Right Triangle Calculator (Sides, Angles, Area and Height from Any Two Values)

Of the eight values of a right triangle (three sides, two angles, height to the hypotenuse, area and perimeter), fill in exactly the two you know. Using the Pythagorean theorem and trig ratios, everything else is calculated with steps.

Side a (leg, opposite angle α)

Side b (leg, opposite angle β)

Side c (hypotenuse)

Height h to the hypotenuse

Angle α (degrees, opposite side a)

Angle β (degrees, opposite side b)

Area S

Perimeter L (sum of the three sides)

Enter only two values (leave the other fields blank). Enter angles in degrees (°); the result also shows radians. Angles α and β together are not enough, because they do not fix the size. Use the same unit for all lengths (for example, all in feet).

Result and figure

Enter the two values you know in the fields on the left and press "Calculate". The result will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formulas and figures
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Of the eight values of a right triangle (the three sides \(a\), \(b\), \(c\), the two angles \(\alpha\), \(\beta\), the height to the hypotenuse \(h\), the area \(S\) and the perimeter \(L\)), enter the two you know and get all the rest on the spot
Any pair works, such as "the other sides from the hypotenuse and an angle", "a side from the area and one side" or "the three sides from the perimeter and one more value" (except two angles alone)
Angles are shown in both degrees (°) and radians. Step-by-step work shows which formulas were used
A plain-language explanation of the Pythagorean theorem and the trig ratios (sin, cos, tan), and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page

This page works only for right triangles (triangles with one angle of exactly 90°). Enter angles in degrees (°). It cannot be used for general triangles without a right angle.

What is this calculation used for?

Roof pitch and rafter length (carpentry and construction)

In the US, the steepness of a roof is given as a pitch such as "4/12" (it rises 4 inches for every 12 inches across). This is exactly a right triangle, and the angle is \(\tan^{-1}(4 \div 12) \approx 18.4^\circ\). The rafter is the hypotenuse, so for a horizontal run of 15 feet (a rise of 5 feet), it needs \(\sqrt{15^2 + 5^2} \approx 15.81\) feet, longer than the run.
The carpenter's framing square even has rafter tables printed on it. Building work has always gone hand in hand with right triangle calculations.

Measuring the height of a tree or building from a distance (surveying and forestry)

Something too tall to measure directly can be measured from the angle you look up at it from a distance. If you stand 30 feet from a tree and the angle of elevation is 40°, the part above your eyes is \(30 \times \tan 40^\circ \approx 25.2\) feet. Add your eye height of 5 feet, and the tree is about 30 feet tall.
Surveying with a theodolite (an instrument that measures angles precisely) and measuring standing trees in forestry are this very calculation: finding the other sides from one side and one angle.

Setting a ladder at a safe angle (workplace safety)

A ladder is dangerous if it is too steep or too flat. OSHA's rule for a ladder leaning on a wall is that the base should be out from the wall by one quarter of the ladder's working length (the 4-to-1 rule), which is an angle of about 75°. A 20-foot ladder set at 75° stands \(20 \times \cos 75^\circ \approx 5.2\) feet from the wall and reaches \(20 \times \sin 75^\circ \approx 19.3\) feet up.
"How high will this ladder reach?" and "How far out should the feet go?" are trig ratio calculations used on job sites every day.

Understanding a road sign's "% grade" as an angle (driving and geography)

A "6% grade" on a road sign means the road rises 6 feet for every 100 feet across, which is the tangent value itself. As an angle, it is \(\tan^{-1} 0.06 \approx 3.4^\circ\). That may sound gentle, but it is a steep hill for heavy trucks, and the warning signs are there for them.
Once you understand how percent grade relates to an angle, you can estimate the steepness of a hill from the elevation changes in a map app, or make sense of railroad grades, where even 2% is steep for a train.

Designing a wheelchair ramp (accessibility)

The ADA Standards limit the slope of a wheelchair ramp to 1:12 (1 inch of rise for every 12 inches across). To climb a 30-inch rise at 1:12, the ramp needs \(30 \times 12 = 360\) inches (30 feet) of horizontal run. The actual ramp surface is the hypotenuse, \(\sqrt{360^2 + 30^2} \approx 361.2\) inches, and the angle is about 4.8°.
"Is there enough room?" and "Is a ready-made ramp long enough?" can be checked on the spot with a right triangle calculation.

Formulas and figures

Finding the legs with trig ratios (when you know an angle and the hypotenuse)

Figure

Standard notation (the usual math form)

\(a\) \(=\) \(c\) \(\times\) \(\sin\alpha\)

\(b\) \(=\) \(c\) \(\times\) \(\cos\alpha\)

In words (symbols replaced with words)

③ \(a\): side opposite angle \(\alpha\) \(=\) ① \(c\): hypotenuse \(\times\) ② \(\sin\alpha\): sine

⑤ \(b\): side adjacent to angle \(\alpha\) \(=\) \(c\): hypotenuse \(\times\) ④ \(\cos\alpha\): cosine

The formula in words

① Multiply the hypotenuse \(c\)

② by the sine \(\sin\alpha\) (the opposite side as a fraction of the hypotenuse)

③ and you get the side \(a\) opposite angle \(\alpha\)

④ Multiply the same hypotenuse by the cosine \(\cos\alpha\) (the adjacent side as a fraction of the hypotenuse)

⑤ and you get the side \(b\) adjacent to angle \(\alpha\)

Quick example

With a hypotenuse of 10 and angle α = 30°, the other two sides are

side \(a\) \(=\) hypotenuse (10) \(\times\) sin 30° (= 0.5)

side \(b\) \(=\) hypotenuse (10) \(\times\) cos 30° (≈ 0.866)

\(a = 10 \times \sin 30^\circ = 10 \times 0.5 = 5\)

\(b = 10 \times \cos 30^\circ = 10 \times 0.8660\cdots \approx 8.66\)

Key idea

The trig ratios (sin, cos, tan) are ratios between the sides of a right triangle. Choosing the angle \(\alpha\) fixes the shape of the triangle, so \(\sin\alpha =\) (opposite side \(a\)) ÷ (hypotenuse \(c\)) and \(\cos\alpha =\) (adjacent side \(b\)) ÷ (hypotenuse \(c\)) each have one value. It helps to remember the common angles. In the special right triangles, the 30-60-90 triangle has side ratio \(1 : \sqrt{3} : 2\), and the 45-45-90 triangle (the isosceles right triangle) has side ratio \(1 : 1 : \sqrt{2}\).

Finding the angles (when you know two sides)

Figure

Standard notation (the usual math form)

\(\tan\alpha\) \(=\) \(a\) \(\div\) \(b\)

\(\beta\) \(=\) \(90^\circ\) \(-\) \(\alpha\)

In words (symbols replaced with words)

③ \(\tan\alpha\): tangent of angle \(\alpha\) \(=\) ① \(a\): opposite side \(\div\) ② \(b\): adjacent side

⑥ \(\beta\): the other angle \(=\) ④ \(90^\circ\): sum of the two acute angles \(-\) ⑤ \(\alpha\): angle alpha

The formula in words

① Divide the side \(a\) opposite angle \(\alpha\)

② by the adjacent side \(b\)

③ and you get the tangent \(\tan\alpha\) (the \(\tan^{-1}\) key on a calculator turns it back into the angle \(\alpha\))

④ Subtract from the sum of the two acute angles, \(90^\circ\)

⑤ the angle \(\alpha\)

⑥ and you get the other angle \(\beta\)

Quick example

When the two legs are 3 and 4, angles α and β are

tangent of angle \(\alpha\) \(=\) opposite side (3) \(\div\) adjacent side (4)

\(\tan\alpha = 3 \div 4 = 0.75\)

\(\alpha = \tan^{-1} 0.75 \approx 36.87^\circ\)

\(\beta = 90^\circ - 36.87^\circ = 53.13^\circ\)

Key idea

\(\tan^{-1}\) (inverse tangent, or arctangent) works backward from a tangent value to the angle. Scientific calculators, Excel and phone calculator apps (in scientific mode) have it. If you know the hypotenuse, \(\alpha = \sin^{-1}(a \div c)\) gives the same angle. The two acute angles always add up to 90° because the angles of a triangle add up to 180°, and taking away the 90° of the right angle leaves 90° shared between them. So once you know one angle, the other is just a subtraction.

Formula for the area

Figure

Standard notation (the usual math form)

\(S\) \(=\) \(a\) \(\times\) \(b\) \(\div\) \(2\)

In words (symbols replaced with words)

④ \(S\): area \(=\) ① \(a\): side a (base) \(\times\) ② \(b\): side b (height) \(\div\) ③ \(2\) (to take half)

The formula in words

① Multiply the leg \(a\) (the base)

② by the other leg \(b\) (the height)

③ divide by \(2\)

④ and you get the area \(S\)

Quick example

When the two legs are 3 and 4, the area is

area \(S\) \(=\) base (3) \(\times\) height (4) \(\div\) 2

\(S = 3 \times 4 \div 2 = 6\)

Key idea

The area formula for a triangle is base × height ÷ 2. In a right triangle, the two legs \(a\) and \(b\) are the base and the height as they are, so these two sides alone give the area (the ÷ 2 is there because the triangle is a rectangle of \(a\) by \(b\) cut in half along its diagonal). If you know the hypotenuse \(c\) and the height \(h\) to the hypotenuse, \(S = c \times h \div 2\) gives the same area.

Formula for the height to the hypotenuse

Figure

Standard notation (the usual math form)

\(h\) \(=\) \(a\) \(\times\) \(b\) \(\div\) \(c\)

In words (symbols replaced with words)

④ \(h\): height to the hypotenuse \(=\) ① \(a\): side a \(\times\) ② \(b\): side b \(\div\) ③ \(c\): hypotenuse

The formula in words

① Multiply the leg \(a\)

② by the other leg \(b\)

③ divide by the hypotenuse \(c\)

④ and you get the height \(h\) with the hypotenuse as the base

Quick example

In the right triangle with sides 3, 4 and 5, the height with the hypotenuse 5 as the base is

height to the hypotenuse \(h\) \(=\) side a (3) \(\times\) side b (4) \(\div\) hypotenuse (5)

\(h = 3 \times 4 \div 5 = 2.4\)

Key idea

Writing the area of the same triangle in two ways gives \(a \times b \div 2 = c \times h \div 2\). Comparing the two sides gives \(a \times b = c \times h\), that is, \(h = a \times b \div c\). This height (the altitude from the right-angle corner to the hypotenuse) cuts the original triangle into two smaller right triangles that are both similar to the original. It is a favorite topic in Geometry (similar right triangles and the geometric mean).

Formula for the perimeter

Figure

Standard notation (the usual math form)

\(L\) \(=\) \(a\) \(+\) \(b\) \(+\) \(c\)

In words (symbols replaced with words)

④ \(L\): perimeter \(=\) ① \(a\): side a \(+\) ② \(b\): side b \(+\) ③ \(c\): hypotenuse

The formula in words

① Add the leg \(a\)

② the other leg \(b\)

③ and the hypotenuse \(c\)

④ and you get the perimeter \(L\)

Quick example

The perimeter of the right triangle with sides 3, 4 and 5 is

perimeter \(L\) \(=\) side a (3) \(+\) side b (4) \(+\) hypotenuse (5)

\(L = 3 + 4 + 5 = 12\)

Key idea

Just add up all three sides. If you know only two values, first find the missing sides with the Pythagorean theorem or trig ratios, then add. Sets like 3, 4, 5 or 5, 12, 13 or 8, 15, 17, where all three sides of a right triangle are whole numbers, are called Pythagorean triples. They show up in math problems and in the old way of laying out a right angle with a rope marked in 3 : 4 : 5 lengths (the 3-4-5 rule builders still use).

A right triangle rests on two pillars, the Pythagorean theorem \(a^2 + b^2 = c^2\) and the trig ratios (sin, cos, tan). Know any two values, and you can calculate all the rest. The area is \(a \times b \div 2\), the height to the hypotenuse is \(a \times b \div c\), and the perimeter is the sum of the three sides.

Symbols and terms

Symbols

\(a\)	a	One of the two sides that form the right angle (a leg). It is the side across from angle \(\alpha\).
\(b\)	b	The other side that forms the right angle (a leg). It is the side across from angle \(\beta\).
\(c\)	c	The hypotenuse, the longest side of a right triangle, across from the right angle.
\(\alpha\)	alpha	The angle across from side \(a\). It is a Greek letter often used to name angles.
\(\beta\)	beta	The angle across from side \(b\). \(\alpha + \beta = 90^\circ\).
\(h\)	h	The height with the hypotenuse as the base: the length of the perpendicular from the right-angle corner to the hypotenuse. \(h = a \times b \div c\).
\(S\)	S	The area. \(S = a \times b \div 2\). (US textbooks often use \(A\) for area.)
\(L\)	L	The perimeter (the sum of the three sides). \(L = a + b + c\). (US textbooks often use \(P\) for perimeter.)
\(\sin\alpha\)	sine alpha	The opposite side as a fraction of the hypotenuse. \(\sin\alpha = a \div c\). (Example - \(\sin 30^\circ = 0.5\))
\(\cos\alpha\)	cosine alpha	The adjacent side as a fraction of the hypotenuse. \(\cos\alpha = b \div c\). (Example - \(\cos 60^\circ = 0.5\))
\(\tan\alpha\)	tangent alpha	The opposite side as a fraction of the adjacent side. \(\tan\alpha = a \div b\). (Example - \(\tan 45^\circ = 1\))
\(\tan^{-1}\)	inverse tangent (arctangent)	The calculation that works backward from a tangent value to the angle. On a scientific calculator, it is the tan⁻¹ key. It is written with "−1", but it means "the inverse calculation", not the reciprocal.

Terms

right triangle	A triangle with one angle of exactly 90° (a right angle). The other two angles are always acute (less than 90°) and add up to 90°.
hypotenuse	The side across from the right angle. It is always the longest of the three sides of a right triangle.
Pythagorean theorem	The relation \(a^2 + b^2 = c^2\) between the three sides of a right triangle. It is taught in Grade 8. If you know two sides, you can calculate the third.
trig ratio	A ratio between the sides of a right triangle (sine, cosine or tangent). The angle decides the ratio, so from an angle and one side you can calculate the other sides. It is taught in Geometry.
radian	Another unit of angle. The central angle whose arc is as long as the radius is 1 radian, and \(180^\circ = \pi\) radians. To convert from degrees, use degrees × π ÷ 180.
Pythagorean triple	A set of three whole numbers that are the sides of a right triangle. Well-known ones are 3, 4, 5 and 5, 12, 13 and 8, 15, 17.
isosceles right triangle	A right triangle whose two legs are equal. Both acute angles are 45°, and the side ratio is \(1 : 1 : \sqrt{2}\). It is also called the 45-45-90 triangle, one of the special right triangles.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.

Area of a triangle (Grade 6)	Knowing that the area of a triangle is base × height ÷ 2 Knowing that in a right triangle, the two legs are the base and the height as they are
Angle sum of a triangle (Grade 8)	Knowing that the three angles of a triangle add up to 180° Being able to explain why the two angles other than the right angle add up to 90° in a right triangle
The Pythagorean theorem and square roots (Grade 8)	Knowing that the three sides of a right triangle satisfy \(a^2 + b^2 = c^2\) (\(c\) is the hypotenuse) Being able to find a square root, the value that gives the original number when squared, as in \(\sqrt{25} = 5\)
Right triangle trigonometry (Geometry)	Being able to explain that sin, cos and tan are ratios between the sides of a right triangle Knowing that from an angle and one side, trig ratios give the other sides Being able to give the trig values of 30°, 45° and 60°, such as \(\sin 30^\circ = 0.5\)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to find the two legs from an angle and the hypotenuse

Hypotenuse c	10
Angle α (degrees)	30
Side a	=B1*SIN(RADIANS(B2))
Side b	=B1*COS(RADIANS(B2))

Table to find the angles from two sides

Side a	3
Side b	4
Angle α (degrees)	=DEGREES(ATAN(B1/B2))
Angle β (degrees)	=90-B3

Table to find the area

Side a	3
Side b	4
Area S	=B1*B2/2

Table to find the height to the hypotenuse

Side a	3
Side b	4
Hypotenuse c	5
Height h	=B1*B2/B3

Table to find the perimeter

Side a	3
Side b	4
Hypotenuse c	5
Perimeter L	=B1+B2+B3

After pasting, the upper rows are your inputs and the bottom row (or rows) is calculated automatically.
Excel's SIN and COS functions take radians, so an angle entered in degrees is converted with the RADIANS function. The other way around, ATAN returns radians, so the DEGREES function converts it back to degrees.
The first table shows 5 for side a and about 8.66 for side b. The second shows about 36.87 for angle α and about 53.13 for angle β. Just replace the inputs in column B with your own numbers.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to find the two legs from an angle and the hypotenuse

Hypotenuse c	10
Angle α (degrees)	30
Side a	=B1*SIN(RADIANS(B2))
Side b	=B1*COS(RADIANS(B2))

Table to find the angles from two sides

Side a	3
Side b	4
Angle α (degrees)	=DEGREES(ATAN(B1/B2))
Angle β (degrees)	=90-B3

Table to find the area

Side a	3
Side b	4
Area S	=B1*B2/2

Table to find the height to the hypotenuse

Side a	3
Side b	4
Hypotenuse c	5
Height h	=B1*B2/B3

Table to find the perimeter

Side a	3
Side b	4
Hypotenuse c	5
Perimeter L	=B1+B2+B3

The same formulas as in Excel (SIN, COS, ATAN, RADIANS, DEGREES) work as is. Copy the whole table, paste it into cell A1, and replace the inputs in column B with your own numbers.

How to calculate it in Python

import math

side_a = 3.0  # leg a
side_b = 4.0  # leg b

side_c = math.sqrt(side_a ** 2 + side_b ** 2)      # hypotenuse (Pythagorean theorem)
angle_alpha = math.degrees(math.atan(side_a / side_b))  # angle α (degrees)
angle_beta = 90 - angle_alpha                       # angle β (degrees)
height = side_a * side_b / side_c                   # height to the hypotenuse
area = side_a * side_b / 2                          # area
perimeter = side_a + side_b + side_c                # perimeter

print(f"Hypotenuse c: {side_c}")
print(f"Angle α: {angle_alpha} degrees = {math.radians(angle_alpha)} radians")
print(f"Angle β: {angle_beta} degrees = {math.radians(angle_beta)} radians")
print(f"Height h: {height}")
print(f"Area S: {area}")
print(f"Perimeter L: {perimeter}")

Runs with just math from the standard library. Trig functions such as math.sin and math.atan work in radians, so the key is to convert with math.degrees (radians to degrees) and math.radians (degrees to radians). Replace the two sides at the top with your own numbers and run it.

How to write it in LaTeX and other math languages (copy and paste)

Finding the legs with trig ratios (when you know an angle and the hypotenuse)

a = c·sin α,  b = c·cos α

a = c \sin\alpha, \quad b = c \cos\alpha

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>a</mi><mo>=</mo><mi>c</mi><mo>&#x2062;</mo><mi>sin</mi><mo>&#x2061;</mo><mi>&#x3B1;</mi>
    <mo>,</mo>
    <mi>b</mi><mo>=</mo><mi>c</mi><mo>&#x2062;</mo><mi>cos</mi><mo>&#x2061;</mo><mi>&#x3B1;</mi>
  </mrow>
</math>

a = c sin(alpha), b = c cos(alpha)

a == c Sin[α] && b == c Cos[α]

a := c*sin(alpha); b := c*cos(alpha);

a = c*sind(alpha); b = c*cosd(alpha);

a = c sin α,  b = c cos α

Finding the angles (when you know two sides)

α = tan⁻¹(a ÷ b),  β = 90° − α

\alpha = \tan^{-1}\frac{a}{b}, \quad \beta = 90^\circ - \alpha

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>&#x3B1;</mi><mo>=</mo>
    <msup><mi>tan</mi><mrow><mo>&#x2212;</mo><mn>1</mn></mrow></msup>
    <mo>&#x2061;</mo>
    <mfrac><mi>a</mi><mi>b</mi></mfrac>
    <mo>,</mo>
    <mi>&#x3B2;</mi><mo>=</mo><mrow><mn>90</mn><mo>&#xB0;</mo></mrow><mo>&#x2212;</mo><mi>&#x3B1;</mi>
  </mrow>
</math>

alpha = arctan(a/b), beta = 90 - alpha

α == ArcTan[a/b] && β == Pi/2 - α

alpha := arctan(a/b); beta := Pi/2 - alpha;

alpha = atand(a/b); beta = 90 - alpha;

α = tan^(-1)(a/b),  β = 90° - α

Formula for the area

S = a × b ÷ 2

S = \frac{1}{2}ab

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>S</mi><mo>=</mo>
    <mfrac><mrow><mi>a</mi><mo>&#x2062;</mo><mi>b</mi></mrow><mn>2</mn></mfrac>
  </mrow>
</math>

S = (a b)/2

a b / 2

S := a*b/2;

S = a*b/2;

S = ab/2

Formula for the height to the hypotenuse

h = a × b ÷ c

h = \frac{ab}{c}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>h</mi><mo>=</mo>
    <mfrac><mrow><mi>a</mi><mo>&#x2062;</mo><mi>b</mi></mrow><mi>c</mi></mfrac>
  </mrow>
</math>

h = (a b)/c

a b / c

h := a*b/c;

h = a*b/c;

h = ab/c

Formula for the perimeter

L = a + b + c

L = a + b + c

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>L</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi>
  </mrow>
</math>

L = a + b + c

a + b + c

L := a + b + c;

L = a + b + c;

L = a + b + c

How to have ChatGPT do the calculation

You are a math calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

In a right triangle, leg a is 3 and the hypotenuse c is 5.
Find each of the following:
1. The other leg b (Pythagorean theorem)
2. Angles α and β (in both degrees and radians)
3. The height h with the hypotenuse as the base
4. The area S and the perimeter L

Show the formulas you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula