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Right Triangle Calculator

Finding out the missing side or angle couldn't be easier than with our great tool - right triangle side and angle calculator. Choose two given values, type them into the calculator and the remaining unknowns will be determined in a blink of an eye! If you are wondering how to find the missing side of a right triangle, keep scrolling and you'll find the formulas behind our calculator.

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Calculators play an integral part in the modern world. The different fields in mathematics and physics are getting lengthy and complex day by day and therefore this is challenging and needs calculators to make work easier and more efficient.

A right triangle calculator is a tool that simplifies your work in finding the area of a right-angle triangle. Finding the lengths of the sides of a right-angled triangle won’t be difficult anymore, this triangle solver is here for you to give you more information about right-angled triangles.

Right-angled triangle

A right-angled triangle simply means one of the angles is 90° and so the total sum of the three angles in a triangle is 180°. Thus 90° is the total sum of the two angles remaining.

The triangle's base is formed from the sides of the triangle with a perpendicular line. So in a right-angled triangle, the longest side is known as the hypotenuse. The hypotenuse is the largest in a right-angled triangle, it is also opposite to the length and height of a right-angled triangle.

Pythagoras theorem explains more on the relationship of these sides of the triangle. This article will elaborate more on the Pythagoras theorem that is associated with right-angled triangles.


Area and the right-angled triangle

Many people are always eager to find the right triangle surface area in a simplified way, well worry no more because you're in the right place, a right triangle calculator is an excellent tool for you to find the accurate surface area of a right-angled triangle.

Looking for an equation of the triangle sides and the hypotenuse? Worry no more as you won't be disappointed because this right triangle calculator has all features that you need at your disposal.

Want to find a triangle in which one of the angles is exactly 90°? The right-angled triangle is the one you are looking for. The keywords "base" and "height" refer to the two legs. Below is a formula of the area of a right-angle triangle that can be simplified by our calculator.

A right triangle's area is equal to half of its base multiplied by its height, (1/2 ×b× h).


The formula for the area of the right triangle

For instance, multiplying two numbers like 3 × 4 gives us a total of 12, and dividing it by two gives us 6. If you multiply your triangle's base and height, divide by two, you get its area.


The area of a right triangle with a base of 14 ft and height of 8 ft is 1/2 × 14 × 8 = 56 ft2.

The area of a triangle with a base of 40 cm and height of 25 cm is 1/2 × 40× 25=500 cm2


The formula for the right triangle area: how to calculate it?

Putting the length and width of a rectangle into consideration. A diagonal distinction needs to be made inside the rectangle. Right triangles can be seen in the rectangle.

Rectangular areas are equal to their length multiplied by their width. There is no doubt about this formula. Triangles inside a rectangle are identical. That's because they're overlapping with each other.

In order to simplify this, the right triangle calculator comes up with a formula: the area of the rectangle is equal to two right triangles' areas.


If the area of the rectangle = l × w = 4 then it means you’ll multiply the answer which is 4 × 1 then divide by two to get the surface area of one of the triangles.

This gives,

The surface area of a right triangle is equal to 1/2 of its length times its width.


Proper Rectangle Area, Hypotenuse Included

Pythagoras theorem is very clear when it comes to a right-angled triangle. This means that the formula for calculating the hypotenuse is (base) 2 + (height) 2.

Although the hypotenuse alone cannot be used to determine the area of a right triangle, the hypotenuse can still be used to determine the area of the triangle if the base and/or height are also known. Before using a Pythagorean Theorem calculator, let's look at an example.

Example:  Find the surface area of a right-angle triangle with 12 inches as the base and 16 inches as the hypotenuse.

Pythagoras theorem application below:

Let’s calculate manually,

 (Hypotenuse) 2 = (base) 2 + (height) 2(get the values of base + height)

202 = 122 + (height) 2 (square both hypotenuse and base)

400 = 144 + (height) 2 (add base + height)

(Height) 2 = 256

Height = √ (256) = 16 units.

So, the area of the given triangle = 1/2 × base × height = 1/2 × 12 × 16 = 96 square units.

Let's look at how the procedure might work if we used the right triangle calculator on this 

Figure below:

In the calculator, enter the values for x and y.

X and y should be squared.

Find the total values of x2 and y2

After taking the square root of the total of x and y a positive and a negative result will be obtained.

Just assume the negative outcome first, the value obtained is the hypotenuse r.

You'll also get the area value for such a triangle as a bonus.


Factors to consider when working with a right triangle

The sides of a triangle have a slope also known as the gradient in mathematics. Now we'll look at some of the various things that can be determined from a right triangle when we use some of the features in a slope calculator.

A triangle's sides have a certain slope or gradient. The slope of each side can be calculated with a slope calculator. If you want to calculate the slope by yourself, here is the formula:

slope = (y - y)/(x - x)

Two or even three side lengths may be missing from a right angle and this may be an issue at times. The right triangle calculator, the hypotenuse calculator, and the method for calculating the right triangle's area are useless in certain situations.

At this point, complex numbers will be needed to fill in the gaps. A right triangle calculator will solve this problem in a simple way.


Unique triangles

These are triangles that are unique compared to the other triangles. Several features that are found in this category of unique triangles make them different from the rest of other triangles.

This is why the right triangle is special: it contains only one hypotenuse and thus a right angle. Apart from the right-angled triangle, there seem to be additional unusual triangles with unique features.

The equilateral triangle, with its three equal sides and 60° angles, is one of the unique triangles known. In this case, an equilateral triangle calculator is used to calculate the parameters of this unique triangles. 

The diagram above is a sample of a unique triangle. Unique triangles are so many, the isosceles triangle is among them and it also comes in different shapes. The isosceles triangle calculator is the best as it is recommended for this type of triangle. The special right triangle also falls under this category with its specific features.


Right triangles (special right triangles)

The diagram below is an illustration of a special right triangle.

Special right triangles include the "45 45 90" triangle, which is one of the most unusual. An isosceles triangle is among this category In addition to being an isosceles triangle, this right-angled triangle also has a hypotenuse. 

If you divide a square by its diagonal, you'll get a right triangle like the one shown above. Both sides of this square have the same length. This is where the right-angled triangle comes in. We encourage the use of the “45 45 90” triangle calculator for anyone interested in learning more about the most unique right triangles.

Another unique triangle from the collection of special right triangles is also known as the "30 60 90" triangle. This triangle has one right angle shown as (90°), the two remaining angles are 30° and 60° respectively. These angles are unique in a way because of the values of their trigonometric functions (the cosine, the sine, the tangent and many more).

The formula below can be used in the “30 60 90” triangle calculator to see and understand the results of this, but we'll summarize some of them here for those who want to know more. Considering that the shorter side is of length a, the triangle shows:

The hypotenuse is 2a, 

The area will be a²√(3/2), 

So the total of the perimeter is a(3 + 3), 

Therefore the second length will be a²√(3),


Parallelograms / Right-angled triangles

A right-angled triangle and a parallelogram have nothing in common but what role does a parallelogram solver have in helping to analyze a triangle? A parallelogram can simply be divided into two or more parts. Below is an example of a rectangle that can be seen easily.

Let’s consider a rectangle of any shape. Now draw a line on the diagonals of this rectangle from one end to the other end (gain more knowledge about this in a rectangle calculator). We get two right-angled triangles that are clearly seen in this rectangle after dividing the rectangle along with a diagonal line.

No need to use a right triangle calculator by seeing that both triangles are equal, as such their areas seem to be equal. In other words, it means that the rectangle has twice the area of each triangle.

For a rectangle with two sides a and b, we get area = a× b, while for the right triangle, we get area = base × height / 2, which in this case would be area = a × b /2. Just cutting the rectangle diagonally reveals the same result.

Even though this is an example of a rectangle, the same logic applies to a square's area. For many other parallelograms, the procedure becomes more difficult to calculate the area using this method (more than four different right triangles of different sizes are necessary), although using right-angled triangles to calculate the area of a parallelogram can still be possible. To save you time and effort, our calculator is here for you.


Mathematics meets Pythagorean triplets and triangles

The relationship for both geometry and polygons, in particular triangles, is unbreakable. Without even a question, non-mathematicians also find it amazing and unbelievable to understand the properties of certain triangles, like the right triangle. Mathematics fans, take note: right-angled triangles are indeed a favorite topic of study.

The integers (a, b, and c) in the above diagram are the three integers that make up the Pythagorean triplets and fulfill the Pythagorean theorem, which further states that a2 + b2 = c2. Consequently, a right-angled triangle can be created, with these angles (a, b, c). Mathematicians have such a great deal of fun looking for new numbers that match this formula above and applying them on a calculator.

The mystery this relationship excites is that it has some useful properties for cryptography. Mathematics has gone even further, using other numbers that satisfy a similar relation which includes, for example, 4, 5… where the sum of any of the squares with the exception of one, yields the square number that is deserted.

The legendary Pierre Fermat used the infamous Fermat's last theorem which states that the relation of an+ bn = cn for n is greater than 2. This theory has been used by mathematicians for many years as it is linked to these Pythagorean triplets. Despite the fact that it has yet to be mathematically proven, this conjecture is widely regarded as one of the century's most important problems in mathematics.


The radius of the earth, shadows, and right triangles 

When an object is lighted, it casts a dark shadow on the surrounding area. Right-angled triangles are easy to spot if you look closely at the shadows, the objects, and the ground collectively.

The object and ground must be horizontal in order for this to be true. If this is the case, it's probably close enough for most of the time, as well. This means that the right triangle calculator can be used to collect data under the sun.

On top of that, the height of a building can't be calculated easily by using measuring tape on top of the building. This can be done easily by using your own vehicle while moving. The length of the shadow cast on the ground can still be measured.

Whatever will determine the angle between the shadow and the ground is an angle measuring tool and a paper. It is possible to calculate an object's height by inputting its side and angle measurements into a right triangle side and angle calculator.

A bright sunny day or any other source of light that can be used when it comes to the measuring of heights of the various objects. This is the technique that was used mostly by the ancient people in calculating distances. To our knowledge, Eratosthenes used right triangles and shadows to determine the Earth's circumferential radius. Here is how he accomplished it:

On the summer solstice, Eratosthenes noted that there was an area on earth where the wells had no shadow during midday, i.e., the sun shined directly down on them. To test this, he positioned a column of specific height at a fixed radius from the well, and at the exact same time of the day and the year, he then measured the length of the shadow cast by the column. Right-angled triangles and trigonometry were also used to measure the angle between cities as well as the earth's radius, as he already knew how far apart they were.

Surprisingly, you can accomplish the same thing with the calculators that have been developed for you. 

To summarize, the calculator has more benefits than limitations. Scientists and mathematicians should become more comfortable with the development of new technology as a result of using calculators. With the formulas given above, now more than ever we can be confident when using a calculator to study mathematics and physics.

A calculator is very beneficial when it comes to solving differential equations in mathematics as well as physics. It is impossible to learn the different formulas in mathematics and physics without a calculator. So the right triangle calculator can be of importance when it comes to mathematics and physics.