What is the ABC Triangle Calculator?

This is a tool for solving right triangles. The three vertices are A, B and C, with the right angle at C. Therefore side c is the hypotenuse while a and b are the legs. Enter any two given conditions sufficient to define the triangle (e.g. two sides, one side and one acute angle or area and one side), and this calculator will solve for the remaining sides, angles, area, perimeter, and radii of the circumscribed and inscribed circles.

How do you find the hypotenuse of a right triangle?

Use the Pythagorean Theorem. The hypotenuse c is equal to the square root of the sum of the squares of both legs, so c = √(a² + b²). If the two legs are 3 and 4, then the result is the square root of 9 + 16 = 25, so c = 5. The hypotenuse is always the longest side and is opposite the right angle.

What is a right triangle?

A right triangle is a triangle with one interior angle (90 degrees). The sides of a right triangle obey the Pythagorean theorem, which states that the square of the longest side is equal to the sum of squares of the other two sides. If the lengths of the sides are 3, 4 and 5, then the square of 5 is 25, and since 9 + 16 also equals 25, a triangle with side lengths of 3-4-5 is a right triangle. These three numbers form a Pythagorean triple.

Can you make a right triangle with 4-5-6?

No. A triangle with side lengths of 4, 5, and 6 does not satisfy the Pythagorean theorem. The square of the longest side, 6, is 36, but the sum of the squares of the other two sides is 16 + 25 = 41. Since 41 is not equal to 36, this combination of side lengths cannot form a right triangle.

What is the relationship between the two acute angles of a right triangle?

The two angles are complementary to each other and their sum is 90 degrees. The sum of the interior angles in any triangle is 180 degrees, so if one angle is already 90 degrees then the sum of the other two must be 90 degrees. So knowing one acute angle allows you to find the other acute angle by subtracting it from 90.

Can a right triangle only be calculated with two acute angles?

No. Two angles define the shape of a triangle but not its size. There are infinitely many right triangles with the same angles. You need to know at least one side length. If you enter a side and an angle, then the calculator will determine all other values based on the similarity ratio.

How do you find the area of a right triangle?

Since the two legs meet at a right angle, either one can be used as the base or the height. The area is half the product of those two sides, so Area = 1/2*a*b. If the area and one of the legs are known, then you can find the other leg by dividing twice the area by the length of the known leg. The other leg is equal to two times the area divided by the length of the known leg.

Are 1-2-3 Pythagorean numbers?

No. The numbers 1, 2 and 3 do not satisfy the Pythagorean theorem. The sum of the squares of the two smaller numbers is 1 + 4 = 5, which is unequal to the square of the largest number, 9. In fact, 1, 2 and 3 cannot form a triangle since 1 + 2 is not greater than 3.

ABC Triangle Calculator

Enter what you know

A right triangle is fixed by any two independent measurements as long as one is a length - two sides, one side and one acute angle, or the area with one side. Fill in what you know and leave the rest blank; the blanks fill themselves in.

Side a (leg)

Side b (leg)

Side c (hypotenuse)

Angle alpha (at A)

Angle beta (at B)

Area

Also worked out

Enter any two measurements - at least one of them a length - and the whole right triangle is solved here. For example two legs, one leg and an acute angle, the hypotenuse and an angle, or the area with one side.

Please input values into the fields

Steps and charts

Show the step-by-step working

Walk through how the Pythagorean theorem and SOH-CAH-TOA produce the answers above.

Show the charts

Compare the three sides and the three angles of your right triangle at a glance.

The ABC triangle calculator can solve any right triangle, which is one with one angle of 90 degrees. The three corners are called A, B and C, for the angles, with the right angle being at C. The sides opposite each corner are named using the corresponding lowercase letter, so a is the side opposite the angle A, b is the side opposite the angle B and c is the side opposite the angle C.

Since the right angle is at C, c is the hypotenuse, the side opposite the right angle, and a and b are the legs, the other two sides of the triangle. The other two angles will be denoted by α for A and β for B, both acute angles that sum to 90 degrees.

If you input any two known values such as two sides, one side and an acute angle or the area and a side this calculator will calculate all remaining values including unknown sides, angles, area, perimeter, hypotenuse height and radii of circumscribed and inscribed circles.

Pythagorean theorem

The Pythagorean theorem states the fundamental relationship between the sides of a right triangle. The square of the hypotenuse is equal to the sum of the squares of the other two sides.

a^2 + b^2 = c^2

By rearranging the equation you can solve for any unknown side. To find the hypotenuse add the squares of both legs and then take the square root. To find either leg subtract the square of the known leg from the square of the hypotenuse and then take the square root.

c = \sqrt{a^2 + b^2} \qquad a = \sqrt{c^2 - b^2} \qquad b = \sqrt{c^2 - a^2}

A set of three positive integers that satisfies the Pythagorean theorem is called a Pythagorean triple. The most common one is 3, 4 and 5. The square of 3 plus the square of 4, which are 9 and 16 respectively, equals 25, which is the square of 5. This does not hold true for every combination of numbers. If the sides were 4, 5 and 6, then the sum of 16 and 25 would be 41, which is not equal to 36, so no right triangle could be formed.

Using Soh-Cah-Toa to find angles.

If two sides are known, the acute angles can be calculated using the ratios of the three basic trigonometric functions. For a given angle, the opposite side is called the opposite leg, the other leg is the adjacent leg and the longest side is the hypotenuse.

\sin\alpha = \frac{a}{c} \qquad \cos\alpha = \frac{b}{c} \qquad \tan\alpha = \frac{a}{b}

The term SOH-CAH-TOA summarizes the three relationships: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse and Tangent is Opposite over Adjacent. To derive an angle from a ratio, inverse trigonometric functions are used such as α = arctan(a / b).

Normally you do not have to calculate the two acute angles separately. This is because they are complementary angles; if you know one of them then you can find the other by subtracting it from 90 degrees.

\alpha + \beta = 90^\circ

Instructions for using the triangle calculator tool.

You must enter two arbitrary, independent measurements, at least one of which must be a length. Two angles alone will determine the shape but not the size. Enter the known values and leave all other fields blank (0).

You enter	The calculator uses	and returns
Two legs a and b	Pythagoras + arctangent	c, alpha, beta, area, and more
A leg and the hypotenuse	Pythagoras + arcsine	the other leg and both angles
A side and one acute angle	SOH-CAH-TOA	the remaining sides and angle
The area and one side	area = one half a b	the other leg, then the rest

If all three sides are given, the calculator will check to see if they actually form a right triangle. This is the same as asking "can 4-5-6 make a right triangle?" Enter the longest side as c.

Example calculation (two known catheti)

Suppose the two legs are a = 3 and b = 4. The hypotenuse is the square root of the sum of squares of 3 and 4, which is the square root of 25, so c = 5. Angle α is arctan(3 / 4) = 36.87 degrees, angle β is the corresponding angle and is 53.13 degrees. The area is half the product of 3 and 4, which is 6. The perimeter is 3 + 4 + 5 = 12.

Example calculation (one side and one angle known)

Suppose the hypotenuse c = 10 and the acute angle α = 30 degrees. Then a = c × sin 30 = 5 and b = c × cos 30 = 8.66. The angle β is 60 degrees, and the area is half of the product of 5 and 8.66, which is 21.65.

Other results that can be obtained with a calculation tool:

If all three sides are known, some common geometric properties can be calculated directly. For right triangles, these calculations are particularly simple.

Quantity	Formula	Note
Area	one half a b	Half the product of the two legs
Perimeter	a + b + c	The distance around
Height to hypotenuse	a b / c	Altitude from the right angle
Circumradius R	c / 2	Exactly half the hypotenuse
Inradius r	(a + b - c) / 2	Largest circle that fits inside

The circumradius is particularly important as a result. The three vertices of a right triangle always lie on a circle whose diameter is the hypotenuse. Thus, the circumradius is half the length of the hypotenuse, and its center is at the midpoint of the hypotenuse. This is Thales' theorem, which also explains why the median to the hypotenuse of a right triangle has exactly half the length of the hypotenuse.

Special right triangles.

There are two particularly well known types of right triangles whose side ratios are worth memorizing. These ratios can be directly derived from the rules mentioned above.

Triangle	Angles	Side ratio a : b : c
Isosceles right	45, 45, 90	1 : 1 : root 2
30-60-90	30, 60, 90	1 : root 3 : 2

In a triangle with sides of 45-45 and 90 the two legs are equal in length and the hypotenuse is root 2 times one of the legs. In a triangle with angles of 30-60 and 90 the opposite side to the 30-degree angle is half the length of the hypotenuse, and the opposite side to the 60-degree angle is root 3 times this shorter side.

Typical Pythagorean numbers

These combinations of whole numbers as side lengths satisfy the Pythagorean theorem exactly, which makes them useful for quick checks and in aligning right angles on construction sites when only a tape measure is available.

a	b	c (hypotenuse)
3	4	5
5	12	13
8	15	17
7	24	25
20	21	29

Any multiple of a Pythagorean triple is also a Pythagorean triple. Thus, 6-8-10 and 9-12-15 are just the results of scaling up the 3-4-5 triangle.

Application examples for right triangles

The trigonometric relationships in a right triangle are important tools that support many practical surveying tasks. Construction workers can use the 3-4-5 rule to square corners of foundations and walls without using an angle measure. Surveyors and navigators can determine heights or distances that cannot be measured directly by measuring an angle and one distance.

The same relationship can be used to calculate the length of a rafter from the height and span of a roof, the height of a tree from its shadow and the angle of elevation of the sun or the required length of diagonal braces for bracing a rectangular frame. As long as one right angle and either a known length or a known angle are present at the same time this calculation tool can be used to make corresponding calculations.

Symbols used:

Symbol	Meaning	Example
a, b	The two legs (opposite A and B)	3, 4
c	The hypotenuse (opposite the right angle at C)	5
alpha, beta	The two acute angles, at A and B	36.87, 53.13 deg
R	Circumradius = c / 2	2.5
r	Inradius = (a + b - c) / 2	1

This calculator is for general education and problem-solving purposes only. It assumes that the desired figure is a right triangle with the right angle at C. Enter the longest side as hypotenuse c.

Frequently asked questions

What is the ABC Triangle Calculator?: This is a tool for solving right triangles. The three vertices are A, B and C, with the right angle at C. Therefore side c is the hypotenuse while a and b are the legs. Enter any two given conditions sufficient to define the triangle (e.g. two sides, one side and one acute angle or area and one side), and this calculator will solve for the remaining sides, angles, area, perimeter, and radii of the circumscribed and inscribed circles.
How do you find the hypotenuse of a right triangle?: Use the Pythagorean Theorem. The hypotenuse c is equal to the square root of the sum of the squares of both legs, so c = √(a² + b²). If the two legs are 3 and 4, then the result is the square root of 9 + 16 = 25, so c = 5. The hypotenuse is always the longest side and is opposite the right angle.
What is a right triangle?: A right triangle is a triangle with one interior angle (90 degrees). The sides of a right triangle obey the Pythagorean theorem, which states that the square of the longest side is equal to the sum of squares of the other two sides. If the lengths of the sides are 3, 4 and 5, then the square of 5 is 25, and since 9 + 16 also equals 25, a triangle with side lengths of 3-4-5 is a right triangle. These three numbers form a Pythagorean triple.
Can you make a right triangle with 4-5-6?: No. A triangle with side lengths of 4, 5, and 6 does not satisfy the Pythagorean theorem. The square of the longest side, 6, is 36, but the sum of the squares of the other two sides is 16 + 25 = 41. Since 41 is not equal to 36, this combination of side lengths cannot form a right triangle.
What is the relationship between the two acute angles of a right triangle?: The two angles are complementary to each other and their sum is 90 degrees. The sum of the interior angles in any triangle is 180 degrees, so if one angle is already 90 degrees then the sum of the other two must be 90 degrees. So knowing one acute angle allows you to find the other acute angle by subtracting it from 90.
Can a right triangle only be calculated with two acute angles?: No. Two angles define the shape of a triangle but not its size. There are infinitely many right triangles with the same angles. You need to know at least one side length. If you enter a side and an angle, then the calculator will determine all other values based on the similarity ratio.
How do you find the area of a right triangle?: Since the two legs meet at a right angle, either one can be used as the base or the height. The area is half the product of those two sides, so Area = 1/2*a*b. If the area and one of the legs are known, then you can find the other leg by dividing twice the area by the length of the known leg. The other leg is equal to two times the area divided by the length of the known leg.
Are 1-2-3 Pythagorean numbers?: No. The numbers 1, 2 and 3 do not satisfy the Pythagorean theorem. The sum of the squares of the two smaller numbers is 1 + 4 = 5, which is unequal to the square of the largest number, 9. In fact, 1, 2 and 3 cannot form a triangle since 1 + 2 is not greater than 3.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Math is Fun: Pythagoras' Theorem
The a^2 + b^2 = c^2 relationship and worked examples.
Khan Academy: Right triangles & trigonometry
SOH-CAH-TOA and solving right triangles from a side and an angle.
Wikipedia: Special right triangle
The 45-45-90 and 30-60-90 side ratios and Pythagorean triples.