Right triangle calculator

To calculate the properties of a right triangle when given certain information, you can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The theorem can be written as:

c² = a² + b²

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

If you know the length of the hypotenuse and one of the other two sides, you can use the Pythagorean theorem to find the length of the remaining side. For example, if you know the length of the hypotenuse is c and the length of one of the legs is a, you can find the length of the other leg by:

b² = c² - a²

Additionally, you can use the Pythagorean theorem to find the measure of the angles in a right triangle. You can use the inverse trigonometric functions such as arctan, arcsin, arccos to find the angles.

If you know the side lengths, you can use the trigonometric functions to find the angles:

sin α = a/c
cos α = b/c
tan α = a/b

It's important to note that the Pythagorean theorem holds true only for right triangles. If the triangle is not a right triangle, this theorem will not work.The right triangle calculator computes angles, sides (adjacent, opposite, hypotenuse), and area of any right-angled triangle for real-world applications. Two independent properties entirely determine any right-angled triangle. The calculator provides a step-by-step explanation for each calculation.

A right triangle is a type of triangle that has one angle measuring C=90°. In a right triangle, the side c that is opposite the C=90° angle is the longest side of the triangle and is called the hypotenuse. The symbols a and b are the lengths of the shorter sides, also called legs or catheti. Symbols for angles are A (or α alpha) and B (or β beta). Symbol h refers to the altitude (height) of the triangle, which is the length of the perpendicular line segment drawn from the vertex C to the hypotenuse.

Examples for right triangle calculations:

• two catheti a and b
• cathetus a and hypotenuse c
• cathetus a and opposite angle A
• cathetus a and adjacent angle B
• hypotenuse c and angle A
• hypotenuse c and altitude h
• area T and hypotenuse c
• area T and cathetus a
• area T and angle A
• circumradius R and cathetus b
• perimeter p and hypotenuse c
• perimeter p and cathetus a
• inradius r and cathetus a
• inradius r and area T
• Medians m_a and m_b

Right triangles in word problems in mathematics:

• Triangle - angles
  ABC triangle, alpha = 54 degrees 32 minutes, beta = 79 degrees. What are the sizes of the exterior angles?
• Triangle and axes
  Draw any triangle. Make the axis of its two sides. Their intersection is point S. (a) Measure the distance of point S from all three vertices (b) Draw the axis of the third side.
• A triangle 10
  A triangle has vertices at (4, 5), (-3, 2), and (-2, 5). What are the coordinates of the vertices of the image after the translation (x, y) arrow-right (x + 3, y - 5)?
• Triangular flowerbed
  A gardener plants one row of tulips around a triangular bed with sides of 5 m, 6 m, and 10 m. How many tulip bulbs does he need if he wants to plant 8 bulbs on a length of 1 m?
• Triangle - ratio
  Change the triangle in a ratio of 3:4 The length of the sides of a triangle: a = 7 cm b = 6 cm c = 5 cm
• An isosceles 2
  An isosceles triangular frame has a measure of 72 meters on its legs and 18 meters on its base. Find the perimeter of the frame.
• The sides 7
  The sides of the triangle are 5.2, 4.6, and x. If the PERIMETER of the triangle is 11.2 feet, what is the length of the unknown side? (hint: draw a picture)
• Angle
  Draw angle |∠ ABC| = 30° and construct its bisector. What is the angle between the bisector and each arm of the angle?
• N points on the side
  An equilateral triangle A, B, and C on each of its inner sides lies N=13 points. Find the number of all triangles whose vertices lie at given points on different sides.
• Similar triangles
  In the triangle DEF is DE = 21 cm, EF = 14.7 cm, DF = 28 cm. The triangle D'E'F' is similar to the triangle DEF. Calculate the lengths of the sides of the triangle D'E'F' if the similarity coefficient is one-seventh.
• Similarity of triangles
  If triangle ABC ~ to triangle XYZ, AC = 24, AB = 15, BC = 17, and XY = 9, what is the perimeter of triangle XYZ? Round all sides to 1 decimal place.
• Hypotenuse - construct problem
  A line segment AA1 of length 6 cm is given. Construct all triangles ABC for which AA1 is the hypotenuse, side length BC is 5 cm, and angle gamma is 60°.
• Triangle angles
  In a triangle, one internal angle measures 36 degrees. The second angle is twice the third angle. Calculate unknown angles.
• A triangle 3
  A triangle has a base of 5 5/6 feet and a height of 7 2/5 feet. Find the area of the triangle as a mixed number.
• Cone projection
  In axonometry, construct a projection of an oblique circular cone with a base in a plane. The stop triangle gives dimension. We know the center of the base S, the radius of the base ra the top of the cone V, Triangle (6,7,6), S (2,0,4), V (-2,7,6), r = 3

more triangle problems »

Also, take a look at our friend's collection of math problems and questions!

• triangle
• law of sines
• right triangle
• law of cosines
• Heron's formula
• Pythagorean theorem
• triangle inequality
• similarity of triangles

See triangle basics on Wikipedia or more details on solving triangles.