Our online tools will provide quick answers to your calculation and conversion needs. On this page, you can solve math problems involving right triangles. You can calculate angle, side (adjacent, opposite, hypotenuse) and area of any right-angled triangle and use it in real world to find height and distances. Download: Use this right-triangle solver offline with our all-in-one calculator app for Android and iOS. Right triangle calculationFormulas used for calculations on this page: Pythagoras' Theorem Trigonometric functions: Area = a*b/2, where a is height and b is base of the right triangle. © everydaycalculation.com a=3 b=4 c=5 ... triangle calc by three sides a,b,c. What do the symbols mean? Additional Information Note that giving the sizes of the two angles, A & B will not allow any
other sizes to be found. At least one dimension, of edge-length or area, has to be supplied. Created by Álvaro Díez, Mateusz Mucha and Piotr Małek Reviewed by Jack Bowater Last updated: Apr 12, 2022 The right triangle calculator will help you find the lengths of the sides of a right-angled triangle. This triangle solver will also teach you how to find the area of a right triangle as well as give plenty of information about the practical uses of a right triangle. What is a right triangle (or right-angled triangle)?First things first, let's explain what a right triangle is. The definition is very simple and might even seem obvious for those who already know it: a right-angled triangle is a triangle where one and only one of the angles is exactly 90°. The other two angles will clearly be smaller than the right angle because the sum of all angles in a triangle is always 180°. In a right angled triangle the sides are defined in a special way. The side opposing the right angle is always the biggest in the triangle and receives the name of "hypotenuse". The other two sides are called catheti. The relationship between the hypotenuse and each of the cathetus is a very simple one, as we will see when we will talk about Pythagoras' theorem. Hypotenuse calculatorIf all you want to calculate is the hypotenuse of a right triangle, this page and its right triangle calculator will work just fine. However, we would also recommend to use the specific tool we have developed at Omni Calculators: the hypotenuse calculator. The hypotenuse is opposite the right angle and can be solved by
using the Pythagorean theorem. In a right triangle with cathetus To solve for Let's now solve a practical example of what it would take to calculate the hypotenuse of a right triangle without using any calculators available at Omni:
Now let's see what the process would be using one of Omni's calculator, for example, the right triangle calculator on this web page:
How to find the area of a right triangleWe have seen already that calculating the area of a right angle triangle is very easy with the right triangle calculator. At Omni Calculators, we have a calculator specifically designed for that purpose as well: area of a right triangle calculator. There is even one for the adventurous amongst you that would like to calculate the area of any triangle: area of a triangle calculator. Let's now see a bit more in-depth how to calculate areas of right triangles. The method for finding the area of a right triangle is quite simple. All that you need are the lengths of the base and the height. In a right triangle, the base and the height are the two sides which form the right angle. Since multiplying these to values together would give the area of the corresponding rectangle, and the triangle is half of that, the formula is:
If you don't know the base or the height, you can find it using the Pythagorean theorem. Use the right triangle calculator to check your calculations or calculate the area of triangles with sides that have larger or decimal value length. Other considerations when dealing with a right triangleNow we're gonna see other things that can be calculated from a right triangle using some of the tools available at Omni. The sides of a triangle have a certain gradient or slope. We can use a slope calculator to determine the slope of each side. The formula for the slope (in case if you want to calculate by hand) is
So if the coordinates are There is an easy way to convert angles from radians to degrees and degrees to radians with the use of the angle conversion:
Sometimes you may encounter a problem where two or even three side lengths are missing. In such cases, the right triangle calculator, hypotenuse calculator and method on how to find the area of a right triangle won't help. You have to use trigonometric functions to solve for these missing pieces. This can be accomplished by hand or by using the triangle calculator. Special trianglesThe right triangle is just one of the many special triangles that exist. These triangles have one or several special characteristics that make them unique. For example, as we have seen, the right triangle has a right angle, and hence a hypotenuse, that makes it a unique kind of triangle. Aside from the right-angled triangle, there are other special triangles with interesting properties. One of the most known special triangles is the equilateral triangle, which has three equal sides and all its angles are 60°. This makes it much more simple to make a triangle solver calculator, such as the equilateral triangle calculator, in which one can calculate different parameters of such a triangle. Another of special triangles is the isosceles triangle, which has 2 sides of equal length, and hence two angles of the same size. As opposed to the equilateral triangle, isosceles triangles come in many different shapes, but all have certain properties that are exploited by the isosceles triangle calculator to obtain all the parameters of these triangles. There are many other special triangles. However, we will now take a look at a few very special right triangles that besides being right-angled triangles, they have other special properties that make them interesting. Special right trianglesAmong all the special right triangles, probably the most special is the so-called "45 45 90" triangle. This is a right-angled triangle that is also an isosceles triangle. Both its catheti are of the same length (isosceles) and it also has the peculiarity that the non-right angles are exactly half the size of the right angle that gives the name to the right triangle. This right triangle is the kind of triangle that you can obtain when you divide a square by its diagonal. That is why both catheti (sides of the square) are of equal length. For those interested in knowing more about the most special of the special right triangles, we recommend checking out the 45 45 90 triangle calculator made for this purpose. Another very interesting triangle from the group of special right triangles is the so-called "30 60 90" triangle. The name comes from having one right angle (90°), then one angle of 30° and another of 60°. These angles are special because of the values of their trigonometric functions (cosine, sine, tangent, etc.). The consequences of
this can be seen and understood with the 30 60 90 triangle calculator, but for those who are too lazy to click the link, we will summarize some of them here. Assuming that the shorter side is of length
Right angled triangles and parallelogramsIt might seem at first glance that a right triangle and a parallelogram do not have anything in common. How can a triangle solver help you with understanding a parallelogram? The reality is that any parallelogram can be decomposed into 2 or more right triangles. Let's take an example of the rectangle which is the easiest one to see it. Imagine a rectangle, any rectangle. Now draw trace on one of the diagonals of this rectangle (you can learn more about this in the diagonal of a rectangle calculator). If we separate the rectangle by the diagonal, what will we obtain is two right-angled triangles. Looking at the triangles, there is no need to use the right triangle calculator to see that both are equal, so their areas will be the same. This means that the area of the rectangle is double that of each triangle. If we think about the equations, it makes sense since the area of a rectangle of sides It was a simple example of a rectangle, but the same applies to the area of a square. For other parallelograms, the process becomes a bit more complicated (it might involve up to 4 right triangles of different sizes), but with a bit of skill, you can use the same idea and calculate the area of a parallelogram using right-angled triangles. You can, of course, be even more efficient and just use our calculator. Pythagorean triplets, triangles meet mathsGeometry and polygons, especially triangles, always come together. The properties of some triangles, like right triangles, are usually interesting and shocking, even for non-mathematicians. We will now have a look at an interesting set of numbers very closely related to right-angled triangles that mathematicians love, and maybe you will too. These sets of numbers are called the Pythagorean triplets and are sets of 3 integers (let's call the Aside from the curiosity factor of this relationship, it has some interesting properties that are exploited in cryptography. Given the applications that one might find for such sets of numbers, mathematicians have explored even beyond, using 4, 5... and more sets of numbers that satisfy a similar relation in which the sum of the squares of all the numbers except for one, give the square of the number that's left. Also very connected to these Pythagorean triplets is the infamous Fermat's last theorem in which the almost legendary cryptic mathematician Pierre Fermat stated that there could not be a set of three integer numbers that would satisfy the relation:
Shadows and right triangles (radius of the Earth)We have talked a lot about triangles, in particular, right triangles and their applications in maths and geometry. What we haven't talked about yet is the usefulness of right triangles for calculating things in real life. It might seem like the applications outside of geometry are limited, but let's have a look at shadows. Yes, shadows. The dark shade projected by an object when it is illuminated. If you were to look at the shape made by the shadow, the object, and the ground, you would notice that it, in fact, a right-angled triangle! At least it is when the object is perfectly vertical, and the ground is horizontal. Most of the time this is the case or at least close enough. This means that we can use the right triangle calculator to find different pieces of information about objects under the sun. Let's see how. Imagine that you have a building of which we want to know the height, but you cannot measure it directly because it's too high to drop a measuring tape from the top. What you can do is measure the length of the shadow on the street. Then, with the help of any angle-measuring tool and a piece of paper you can find out the angle between the shadow and the ground. Knowing that the angle between the building and the ground is 90°, you can input these data values into the right triangle side and angle calculator and obtain the value of the height of the building. Using this technique, you can measure the height of many objects as long as you have a bright sunny day or other sources of light to illuminate the object. In fact, this use to be a very common measuring technique in the olden days. Probably the most interesting and mind-blowing use of right triangles is that of Eratosthenes, who managed to use right-angled triangles and shadows to measure the radius of the Earth, and now we are gonna explain how he did it. Eratosthenes noticed that on the summer solstice there was a place on Earth where the wells did not have a shadow at midday, i.e., the sun shone straight down onto them. Noting this, he set up a column of a known height at a known distance from that well and measured the size of the shadow at the same time of the day and the same day of the year in both places. Then using right-angled triangles and trigonometry, he was able to measure the angle between the two cities and also the radius of the Earth, since he knew the distance between the cities. It was quite an astonishing feat, that now you can do much more easily, by just using the Omni calculators that we have created for you. FAQWhich side lengths form a right triangle?Side lengths Do 2, 3, and 4 make a right triangle?We have How do I find the circumcenter of a right angle triangle?For a right-angled triangle, the circumcenter, i.e., the center of the circle circumscribed on the triangle, coincides with the midpoint of the triangle's longest side (its hypotenuse). How do I find the orthocenter of a right angle triangle?The orthocenter of a right-angled triangle, i.e., the point where the triangle's altitudes intersect, coincides with the triangle's vertex of the right angle. Álvaro Díez, Mateusz Mucha and Piotr Małek 30 60 90 triangle45 45 90 triangleArea of a right triangle… 15 more How do you determine if a triangle is a right triangle?If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
Does 9 12 15 form a right triangle?Which set of sides could make a right triangle? Explanation: By virtue of the Pythagorean Theorem, in a right triangle the sum of the squares of the smaller two sides equals the square of the largest side. Only 9, 12, and 15 fit this rule.
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