Download your angle as a plain text file. Change any field and press Calculate to refresh it first.

An angle calculator works out the size of an angle in degrees, then converts it to radians, gradians, turns, degrees-minutes-seconds and mils.

For a right triangle, the angle is the arctangent of the opposite side divided by the adjacent side. A triangle with sides 3 and 4 around the right angle gives arctan(3 ÷ 4) = 36.87 degrees, which is 0.6435 radians, 40.97 gradians and 36° 52′ 11.6″. To convert degrees to radians, multiply by pi and divide by 180. To go back, multiply by 180 and divide by pi.

Rule of thumb: an angle under 90 degrees is acute, exactly 90 is a right angle, between 90 and 180 is obtuse, exactly 180 is straight, and over 180 is reflex.

An angle is the amount of turn between two lines that meet at a point. It is measured by how far one arm has swung away from the other, not by how long the arms are. A full turn all the way round is 360 degrees, so a quarter turn, the square corner you see on a page or a wall, is 90 degrees.

Two arms and a shared corner, called the vertex, are all you need. Stretch the arms longer and the angle does not change, which is the single most useful thing to know about angles: size of turn, not size of shape.

3.1 A worked one-liner

A right triangle with a side of 3 across from the angle and a side of 4 next to it has an angle of arctan(3 ÷ 4) = 36.87 degrees.

3.2 What the units actually mean

3.3 Who uses angles and why

• Builders and roofers setting a roof pitch, a ramp gradient or a staircase.
• Carpenters cutting mitres, where two 45 degree cuts make a 90 degree corner.
• Surveyors and navigators working in degrees, minutes and seconds.
• Physiotherapists recording joint range of motion with a goniometer.
• Engineers and programmers working in radians, because calculus and code expect them.
• Students solving triangles with sine, cosine and tangent.

3.4 Angles against the nearest alternatives

The trap here is treating percent grade as degrees. A 100% grade is not a vertical wall, it is 45 degrees, because rise equals run. A vertical wall has no percent grade at all.

Your angle tells you how far one line has turned from another. Under 90 degrees is acute, a sharp corner. Exactly 90 is a right angle, a square corner. Between 90 and 180 is obtuse, an open corner. Exactly 180 is a straight line, and anything over 180 is reflex, the big way round. The unit matters as much as the number, so always say which one you mean.

4.1 What your specific angle means

4.2 Where it sits against the common angles

4.3 The trigonometric ratios for your angle

4.4 Your complement and supplement

4.5 Why rounding an angle matters more than it looks

Rounding 36.87 degrees to 37 is a change of 0.13 degrees, which sounds like nothing. Over a 10 metre run that moves the far end by about 23 millimetres. Angular error grows with distance, which is why surveyors keep seconds and why a small aiming error matters more the further you are from the target.

4.6 What changes your angle

• Which side you call opposite. Swap opposite and adjacent and you get the complement instead, 53.13 degrees rather than 36.87.
• Your reference line. An angle from the horizontal and the same angle from the vertical differ by 90 degrees.
• Direction of turn. Clockwise and anticlockwise give the same size but opposite signs in most software.
• Going past 360. 405 degrees and 45 degrees point the same way. They are coterminal.
• Degrees against radians. A calculator left in the wrong mode gives an answer that looks plausible and is completely wrong.

4.7 What to do next

1. Write the angle down with its unit, never as a bare number.
2. Sketch it and check the sketch matches the number.
3. If you are cutting, measure the angle again from the other side as a check.
4. Decide the precision you need. Whole degrees for a garden fence, seconds for a survey.
5. Keep the full decimal value for any further calculation, and round only at the end.

4.8 Common mistakes this calculator prevents

• Leaving a calculator in radian mode and reading the answer as degrees.
• Writing 36.87 degrees as 36 degrees 87 minutes. It is 36° 52′ 11.6″, because minutes are sixtieths.
• Treating a 100% grade as vertical. It is 45 degrees.
• Swapping opposite and adjacent, which returns the complement rather than the angle.
• Assuming three sides always make a triangle. If two sides do not exceed the third, no triangle exists.
• Adding a reflex and a non-reflex angle without noticing they were measured different ways round.

4.9 A sanity check you can do in your head

Use your angle three ways: to set or cut something physical, to feed a further calculation, and to record what you did. Set a saw or a bevel gauge from the degree value, use the radian value in any code or spreadsheet trigonometry, and record degrees-minutes-seconds when the work has to be repeatable by someone else. Always carry the unit with the number.

Five ready to copy blocks. Each one fills in with your own numbers once you press Calculate.

There is no single angle formula, because it depends on what you can measure. From two sides of a right triangle, the angle is arctan(opposite ÷ adjacent). From three sides of any triangle it is the law of cosines. From two known angles it is 180 minus their sum. Every one of these gives degrees, and every other unit is a conversion from there.

Ten steps, carried through one worked example: a right triangle with a side of 3 opposite the angle and a side of 4 next to it.

1. Choose what you want to work out. The dropdown has ten modes in four groups. Ours is "Right triangle angle from two sides".
2. Read the field labels carefully. Opposite is the side across from the angle you want. Adjacent is the side touching it, not the long sloping one.
3. Type the opposite side. Enter 3. Units do not matter as long as both sides use the same one.
4. Type the adjacent side. Enter 4.
5. Press Calculate Angle. The result panel shows 36.87 degrees and calls it an acute angle straight away.
6. Check the protractor diagram. Section 2 draws your angle. If the drawing does not look like the number, you have mixed up a field.
7. Read the other units. 0.6435 radians, 40.97 gradians, 0.1024 turns, 655.46 mils and 36° 52′ 11.6″.
8. Use the trig ratios if you need another side. sin is 0.6, cos is 0.8 and tan is 0.75 for this angle.
9. Switch modes to cross-check. Put 36.87 into the complement and supplement mode and you get 53.13 and 143.13.
10. Download it. Use Download TXT for the short version, or Download Doc and Download PDF in Section 2 for the full report.

This section shows how to calculate an angle in Excel or Google Sheets, using the same triangle as everywhere else on this page: opposite 3, adjacent 4, giving 36.87 degrees.

8.1 The functions you need

There is no ATAND in Excel. Every inverse trig function returns radians, so DEGREES() around the outside is not optional.

8.2 Step by step

8.3 The awkward cases

• ATAN2 argument order. Excel is ATAN2(x, y). R, Python, JavaScript and almost everything else are atan2(y, x). Swapping them gives you the complement instead of the angle.
• Angles over 90 degrees. ATAN only returns minus 90 to plus 90. Use ATAN2 when the angle can land in any quadrant.
• Rounding errors in ACOS. A ratio that should be exactly 1 can come out as 1.0000000000000002 and throw #NUM!. Wrap it: =ACOS(MIN(1,MAX(-1,ratio))).
• Time formatting hijack. Typing 36:52:12 into a plain cell is read as a time, not an angle. Keep degrees, minutes and seconds in three separate numeric columns.
• Negative DMS. For a negative angle, take the sign off first, split it, then put the minus sign on the degrees only. Never on the minutes and seconds as well.

8.4 Why answers differ across software

• ATAN2 argument order. Excel takes x first; R, Python and JavaScript take y first. This is the single biggest source of disagreement between a spreadsheet and a script.
• Degrees against radians. Excel, R and Python all work in radians internally. This calculator reports degrees by default because that is what people measure in.
• Value of pi. PI(), R's pi and Python's math.pi all carry the same 15 significant digits, so they agree exactly. A page using 3.14 is 0.05% out.
• Where the seconds differ. The Excel sheet above starts from the already-rounded 36.87, which splits to 36° 52′ 12.0″. The R and Python scripts carry the full precision value 36.86990, which splits to 36° 52′ 11.6″. That is a difference in the input, not in the formula. Round once, at the end.

8.5 Charting it in Excel

To draw an angle, use a Scatter with Straight Lines chart. Put the vertex at (0,0), the first arm at (1,0), and the second arm at (cosθ, sinθ) using =COS(RADIANS(D2)) and =SIN(RADIANS(D2)). Set both axes to the same minimum and maximum, otherwise Excel stretches one axis and your 45 degree angle will not look like 45 degrees. For a share-of-a-turn view, a doughnut chart with two slices, the angle and 360 minus the angle, works well.

8.6 Error messages and what they mean

Google Sheets: every formula above works unchanged, including ATAN2 with the same x-then-y order.

One script, complete and runnable from top to bottom. It uses the same 3-4-5 triangle as Section 8, so the angle comes out at 36.87 degrees.

9.1 The complete script

# ============================================================
# Angle Calculator in R
# Stats Unlock - statsunlock.com
# Change only the INPUTS block below, then run the whole file.
# ============================================================

# ---- 1. INPUTS (change these) -------------------------------
opposite <- 3     # side across from the angle
adjacent <- 4     # side next to the angle

# Other things this script also reports:
sides    <- c(a = 5, b = 6, c = 7)   # any triangle, for the law of cosines
rise     <- 6                        # for slope, pitch and grade
run      <- 12
n_sides  <- 8                        # for a regular polygon

# Reading from a file instead:
# dat <- read.csv("angles.csv")
# dat <- dat[complete.cases(dat), ]   # drop rows with missing values

# ---- 2. CORE ANGLE ------------------------------------------
# R works in radians, so convert on the way out.
# NOTE: R is atan2(y, x). Excel is ATAN2(x, y). They are opposite.
angle_rad <- atan2(opposite, adjacent)
angle_deg <- angle_rad * 180 / pi

# ---- 3. EVERY UNIT ------------------------------------------
gradians <- angle_deg * 10 / 9
turns    <- angle_deg / 360
arcmin   <- angle_deg * 60
arcsec   <- angle_deg * 3600
mils     <- angle_deg * 6400 / 360

to_dms <- function(deg) {
  s  <- sign(deg); a <- abs(deg)
  d  <- floor(a)
  mn <- floor((a - d) * 60)
  sc <- (a - d - mn / 60) * 3600
  sprintf("%s%d deg %d min %.1f sec", ifelse(s < 0, "-", ""), d, mn, sc)
}

# ---- 4. THE OTHER MODES -------------------------------------
law_of_cosines <- function(a, b, c) {
  # angle opposite side a
  acos((b^2 + c^2 - a^2) / (2 * b * c)) * 180 / pi
}
A <- law_of_cosines(sides["a"], sides["b"], sides["c"])
B <- law_of_cosines(sides["b"], sides["a"], sides["c"])
C <- 180 - A - B

slope_deg <- atan2(rise, run) * 180 / pi
grade_pct <- rise / run * 100
pitch_12  <- rise / run * 12

interior  <- (n_sides - 2) * 180 / n_sides
exterior  <- 360 / n_sides

# ---- 5. PLAIN ENGLISH VERDICT -------------------------------
band <- as.character(cut(angle_deg,
  breaks = c(-Inf, 0, 90, 180, 360, Inf),
  labels = c("a negative angle", "an acute angle", "an obtuse angle",
             "a reflex angle", "more than a full turn"),
  right  = FALSE))
if (isTRUE(all.equal(angle_deg, 90)))  band <- "a right angle"
if (isTRUE(all.equal(angle_deg, 180))) band <- "a straight angle"

cat("\n=== ANGLE RESULT ===\n")
cat(sprintf("Angle          : %.4f degrees\n", angle_deg))
cat(sprintf("That is %s.\n", band))
cat(sprintf("Deg min sec    : %s\n", to_dms(angle_deg)))
cat(sprintf("Radians        : %.6f\n", angle_rad))
cat(sprintf("Gradians       : %.4f\n", gradians))
cat(sprintf("Turns          : %.6f\n", turns))
cat(sprintf("Arcminutes     : %.2f\n", arcmin))
cat(sprintf("Arcseconds     : %.1f\n", arcsec))
cat(sprintf("Mils (NATO)    : %.2f\n", mils))
cat(sprintf("Complement     : %.4f degrees\n", 90 - angle_deg))
cat(sprintf("Supplement     : %.4f degrees\n", 180 - angle_deg))
cat(sprintf("sin cos tan    : %.4f  %.4f  %.4f\n",
            sin(angle_rad), cos(angle_rad), tan(angle_rad)))
cat(sprintf("\nTriangle %g-%g-%g : A=%.2f B=%.2f C=%.2f (sum %.2f)\n",
            sides["a"], sides["b"], sides["c"], A, B, C, A + B + C))
cat(sprintf("Slope %g in %g   : %.2f degrees, %.2f%% grade, %.2f in 12\n",
            rise, run, slope_deg, grade_pct, pitch_12))
cat(sprintf("Polygon n=%d    : interior %.2f, exterior %.2f degrees\n",
            n_sides, interior, exterior))

# ---- 6. FIGURE AT 300 DPI -----------------------------------
png("angle.png", width = 2000, height = 2000, res = 300)
op <- par(mar = c(2, 2, 3, 2), pty = "s")
plot(NA, xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2), asp = 1,
     axes = FALSE, xlab = "", ylab = "",
     main = sprintf("Angle = %.2f degrees", angle_deg))
th <- seq(0, 2 * pi, length.out = 400)
lines(cos(th), sin(th), col = "grey80")
arc <- seq(0, angle_rad, length.out = 120)
polygon(c(0, 0.42 * cos(arc), 0), c(0, 0.42 * sin(arc), 0),
        col = "#bbf7d0", border = NA)
lines(0.42 * cos(arc), 0.42 * sin(arc), col = "#16a34a", lwd = 3)
arrows(0, 0, 1, 0, col = "#1d4ed8", lwd = 3, length = 0.12)
arrows(0, 0, cos(angle_rad), sin(angle_rad), col = "#7c3aed", lwd = 3, length = 0.12)
text(0.62 * cos(angle_rad / 2), 0.62 * sin(angle_rad / 2),
     sprintf("%.2f deg", angle_deg), col = "#166534", font = 2, cex = 0.9)
points(0, 0, pch = 19, cex = 0.9)
par(op)
dev.off()
cat("\nSaved angle.png at 300 dpi\n")
cat("Generated by STATS UNLOCK - statsunlock.com\n")

There is no install.packages() step. This script runs in any R installation.

9.2 What each part of the script does

9.3 What the figure shows

A unit circle with the angle drawn on it: a blue arrow along the 0 degree line, a purple arrow at your angle, a green filled wedge between them, and the value printed inside the wedge. asp = 1 and pty = "s" together force a square plotting region, which matters more than it sounds: without them R stretches one axis and a 45 degree angle does not look like 45 degrees.

9.4 Problems and fixes

9.5 One-liners

The same 3-4-5 triangle again, so the angle is 36.87 degrees and the full precision DMS is 36° 52′ 11.6″.

10.1 The complete script

# ============================================================
# Angle Calculator in Python
# Stats Unlock - statsunlock.com
# Change only the INPUTS block below, then run the whole file.
# pip install matplotlib
# ============================================================

import math

import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt

# ---- 1. INPUTS (change these) -------------------------------
opposite = 3      # side across from the angle
adjacent = 4      # side next to the angle

sides   = (5, 6, 7)   # any triangle, for the law of cosines
rise    = 6           # for slope, pitch and grade
run     = 12
n_sides = 8           # for a regular polygon

# Reading from a file instead:
# import pandas as pd
# dat = pd.read_csv("angles.csv").dropna()

# ---- 2. CORE ANGLE ------------------------------------------
# Python works in radians. NOTE: math.atan2(y, x) takes y FIRST,
# which is the opposite of Excel's ATAN2(x, y).
angle_rad = math.atan2(opposite, adjacent)
angle_deg = math.degrees(angle_rad)

# ---- 3. EVERY UNIT ------------------------------------------
gradians = angle_deg * 10 / 9
turns    = angle_deg / 360
arcmin   = angle_deg * 60
arcsec   = angle_deg * 3600
mils     = angle_deg * 6400 / 360


def to_dms(deg):
    sign = "-" if deg < 0 else ""
    a = abs(deg)
    d = int(a)
    mn = int((a - d) * 60)
    sc = (a - d - mn / 60) * 3600
    return f"{sign}{d} deg {mn} min {sc:.1f} sec"


# ---- 4. THE OTHER MODES -------------------------------------
def law_of_cosines(a, b, c):
    """Angle opposite side a, in degrees. Clamped so rounding cannot break acos."""
    ratio = (b * b + c * c - a * a) / (2 * b * c)
    return math.degrees(math.acos(max(-1.0, min(1.0, ratio))))


a, b, c = sides
A = law_of_cosines(a, b, c)
B = law_of_cosines(b, a, c)
C = 180 - A - B

slope_deg = math.degrees(math.atan2(rise, run))
grade_pct = rise / run * 100
pitch_12  = rise / run * 12

interior = (n_sides - 2) * 180 / n_sides
exterior = 360 / n_sides

# ---- 5. PLAIN ENGLISH VERDICT -------------------------------
if math.isclose(angle_deg, 90):
    band = "a right angle"
elif math.isclose(angle_deg, 180):
    band = "a straight angle"
elif angle_deg < 0:
    band = "a negative angle"
elif angle_deg < 90:
    band = "an acute angle"
elif angle_deg < 180:
    band = "an obtuse angle"
elif angle_deg < 360:
    band = "a reflex angle"
else:
    band = "more than a full turn"

print("\n=== ANGLE RESULT ===")
print(f"Angle          : {angle_deg:.4f} degrees")
print(f"That is {band}.")
print(f"Deg min sec    : {to_dms(angle_deg)}")
print(f"Radians        : {angle_rad:.6f}")
print(f"Gradians       : {gradians:.4f}")
print(f"Turns          : {turns:.6f}")
print(f"Arcminutes     : {arcmin:.2f}")
print(f"Arcseconds     : {arcsec:.1f}")
print(f"Mils (NATO)    : {mils:.2f}")
print(f"Complement     : {90 - angle_deg:.4f} degrees")
print(f"Supplement     : {180 - angle_deg:.4f} degrees")
print(f"sin cos tan    : {math.sin(angle_rad):.4f}  "
      f"{math.cos(angle_rad):.4f}  {math.tan(angle_rad):.4f}")
print(f"\nTriangle {a}-{b}-{c} : A={A:.2f} B={B:.2f} C={C:.2f} (sum {A + B + C:.2f})")
print(f"Slope {rise} in {run}   : {slope_deg:.2f} degrees, "
      f"{grade_pct:.2f}% grade, {pitch_12:.2f} in 12")
print(f"Polygon n={n_sides}    : interior {interior:.2f}, exterior {exterior:.2f} degrees")

# ---- 6. FIGURE AT 300 DPI -----------------------------------
fig, ax = plt.subplots(figsize=(6, 6), dpi=300)
th = [i * 2 * math.pi / 400 for i in range(401)]
ax.plot([math.cos(t) for t in th], [math.sin(t) for t in th], color="0.8", lw=1)

arc = [i * angle_rad / 120 for i in range(121)]
ax.fill([0] + [0.42 * math.cos(t) for t in arc] + [0],
        [0] + [0.42 * math.sin(t) for t in arc] + [0],
        color="#bbf7d0", zorder=1)
ax.plot([0.42 * math.cos(t) for t in arc], [0.42 * math.sin(t) for t in arc],
        color="#16a34a", lw=2.5, zorder=2)

ax.annotate("", xy=(1, 0), xytext=(0, 0),
            arrowprops=dict(color="#1d4ed8", lw=2.5, arrowstyle="-|>"))
ax.annotate("", xy=(math.cos(angle_rad), math.sin(angle_rad)), xytext=(0, 0),
            arrowprops=dict(color="#7c3aed", lw=2.5, arrowstyle="-|>"))
ax.text(0.62 * math.cos(angle_rad / 2), 0.62 * math.sin(angle_rad / 2),
        f"{angle_deg:.2f} deg", color="#166534", fontweight="bold", ha="center")
ax.plot(0, 0, "ko", ms=5)

ax.set_xlim(-1.2, 1.2); ax.set_ylim(-1.2, 1.2)
ax.set_aspect("equal"); ax.axis("off")
ax.set_title(f"Angle = {angle_deg:.2f} degrees")
fig.tight_layout()
fig.savefig("angle.png", dpi=300)
plt.close(fig)

print("\nSaved angle.png at 300 dpi")
print("Generated by STATS UNLOCK - statsunlock.com")

Install the one third-party package with pip install matplotlib.

10.2 What each part of the script does

10.3 What the figure shows

The same protractor drawing as the R script: a unit circle, a blue arrow at 0 degrees, a purple arrow at your angle, a green wedge between them and the value printed inside. set_aspect("equal") is what keeps the circle a circle.

10.4 Problems and fixes

10.5 One-liners

Nine lookup tables. This is the angle chart section: use it to check an answer, convert a unit, or read off a pitch without typing anything.

Table 11.1 Degrees to radians, gradians, turns and mils

What to conclude: every unit is just a different slicing of the same full turn. To go from degrees to radians multiply by 0.0174533; to come back multiply by 57.2958.

Table 11.2 The common angles and where you meet them

What to conclude: tan is undefined at 90° and 270° because the adjacent side is zero and you cannot divide by it. Every calculator, including this one, reports that rather than a number.

Table 11.3 Roof pitch, angle and percent grade

What to conclude: a 12 in 12 roof is 45 degrees and a 100% grade. If you ever see 100% described as vertical, that page is wrong.

Table 11.4 Percent grade to degrees

What to conclude: grade and angle only agree at 0. They diverge fast, and there is no grade at all for a vertical face, because you would be dividing by a run of zero.

Table 11.5 Regular polygon angles

What to conclude: the exterior angles of any polygon always add to exactly 360°, however many sides it has. That is why the exterior column is simply 360 divided by the number of sides.

Table 11.6 Degrees, minutes and seconds

What to conclude: minutes and seconds are sixtieths, exactly like clock time. 36.87° is 36° 52′, never 36° 87′. The last row is the latitude of London, which is how DMS is normally used.

Table 11.7 Which unit each field uses

What to conclude: the unit is part of the answer. An angle written as a bare number is ambiguous, and that ambiguity is the single most common cause of a wrong result.

Table 11.8 What every mode in this calculator needs

What to conclude: pick the mode by what you can actually measure. If you can measure two sides, you never need a protractor.

Table 11.9 Angle measurement accuracy in practice

What to conclude: these ranges come from the measurement literature in Section 22. Note the clinical goniometer row in particular: studies repeatedly find that two clinicians measuring the same joint can differ by around 5 degrees, which is why joint range is tracked by the same person where possible. Report an angle to the precision your instrument can actually support, not to the precision this calculator can print.

Eight worked examples covering the whole range: a textbook triangle, an exact right angle, a slope, a polygon, a reflex angle, a case where the obvious reading is wrong, an invalid input, and one real world puzzle.

Measure an angle by fixing a reference line first, then reading how far the second line has turned from it. Zero the instrument on a surface you know is flat or square, keep it in the same plane as the angle, read it twice from the same side, and write the unit down with the number. Most wrong angles come from a wrong reference line, not from a wrong instrument.

Twelve steps. They take a couple of minutes and they remove almost every source of error.

1. Decide the reference line first. Horizontal, vertical, or one edge of the workpiece. Write it down. An angle without a stated reference is not a measurement.
2. Zero the instrument. Put a digital angle finder on a surface you trust and check it reads 0.00. Most drift, and most have a zero button.
3. Work in the same plane. Tilting the tool out of the plane of the angle always reads low. This is the commonest error with phone apps.
4. Put the vertex where it belongs. On a protractor, the centre mark goes exactly on the corner, not near it.
5. Read the correct scale. A protractor has two scales running opposite ways. Check your reading against whether the angle looks acute or obtuse.
6. Measure from the same side both times. Reading from the other side gives the supplement, which is a plausible looking wrong answer.
7. Prefer sides over a protractor. If you can measure two lengths, arctan of the ratio beats any protractor for accuracy.
8. Use a longer baseline. Measuring a rise over 12 units of run is far more precise than over 1 unit, because your length error is spread over more distance.
9. Write down the unit. Degrees, radians or gradians. Bare numbers cause more trouble here than anywhere else in measurement.
10. Match precision to the instrument. A phone app cannot support two decimal places. Round to what the tool can actually deliver.
11. Cross-check with a known angle. A square corner is 90, a mitre is 45, an equilateral corner is 60. If your reading is near one of those, verify it.
12. Measure again. A second reading takes ten seconds and catches the wrong scale, the wrong side and the drifted zero.

13.1 The inputs, their units and their typical ranges

13.2 Common measuring mistakes

• Leaving a calculator or a script in radian mode and reading the answer as degrees.
• Reading the wrong scale on a protractor, giving the supplement instead of the angle.
• Tilting a phone or angle finder out of the plane of the angle.
• Swapping opposite and adjacent, giving the complement.
• Measuring the rise over a run of 1, so a 1 mm error becomes a large angular error.
• Recording 36.87 degrees as 36 degrees 87 minutes.
• Comparing a joint angle measured by one clinician with one measured by another, when the two can differ by about 5 degrees.

Use an angle calculator whenever you have measurements but not the angle, or the angle but not in the unit you need. It converts between degrees, radians, gradians, DMS and mils, and finds an angle from two sides, three sides, a rise and run, a polygon, two vectors or two clock hands. It is not a substitute for a calibrated instrument on safety critical work.

14.1 Four quick examples

1. Cutting a mitre for an octagonal frame. Polygon mode, 8 sides, gives 135 degrees interior, so each cut is 22.5 degrees. Right tool.
2. Converting a slope for a drainage spec. Slope mode turns a 1 in 80 fall into 0.72 degrees. Right tool.
3. Recording a patient's knee range of motion. Wrong tool for the measurement. Use a goniometer, and note that two clinicians can differ by about 5 degrees.
4. Setting out a property boundary. Wrong tool. That needs a total station and a licensed surveyor, working in arcseconds.

14.2 Which tool you actually need

16.1 Assumptions

1. The geometry is flat. On a sphere a triangle's angles add to more than 180 degrees, so every triangle mode here would read low over long distances.
2. Your reference line is what you think it is. If the baseline is off by 2 degrees, every angle from it is off by 2 degrees, in the same direction.
3. Both lengths are in the same unit. Mixing inches and centimetres in the two sides gives an angle that looks reasonable and is wrong.
4. The right angle really is 90 degrees. The right triangle mode assumes it. If the corner is out, the answer is out.
5. Your measurements are accurate to about the instrument's rating. A phone app rated at 1 degree cannot support a two decimal answer.
6. The polygon is regular. The interior angle formula only holds if every side and every corner is identical.
7. The angle is measured in one plane. A tool tilted out of plane always reads low.
8. You know which direction you turned. The size is the same either way; the sign and the reflex reading are not.

16.2 Limitations, and what to use instead

If you used this angle calculator in a report, an assignment, a drawing set or a dissertation, cite it like this.

19.1 APA 7th edition

19.2 BibTeX

19.3 A plain reference for a document

19.4 How this calculator was built and checked

Trust in a calculator comes from being able to see how it works, so here is the method in full.

• Formula sources. Every formula here is standard plane trigonometry and geometry, not a proprietary variant. Right triangles use arctangent, arcsine and arccosine. Any triangle uses the law of cosines. Regular polygons use (n minus 2) times 180 divided by n. Vectors use the normalised dot product. None of it is disputed.
• Conversion factors. Exact values only, and all of them defined rather than measured: 360 degrees, 2π radians, 400 gradians and 6,400 NATO mils in a full turn; 60 arcminutes in a degree; 3,600 arcseconds in a degree. Pi comes from the language's own constant, carrying the full 15 significant digits, not a rounded 3.14.
• How the arithmetic is handled. Everything is computed in radians at full double precision and rounded only for display. The arccosine input in the law of cosines is clamped to the range minus 1 to plus 1, because floating point can push a value that should be exactly 1 just past it and break the function.
• Honest undefined values. Tangent is reported as undefined at 90 and 270 degrees rather than as a very large number. That is the mathematically correct answer, and a calculator that prints 16331239353195370 instead is misleading you.
• Verification. All ten modes and all seven units were checked against independent hand calculations before release. The worked example of a 3-4-5 triangle resolves to 36.87 degrees in this tool, in the Excel formulas in Section 8, in the R script in Section 9 and in the Python script in Section 10. Where the three genuinely disagree, such as Excel's ATAN2 taking x before y, or seconds differing because one started from a rounded input, the disagreement is named in Sections 8.3, 8.4 and 10.4 rather than hidden.
• What it will not do. This tool reports an angle from the numbers you type. It is not a calibrated instrument and carries no legal standing for a boundary survey, a structural certification or a clinical record. Section 16 lists every assumption and the direction of the error when each one fails.
• Evidence base. The accuracy figures in Table 11.9 are drawn from the twenty peer-reviewed studies in Section 22, covering clinical goniometry, digital inclinometers, smartphone angle apps and geotechnical tilt sensors.

Maintained by Stats Unlock. Last reviewed 26 August 2026. Found an error? Every formula and every worked number is on this page, so any result can be checked by hand.

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Twenty peer-reviewed studies on angle measurement accuracy, in APA 7th style. They cover clinical goniometry, digital inclinometers, smartphone angle apps and geotechnical tilt sensors, and they are the evidence behind the accuracy figures in Table 11.9.

1. Hancock, G. E., Hepworth, T., & Wembridge, K. (2018). Accuracy and reliability of knee goniometry methods. Journal of Experimental Orthopaedics, 5(1). View paper
2. Kiatkulanusorn, S., et al. (2023). Analysis of the concurrent validity and reliability of five common clinical goniometric devices. Scientific Reports, 13. View paper
3. Shamsi, M., et al. (2019). Universal goniometer and electro-goniometer intra-examiner reliability in measuring the knee range of motion. BMC Sports Science, Medicine and Rehabilitation, 11. View paper
4. Lind, V., et al. (2021). Reliability and validity of a digital goniometer for measuring knee joint range of motion. Measurement in Physical Education and Exercise Science, 25(4). View paper
5. Ishida, T., et al. (2023). Validity and reliability of a wearable goniometer sensor for measuring knee flexion and extension angle during the gait cycle. Sensors, 23. View paper
6. Ishii, K., et al. (2021). Accuracy and reliability of a smartphone application for measuring the knee joint angle. Journal of Physical Therapy Science, 33(5). View paper
7. Gogia, P. P., Braatz, J. H., Rose, S. J., & Norton, B. J. (1987). Reliability and validity of goniometric measurements at the knee. Physical Therapy, 67(2). View paper
8. Cunha, A. B., et al. (2020). Assessing the validity and reliability of a new video goniometer app for measuring joint angles in adults and children. Archives of Physical Medicine and Rehabilitation, 101(2). View paper
9. Ferriero, G., et al. (2013). Reliability of a smartphone-based goniometer for knee joint goniometry. International Journal of Rehabilitation Research, 36(2). View paper
10. Milanese, S., et al. (2014). Reliability and concurrent validity of knee angle measurement: Smart phone app versus universal goniometer. Manual Therapy, 19(6). View paper
11. Pongkunakorn, A., et al. (2026). Accuracy of digital inclinometers for measuring knee extension during total knee arthroplasty. Arthroplasty, 8. View paper
12. Pei, H., et al. (2021). Development of a novel Hall element inclinometer for slope displacement monitoring. Measurement, 181. View paper
13. Komarizadehasl, S., et al. (2022). A novel wireless low-cost inclinometer made from combining the measurements of multiple MEMS gyroscopes and accelerometers. Sensors, 22. View paper
14. Zheng, G., et al. (2024). Comprehensive calibration and laboratory validation of a MEMS sensor-based flexible inclinometer. Measurement Science and Technology, 35. View paper
15. Yang, Y., et al. (2021). Research on electronic inclinometer calibration method and uncertainty budget based on the GUM method. Journal of Physics: Conference Series. View paper
16. Yang, W., et al. (2013). A robust inclinometer system with accurate calibration of tilt and azimuth angles. IEEE Sensors Journal, 13(6). View paper
17. Celik, A., et al. (2022). Biomedical wireless inclinometer device design and comparison of its measurements with an image processing method. European Journal of Science and Technology. View paper
18. Yasoveev, V., et al. (2025). On the issue of assessing methodological errors in determining the static characteristics of inclinometric transducers. Electrical and Data Processing Facilities and Systems. View paper
19. Liu, J., et al. (2025). Simultaneous inclination and azimuth sensing based on a multi-core fiber Fabry-Perot interferometer with the Vernier effect. Photonics, 12. View paper
20. Kusuma, H. A., et al. (2025). Measurement of beach slope inclination based on the MPU6050 inertial sensor. IOP Conference Series: Earth and Environmental Science. View paper