45 45 90 Triangle Calculator
The 45 45 90 Triangle Calculator is a free and easy-to-use online tool that helps you quickly calculate all sides and properties of a 45 45 90 triangle based on a single known value. Whether you’re working with the legs (perpendicular & base), hypotenuse, area, or perimeter, simply enter any one value, and the calculator will instantly determine all the remaining dimensions with accuracy and ease.
It also provides clear, step-by-step explanations, making it ideal for learning geometric concepts, verifying your work, or using it as a quick reference tool—conveniently available on CalculationClub.com.
How to Use the Online 45 45 90 Triangle Calculator
1. Enter Any One Value
Start by entering any one known value of the triangle—such as side a (leg), side b (leg), side c (hypotenuse), h (altitude), Area, or Perimeter. The calculator only needs one input to compute the rest.
2. Choose Decimal Precision and Units
Select the number of decimal places you’d like the results to display (e.g., 3), and choose your preferred unit of measurement, such as meters (m) :
Then, choose your preferred unit of measurement from the following options:
- Meters (m)
- Centimeters (cm)
- Millimeters (mm)
- Yards (yd)
- Feet (ft)
- Inches (in)
3. View Instant Results
As soon as you enter a value, the calculator automatically computes all the missing dimensions of the triangle using the fixed 45 45 90 side ratios.
4. Show or Hide Steps
Click “Show Steps” to view a detailed explanation of how each value was calculated. You can also click “Hide Steps” for a cleaner, summary-only view.
5. Reset for a New Calculation
Click the “Reset” button to clear all fields and enter a new set of values for another triangle calculation.
45 45 90 Triangle Calculator
What is a 45 45 90 Triangle?
A 45 45 90 triangle is a special type of right-angled triangle in geometry, characterized by its two equal angles of 45° and the right angle (90°). Since it has a 90° angle, it is a right triangle, and we can apply the Pythagorean Theorem to this triangle.
Sides of the 45 45 90 Triangle
In every 45-45-90 triangle, the three sides are always in a fixed ratio. Once you know the length of any one side, you can calculate the other two using this ratio.
Let’s assign the sides their standard names:
- Leg (opposite 45° angle): a
- Hypotenuse (opposite 90° angle): a√2
Explanation of Each Side:
Leg (Perpendicular and Base): The legs of the 45 45 90 triangle are always equal and are opposite the two 45° angles. These two sides are congruent.
Hypotenuse: The hypotenuse is opposite the 90° angle and is always √2 times longer than the length of the leg.
Leg : Leg : Hypotenuse = 1 : 1 : √2
Altitude of a 45 45 90 Triangle:
In a triangle, an altitude is a line segment drawn from a vertex perpendicular to the opposite side (or its extension). In a 45-45-90 triangle, we often draw the altitude from the right angle (90°) to the hypotenuse. This splits the triangle into two smaller right-angled triangles.
Formula to Calculate Altitude:
Leg = a
Hypotenuse = a√2
Step-by-Step Explanation:
Step 1: Area Expressions- We can calculate the area of a triangle in two ways:
a. Using the base and height (altitude):
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
$\text{Area} = \frac{1}{2} \times c \times h$
Where:
- c = Hypotenuse (for formula it is base)
- h = Altitude from the right angle to the hypotenuse (for formula it is height)
- c = Hypotenuse (for formula it is base)
b. Using the two legs (leg and leg ):
$\text{Area} = \frac{1}{2} \times a \times b$
Where:
- a =leg (Perpendicular)
- =leg (Base)
- a =leg (Perpendicular)
Step 2: Solving for Altitude (h): By setting both area expressions equal to each other, we can solve for the altitude (h).
$\text{Area}$ = $\text{Area}$
$\frac{1}{2} \times c \times h$= $\frac{1}{2} \times a \times b$
Simplifying this equation:
h = $\frac{a \times b}{c} $
- c = Hypotenuse (for formula it is base)
- h = Altitude from the right angle to the hypotenuse (for formula it is height)
- a = leg (Perpendicular)
- = leg (Base)
Step 3: Substituting Values
Now, let’s substitute the values for the sides of the triangle:
- a = x
- b = x
- c = x√2
Simplifying the equation:
So, the altitude (h) from the right angle to the hypotenuse is:
$\mathbf{h = \frac{x}{\sqrt{2}}}$
What is the Pythagorean Theorem?
The Pythagorean Theorem states:
$\text{(Hypotenuse)}^2 = (\text{Base})^2 + (\text{Height})^2 $
or c2 = a2 + b2
This theorem applies only to right-angled triangles, and since the 30-60-90 triangle has a 90° angle, it qualifies.
Summary of a 45 45 90 Triangle:
Key Properties of 45 45 90 Triangle:
- It’s always a right triangle because it includes a 90° angle; that’s why we can apply the Pythagoras Theorem to this triangle.
- The legs are equal, and the hypotenuse is √2 times longer than either leg.
Side Length Ratios of 45 45 90 Triangle:
A 45-45-90 triangle has a fixed ratio of sides: 1 : 1 : √2
If the leg (opposite the 45° angle) is of length a, then:
- The other leg is also a.
- The hypotenuse (opposite the 90° angle) is a√2.
- If any one side is known, all other sides can be calculated.
- The Pythagorean Theorem holds true because the triangle has a right angle.
Example of 45 45 90 Triangle:
If the leg (opposite 45°) is 5cm:
- The other leg = 5cm (since they are equal)
- Hypotenuse = 5√2 ≈ 7.07cm
Formulas: 45°-45°-90° Triangle
Side:
In a 45°-45°-90° triangle, the sides are always in the ratio:
$\text{Leg} : \text{Leg} : \text{Hypotenuse} = 1 : 1 : \sqrt{2}$
Let each leg be x, then:
Leg 1 = x
Leg 2 = x
Hypotenuse = $x\sqrt{2}$
Altitude (h) from Right Angle to Hypotenuse
Using the area expression:
$\text{Area} = \frac{1}{2} \times a \times b = \frac{1}{2} \times c \times h$
Solving for $h$:
$h = \frac{a \times b}{c} = \frac{x \cdot x}{x\sqrt{2}} = \frac{x}{\sqrt{2}}$
So,
$\mathbf{h = \frac{x}{\sqrt{2}}}$
Perimeter (P)
Add all sides:
$P = x + x + x\sqrt{2} = x(2 + \sqrt{2})$
$\mathbf{P = x(2 + \sqrt{2})}$
Area (A)
Using the legs as base and height:
$A = \frac{1}{2} \cdot x \cdot x = \frac{x^2}{2}$
$\mathbf{A = \frac{x^2}{2}}$
Problem: 45°-45°-45° Triangle
Example.1 = Given a 45°-45°-90° triangle with one leg $x = 5$ units, we need to find the other leg, hypotenuse, altitude, perimeter, and area of the triangle.
Solution: The sides of a 45°-45°-90° triangle follow the ratio $1 : 1 : \sqrt{2}$, with the leg representing $x$. Since the leg is given as $x = 5$, we can calculate the other properties.
The other leg is also $x$, which gives:
$\text{Other Leg} = 5$ $ \text{ units}$
The hypotenuse is $x\sqrt{2}$, so:
$\text{Hypotenuse} = 5\sqrt{2}$ $ \approx 7.07 \text{ units}$
The altitude $h$ from the right angle to the hypotenuse can be calculated using the formula:
$h = \frac{x^2}{\sqrt{2}x} = \frac{x}{\sqrt{2}}$ $ = \frac{5}{\sqrt{2}}$ $ \approx 3.54 \text{ units}$
The perimeter $P$ is the sum of all three sides:
$P = x + x + x\sqrt{2}$ $ = 5 + 5 + 7.07$ $ = 17.07 \text{ units}$
Finally, the area $A$ of the triangle can be calculated using the base and height (legs):
$A = \frac{1}{2} \times 5 \times 5$ $ = \frac{1}{2} \times 25$ $ = 12.5 \text{ square units}$
Example.2 = Given a 45°-45°-90° triangle with the hypotenuse $10$ units, we need to find the legs, altitude, perimeter, and area of the triangle.
Solution: The sides of a 45°-45°-90° triangle follow the ratio $1 : 1 : \sqrt{2}$, with the hypotenuse representing $x\sqrt{2}$. To find the leg $x$, solve:
$x = \frac{10}{\sqrt{2}}$ $ \approx 7.07 \text{ units}$
Now we can calculate the other properties.
The legs are:
$\text{Legs} = x = \frac{10}{\sqrt{2}}$ $ \approx 7.07 \text{ units}$
The altitude $h$ from the right angle to the hypotenuse:
$h = \frac{x^2}{\sqrt{2}x} = \frac{x}{\sqrt{2}} = \frac{10}{2} = 5 \text{ units}$
Perimeter $P$ is the sum of all three sides:
$P = x + x + x\sqrt{2} = 7.07 + 7.07 + 10$ $ = 24.14 \text{ units}$
Finally, Area $A$ of the triangle using base and height (legs):
$A = \frac{1}{2} \times x \times x = \frac{1}{2} \times 7.07 \times 7.07$ $ \approx 25 \text{ square units}$
Example.3 = Given a 45°-45°-90° triangle with an altitude $h = 6$ units from the right angle to the hypotenuse, we need to find the legs, hypotenuse, perimeter, and area of the triangle.
Solution: The altitude from the right angle to the hypotenuse in a 45°-45°-90° triangle is given by:
$h = \frac{x}{\sqrt{2}}$
Solving for $x$:
$x = h \sqrt{2} = 6\sqrt{2} \approx 8.49 \text{ units}$
The legs are:
$\text{Legs} = x = 6\sqrt{2} \approx 8.49 \text{ units}$
Hypotenuse is:
$\text{Hypotenuse} = x\sqrt{2}$ $ = 6\sqrt{2} \cdot \sqrt{2}$ $ = 6 \cdot 2 = 12 \text{ units}$
Perimeter $P$ is:
$P = x + x + x\sqrt{2}$ $ = 8.49 + 8.49 + 12$ $ = 28.98 \text{ units}$
Area $A$ is:
$A = \frac{1}{2} \cdot x \cdot x $ $= \frac{1}{2} \cdot 8.49 \cdot 8.49$ $ \approx 36 \text{ square units}$
Example.4 = Given a 45°-45°-90° triangle with perimeter $P = 20$ units, we need to find the leg $x$, hypotenuse, altitude, and area.
Solution: The perimeter is:
$P = x + x + x\sqrt{2} = x(2 + \sqrt{2})$
Solving for $x$:
$x = \frac{20}{2 + \sqrt{2}}$ $ \approx 5.36 \text{ units}$
The legs are:
$\text{Legs} = x = 5.36 \text{ units}$
Hypotenuse is:
$\text{Hypotenuse} = x\sqrt{2}$ $= 5.36\sqrt{2}$ $ \approx 7.58 \text{ units}$
Altitude $h$ is:
$h = \frac{x}{\sqrt{2}} = \frac{5.36}{\sqrt{2}}$ $ \approx 3.79 \text{ units}$
Area $A$ is:
$A = \frac{1}{2} \cdot x \cdot x = \frac{1}{2} \cdot 5.36 \cdot 5.36$ $ \approx 14.36 \text{ square units}$
Example.5 = Given a 45°-45°-90° triangle with area $A = 32$ square units, we need to find the legs, hypotenuse, altitude, and perimeter.
Solution: The area is:
$A = \frac{1}{2} \cdot x \cdot x$ $ = \frac{x^2}{2}$
Solving for $x$:
$x^2 = 2A = 64 \Rightarrow x = \sqrt{64} = 8 \text{ units}$
The legs are:
$\text{Legs} = x = 8 \text{ units}$
Hypotenuse is:
$\text{Hypotenuse} = x\sqrt{2} = 8\sqrt{2}$ $ \approx 11.31 \text{ units}$
Altitude $h$ is:
$h = \frac{x}{\sqrt{2}}$ $ = \frac{8}{\sqrt{2}}$ $ \approx 5.66 \text{ units}$
Perimeter $P$ is:
$P = x + x + x\sqrt{2}$ $ = 8 + 8 + 11.31 = 27.31 \text{ units}$
Frequently Asked Questions (FAQs) on 45 45 90 Triangle Triangle Calculator
What is a 45-45-90 triangle?
A 45-45-90 triangle is a special right triangle with two angles of 45° and one right angle of 90°. The sides of this triangle have a consistent ratio of 1 : 1 : √2.
How does the 45-45-90 triangle calculator work?
The calculator uses the known side ratios of the triangle. If you input one side (leg or hypotenuse), the calculator will automatically compute the other two using the 1 : 1 : √2 ratio.
Which side is which in a 45-45-90 triangle?
- The legs are the two equal sides, opposite the two 45° angles.
- The hypotenuse is opposite the right angle (90°) and is the longest side.
What is the ratio of the sides in a 45-45-90 triangle?
The side lengths follow the ratio:
Leg : Leg : Hypotenuse = 1 : 1 : √2
Can I use any unit for input?
Yes, as long as you use the same unit for all sides. The calculator preserves the unit in its output.
Is this triangle always a right triangle?
Yes, by definition, a 45-45-90 triangle is always a right triangle because it includes a 90° angle.
What if I enter one of the legs or the hypotenuse instead of the other leg?
The calculator can work with any one side. If you provide one of the legs or the hypotenuse, it will reverse-calculate the other side and use the ratio to find the third side.
Can this calculator show steps or formulas used?
Some versions of the calculator display the formulas used in the background. If yours doesn’t, you can manually apply:
- Leg = Hypotenuse / √2
- Hypotenuse = Leg × √2
Why is this triangle useful in geometry or trigonometry?
It simplifies calculations due to its predictable ratios and is commonly used in trigonometric problems, construction, and design work where symmetry is key.
Can I use this calculator for solving real-life problems?
Absolutely. It’s helpful for solving construction problems, ramp angles, and any application involving a 45-45-90 triangle, such as designing roofs, ramps, or diagonals in square structures.
Final Thoughts: The 45-45-90 Triangle Calculator is a quick and easy tool that helps you find all sides and properties of the triangle with just one known value. Whether you’re a student, teacher, or professional, this tool is fast, accurate, and always available on CalculationClub.com.
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