Obtuse Triangle: Definition, Examples And How To Spot One
The short version
- An obtuse triangle has one angle bigger than 90° and smaller than 180°. The other two angles are always acute.
- A triangle can only ever have one obtuse angle, because the three angles add to exactly 180°.
- If you only know the side lengths, square them. When the longest side squared is bigger than the other two squares added together, the triangle is obtuse.
- Area is still ½ × base × height, but the height of an obtuse triangle often falls outside the triangle.
- An obtuse triangle can be scalene or isosceles. It can never be equilateral or right.
An obtuse triangle is a triangle with one obtuse angle, meaning one angle that measures more than 90° but less than 180°. A triangle with angles of 120°, 35° and 25° is obtuse. Since all three angles must add to 180°, the two remaining angles together make up less than 90°, so both of them are acute.
Below: how to tell an obtuse triangle apart from acute and right triangles, how to check using side lengths alone, how to find missing angles and area, the mistakes kids make most, and practice problems with answers.
What makes a triangle obtuse
Triangles get sorted by their biggest angle. That one angle decides the name.
| Type | Biggest angle | Example angles |
|---|---|---|
| Acute triangle | Less than 90° | 70°, 60°, 50° |
| Right triangle | Exactly 90° | 90°, 55°, 35° |
| Obtuse triangle | More than 90° | 130°, 30°, 20° |
A quick picture test: if one corner looks wider than the corner of a sheet of paper, it’s obtuse. If it’s hard to tell by eye, measure it with a protractor or use the angle sum.
Why a triangle can’t have two obtuse angles
Say a triangle had two angles of 91°. That’s already 182°, which is more than the 180° every triangle adds up to, and there’d be nothing left for the third angle. So one obtuse angle is the limit. The same logic rules out a right angle and an obtuse angle together: 90° plus anything over 90° blows past 180°.
What types of obtuse triangles are there?
Triangles also get sorted by sides, and obtuse triangles come in two of those kinds.
- Obtuse scalene: no sides equal, no angles equal. Example: 110°, 45°, 25°.
- Obtuse isosceles: two equal sides and two equal angles. The obtuse angle sits between the equal sides. Example: 140°, 20°, 20°.
An equilateral triangle can’t be obtuse, since all its angles are 60°.
How to find a missing angle in an obtuse triangle
Subtract the two angles you know from 180°.
Worked example 1
A triangle has angles of 115° and 38°. Find the third angle.
115 + 38 = 153
180 – 153 = 27
The third angle is 27°.
Worked example 2: obtuse isosceles
An isosceles triangle has a 130° angle at the top. Find the two base angles.
The 130° angle has to be the odd one out, since two 130° angles would add to 260°.
180 – 130 = 50
50 ÷ 2 = 25
Each base angle is 25°.
This connects to supplementary angles. The obtuse angle and its outside (exterior) angle at the same corner add to 180°. So a 130° inside angle has a 50° exterior angle, which equals the sum of the other two inside angles (25° + 25°). If that idea is new, our post on supplementary angles walks through it.
How to tell if a triangle is obtuse from its sides
You don’t always get angles. With three side lengths, use this test. Call the longest side c and the other two a and b.
| If | The triangle is |
|---|---|
| c² > a² + b² | Obtuse |
| c² = a² + b² | Right |
| c² < a² + b² | Acute |
The middle row is the Pythagorean theorem. The obtuse angle is always across from the longest side.
Worked example 3
Sides of 4, 5 and 7. Is it obtuse?
Longest side: 7, and 7² = 49
Other two: 4² + 5² = 16 + 25 = 41
49 > 41, so the triangle is obtuse.
Worked example 4
Sides of 6, 7 and 8.
8² = 64
6² + 7² = 36 + 49 = 85
64 < 85, so it’s acute. It looks a bit lopsided but every angle is under 90°.
One check first: the sides have to make a triangle at all. The two shorter sides must add to more than the longest. Sides 2, 3 and 6 fail, since 2 + 3 = 5 is less than 6.
Area of an obtuse triangle
The formula doesn’t change: Area = ½ × base × height. What changes is where the height goes.
The height is the perpendicular distance from the base to the opposite corner. In an obtuse triangle, if you pick one of the short sides as the base, that perpendicular line lands outside the triangle. You extend the base with a dotted line and drop the height down to it.
Worked example 5
An obtuse triangle has a base of 8 cm. The height, measured to the extended base, is 5 cm.
Area = ½ × 8 × 5 = 20 cm²
Kids often grab the slanted side length instead of the height. The slanted side is always longer than the height, so that mistake gives too big an answer.
Obtuse triangles in real life
Low-pitched roofs seen from the end often form obtuse isosceles triangles, since the peak is wide and flat. Coat hangers usually have a wide top angle. Anywhere a triangle looks squat and stretched, check the widest corner.
Common mistakes with obtuse triangles
- Calling a right triangle obtuse. 90° isn’t more than 90°. It needs to be past 90°.
- Naming by the smallest angle. A triangle with 100°, 50° and 30° has two acute angles, but it’s obtuse. Only the biggest angle counts.
- Using the wrong side as c. In the side test, c must be the longest side. Plug a short side in as c and you’ll call every triangle acute.
- Forgetting to square. Comparing 7 with 4 + 5 = 9 tests whether a triangle exists. It doesn’t tell you the angle type.
- Measuring the height along a side. The height has to be perpendicular to the base, even if it’s outside the triangle.
- Protractor misreads. Reading the wrong scale turns 120° into 60°. If the angle is clearly wide, the answer should be over 90.
Practice problems
- A triangle has angles of 100°, 45° and 35°. Is it acute, right or obtuse?
- Two angles of a triangle are 120° and 25°. Find the third.
- Can a triangle have angles of 95° and 90°? Explain.
- An isosceles triangle has a 110° angle. Find the other two angles.
- Sides of 3, 4 and 6. Acute, right or obtuse?
- Sides of 7, 24 and 25. Acute, right or obtuse?
- An obtuse triangle has a base of 10 in and a height of 6 in. Find its area.
- Sides of 8, 9 and 12. Acute, right or obtuse?
Answers
- Obtuse. 100° is more than 90°, and 100 + 45 + 35 = 180.
- 35°. 180 – 120 – 25 = 35.
- No. 95 + 90 = 185, which is already more than 180.
- 35° and 35°. 180 – 110 = 70, and 70 ÷ 2 = 35.
- Obtuse. 6² = 36, and 3² + 4² = 9 + 16 = 25. 36 > 25.
- Right. 25² = 625, and 7² + 24² = 49 + 576 = 625.
- 30 in². ½ × 10 × 6 = 30.
- Acute, just barely. 12² = 144, and 8² + 9² = 64 + 81 = 145. 144 < 145.
If your child is stuck on triangles
Classifying triangles by angle is a 4th and 5th grade skill, and using the 180° rule shows up by middle school. When a kid can’t sort triangles, the usual gap is angle sense itself: what 90° looks like, how to read a protractor, and why angles on a straight line add to 180°. If the side-length test is the sticking point, it’s squaring and comparing numbers.
Those are quick fixes once someone spots them. Our guide to math skills by grade level shows where angle and shape skills land in grades 3 to 5. And our parent’s guide to math assessment explains how to find which earlier skill is actually missing.
Frequently asked questions
How many obtuse angles can a triangle have?
One. Two obtuse angles would add to more than 180°, and a triangle’s three angles add to exactly 180°.
Can an obtuse triangle be a right triangle?
No. A right angle plus an obtuse angle is already more than 180°. A triangle is one or the other.
Can an obtuse triangle be isosceles?
Yes. The obtuse angle sits between the two equal sides, and the two base angles are equal and acute. For example: 120°, 30°, 30°.
Can an obtuse triangle be equilateral?
No. Every angle in an equilateral triangle is 60°.
What do you call triangles with obtuse angles?
Obtuse triangles, sometimes called obtuse-angled triangles. Both names mean a triangle with one angle between 90° and 180°.
Where is the height of an obtuse triangle?
It depends on the base you pick. With the longest side as the base, the height is inside. With either shorter side as the base, the height falls outside, so you extend the base to meet it.
Find the actual gap
Trouble with obtuse triangles usually points at something earlier, like angle measurement or squaring numbers. Math Anon starts with a short adaptive assessment that finds exactly which skills are missing, then gives you a written report so you’re not guessing.
After that, your kid practices about 20 minutes a day on just those skills, and you get weekly progress reports. The assessment is $49 and is credited in full to your first month. Start the assessment or look over pricing.
