the leg length, hypotenuse, and perimeter:

1. Leg: $a = \sqrt{2 \times 18} = \sqrt{36} = 6$ cm
2. Hypotenuse: $c = 6\sqrt{2} \approx 8.49$ cm
3. Perimeter: $\text{P} = 2 \times 6 + 6\sqrt{2} \approx 20.49$ cm

Example 4: Known perimeter

Suppose the perimeter of a 45 45 90 triangle is 24 cm. Find the lengths of the leg, hypotenuse, and area:

1. Leg: $a = \frac{24}{2 + \sqrt{2}} \approx 7.03$ cm
2. Hypotenuse: $c = 7.03 \cdot \sqrt{2} \approx 9.94$ cm
3. Area: $\text{A} = \frac{7.03^2}{2} \approx 24.71$ sq cm

Notes

• The 45 45 90 triangle is a foundational element in geometry and trigonometry, often used in problem solving and model construction.
• Due to its simple relationships and proportions, this triangle is frequently seen in architecture and design, as well as in natural forms and structures.

Frequently asked questions

How to find a leg if the hypotenuse is known?

If the hypotenuse $c$ is known, the leg $a$ can be found using the formula: $a = \frac{c}{\sqrt{2}}$.

Why is the hypotenuse equal to $a\sqrt{2}$?

The hypotenuse is equal to $a\sqrt{2}$ due to the application of the Pythagorean theorem and the equality of the legs. The theorem states: $c^2 = a^2 + a^2 = 2a^2$, hence $c = a\sqrt{2}$.

How to find the area of the triangle if a leg is known?

If a leg $a$ is known, the area can be found using the formula: $\text{A} = \frac{a^2}{2}$.

Is there any triangle with angles different from 45 45 90, having the same properties?

No, only the 45 45 90 triangle has such unique properties of equal legs and simple relationships between the hypotenuse and the legs.

Can the 45 45 90 triangle be used in practical applications?

Yes, due to its symmetry and easy calculations, the 45 45 90 triangle is commonly used in construction, design projects, and various engineering tasks.