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Mathematics · Angles and geometry

Interior angles of a triangle

Tiny or huge, pointy or flat: why do the interior angles of every triangle add up to 180°? Find out with a paper triangle experiment, and learn to find a missing angle and spot impossible triangles.

  • 20 min
  • 1 simulation
  • Mini quiz

Updated:

Preview

The question

A carpenter is making triangular wooden supports for a roof. He has measured the angles at two corners of a support: 40∘40^\circ and 75∘75^\circ. He can't find his protractor to measure the third corner. Do you think he can work out the third angle without measuring it? And can a triangle have two angles of 100∘100^\circ and 90∘90^\circ?

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No account needed: change the values and check your answer. Some simulations ask for a prediction before they start; it doesn't matter whether it's right or wrong.

Protractor

The missing angle of a triangle

Turn the arm to set the angle and press Check.

Prediction

Predict first

If two angles of a triangle are 55° and 70°, what kind of angle is the third one?

It doesn’t matter whether you’re right; the simulation opens once you choose.

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Frequently asked

5 questions

Do the angles of a bigger triangle add up to more?

No. Angles measure how much the sides turn relative to each other, not how long the sides are. Enlarging a triangle makes the sides longer but leaves the angles unchanged; a tiny triangle in your notebook and a giant one in the schoolyard both add up to 180°.

Can a triangle have two right angles?

No. Two right angles already make 90° + 90° = 180°, leaving no room for a third angle. For the same reason a triangle can have at most one right angle or one obtuse angle; a triangle with angles of 100° and 90° is impossible too.

How can I see for myself that the total is 180°?

Cut a triangle out of paper, tear off its three corners and place them side by side so that their tips meet at one point. Together the three angles form a straight line, a straight angle of 180°. Whether the triangle is pointy, flat or large, the result does not change.

Why do the two acute angles of a right triangle add up to 90°?

The right angle is 90°, so 180° − 90° = 90° is left for the other two angles to share. That is why, if one acute angle is 28°, the other is 90° − 28° = 62°.

How do I know whether three given angles can form a triangle?

Add the three angles: if the total is not exactly 180°, the triangle cannot be drawn. 50° + 60° + 80° = 190°, so these angles do not make a triangle. When you calculate a missing angle, also check your answer by adding all three angles.