gizmobench

Triangle Angle Sum Calculator

Give any two angles of any triangle and the third is what is left of 180 degrees. Type them in degrees or radians, and the page names the third angle, says whether the triangle is right, acute or obtuse, and draws the shape at those angles. No side length is asked for and none is invented: with angles alone a triangle has a shape but no size, so the figure is drawn in proportion only. Two angles that already add to 180, or past it, are refused with the arithmetic shown rather than answered with a negative angle.

abcABC
Solvedshape only, no sizes given
A
65°
as typed
B
40°
as typed
C
75°
180° less the other two
shape
acute
from the largest angle
Known
A (°)
B (°)
Angles in

Give any two angles of any triangle and the third is what is left of 180 degrees. No side lengths are needed, and none are guessed: the figure shows the shape, not the size.

  • Three sides8, 15, 17
    right angled · 28.0725° and 61.9275° · area 60
  • Leg and anglea = 10, A = 30°
    b = 17.3205, c = 20 · area 86.6025
  • Angle sumone acute angle is 34°
    90° + 34° + 56° = 180°
How the three modes work. A right triangle has five values: the legs a and b, the hypotenuse c, and the acute angles A and B, with C fixed at a right angle. Any two of them fix the triangle, so two boxes are what the tool asks for: sides come from Pythagoras, angles from the inverse sine, cosine and tangent, and the altitude is the perpendicular dropped onto the hypotenuse, a times b over c. Two angles are refused, because they set the shape and not the size. Three sides go to the law of cosines instead, which works on any triangle and says whether it is right, acute or obtuse, with the area from Heron’s formula. Two angles of any triangle give the third from the 180 degree sum. Exact trigonometry to the precision of the numbers you enter, with every side and angle shown in degrees or radians. It cannot know your units or your measurement error, and it is geometry only, not a structural calculation.

Common questions

Why do the angles of a triangle add to 180 degrees?
Draw a line through one corner parallel to the opposite side. The two angles it makes with the other two sides are equal to the far corners of the triangle, and the three angles along that straight line make a half turn, which is 180 degrees. It holds for every flat triangle, big or small, right or obtuse, which is why two angles are always enough to find the third: 90 and 45 leave 45, 65 and 40 leave 75, 100 and 40 leave 40.
What is the third angle of a right triangle?
If one angle is 90 degrees, the other two share what is left, so they add to 90 and each is acute. Enter 90 and 34 and the third angle is 56. That is why the right triangle page asks for only one acute angle: the other follows from it. An angle of 90 or more is refused on that page for the same reason, since two right angles would use the whole 180 with nothing left over.
Can I use radians here?
Yes. Switch Angles in to Radians and the sum becomes pi rather than 180: a right angle of 1.5708 and an eighth turn of 0.785398 leave 0.785398. Anything already typed is converted as you switch, so the triangle on the stage stays the same triangle. The label on each box says which unit it is reading, and the third angle comes back in the same unit.
Why can I not enter two angles that add to 180 or more?
Because there would be nothing left for the third corner. Two angles adding to exactly 180 would close the shape along a straight line, and anything more would need a negative angle, which no triangle has. The tool says which two numbers you gave and what they add to, so you can see which one to reduce. This is plane geometry: on a sphere the angles of a triangle add to more than 180, and this page does not do spherical trigonometry.
Do I need the side lengths?
Not for the third angle. The angle sum needs nothing but the other two angles, so the sizes never enter it. That also means this page cannot give you an area or a perimeter: the figure it draws is the correct shape at an arbitrary size, with the sides in proportion to the sines of the angles across from them. If you have the three sides, the side-side-side page gives the angles, the area and the classification instead, and if you have a right angle and two values the main page solves the whole triangle.

Exact trigonometry to the precision of the numbers you enter, with every side and angle shown in degrees or radians. It cannot know your units or your measurement error, and it is geometry only, not a structural calculation.