What Is an Obtuse-Angled Triangle? | Spot It In Seconds

An obtuse-angled triangle has one angle greater than 90° and two acute angles, with the longest side opposite the wide angle.

An obtuse-angled triangle is one of the first triangle types students meet in geometry, yet it keeps showing up later in harder topics too. You’ll see it in angle classification, side-length comparisons, area work, and proofs. If you can identify it fast, many triangle questions get easier right away.

The core idea is simple: one interior angle is more than 90°. That single fact tells you a lot. It tells you the other two angles must be acute. It also tells you the side across from the obtuse angle is the longest side in the triangle.

This article gives you a clean definition, the properties that matter in classwork, quick ways to test whether a triangle is obtuse, and common mistakes that trip students up. You’ll also get worked checks using both angles and side lengths so you can spot the triangle type with confidence.

What Is an Obtuse-Angled Triangle? In Plain Terms

A triangle is called obtuse-angled when one of its interior angles is greater than 90° but less than 180°. Since every triangle has a total angle sum of 180°, only one angle can be obtuse. If two angles were both greater than 90°, their sum would already pass 180°, which is not possible in a triangle.

That means an obtuse triangle always has this pattern:

• One obtuse angle (> 90°)
• Two acute angles (< 90° each)
• Total of all three angles = 180°

Students sometimes call it an “obtuse triangle,” and that is correct. “Obtuse-angled triangle” is just the fuller name. Both refer to the same shape.

How It Differs From Acute And Right Triangles

Triangles can be grouped by angles in three ways. An acute triangle has three acute angles. A right triangle has one 90° angle. An obtuse triangle has one angle greater than 90°. These groups do not overlap. A triangle can belong to only one of them.

If you’re checking a triangle from a diagram, the fastest path is to inspect the widest angle first. If the angle is square-cornered, it is right. If the widest angle opens wider than a right angle, it is obtuse. If all three look smaller than a right angle, it is acute.

Why Students Mix It Up

The confusion usually starts when a drawing is not to scale. A sketch may look obtuse even when the labeled angles are not. In geometry, labels and measurements beat appearance. If the diagram says 88°, 47°, and 45°, it is acute even if the sketch looks wide.

Another mix-up happens with side names. Some learners think “longest side” means “right triangle hypotenuse.” The hypotenuse exists only in a right triangle. In an obtuse triangle, there is no hypotenuse. You still have a longest side, but it is simply the side opposite the largest angle.

Core Properties That Help You Solve Questions

Once you know a triangle is obtuse, several facts fall into place. These facts help in classification, unknown-angle problems, and side comparison questions.

Only One Angle Can Be Obtuse

This comes straight from the 180° angle sum rule. If one angle is 100°, the remaining two must add to 80°. That makes both of them acute. This rule is a quick error check when your answer gives two wide angles.

The Side Opposite The Obtuse Angle Is The Longest

In any triangle, larger angles sit across from longer sides. So if angle A is the obtuse angle, side BC (the side opposite angle A) is the longest side. This helps when you need to compare side lengths without measuring them.

Khan Academy’s triangle classification material reinforces the angle-based grouping used in school geometry and is a handy classroom-aligned reference for this topic: types of triangles review.

An Obtuse Triangle Can Be Scalene Or Isosceles

An obtuse triangle can have all sides different (scalene), or two equal sides (isosceles). It cannot be equilateral, since an equilateral triangle has three 60° angles, and none of those is obtuse.

Height May Fall Outside The Triangle

This point matters in area problems. In many obtuse triangles, the perpendicular height from a vertex lands on an extension of the base, not on the base segment itself. The area formula still works, but you may need to extend a side to draw the height.

Fast Ways To Tell If A Triangle Is Obtuse

You can classify an obtuse triangle in more than one way. The best method depends on what the question gives you: angle measures, side lengths, or a diagram.

Method 1: Check The Angles

This is the most direct method. If any one interior angle is greater than 90°, the triangle is obtuse. Then verify that all three angles add to 180°.

Quick checks:

• 95°, 35°, 50° → obtuse (one angle is greater than 90° and total is 180°)
• 90°, 45°, 45° → right (not obtuse)
• 89°, 46°, 45° → acute (largest angle is still below 90°)

Method 2: Use Side Lengths

If you only have side lengths, compare the square of the longest side to the sum of the squares of the other two sides. This is a converse-style test tied to the Pythagorean relationship.

Let the longest side be c. Then:

• If c² > a² + b², the triangle is obtuse
• If c² = a² + b², the triangle is right
• If c² < a² + b², the triangle is acute

This method is great for exam questions because it avoids measuring a sketch. It uses only arithmetic.

Method 3: Use The Widest Angle In A Diagram

If the task is visual and no numbers are shown, look for the widest interior angle. This is a rough identification method, not a proof. Use it for sorting shapes quickly, then switch to measurements if the question asks for certainty.

Ways To Identify An Obtuse-Angled Triangle

What You Are Given	What To Check	Outcome For Obtuse Triangle
Three angle measures	Any angle > 90° and total = 180°	Yes, if one angle is obtuse
Two angles and one missing angle	Find missing angle using 180° total	Yes, if the missing angle or a given angle > 90°
Three side lengths	Let longest side be c, compare c² with a² + b²	Obtuse when c² > a² + b²
Labeled diagram with angle marks	Read labels, not sketch shape	Obtuse if one labeled angle > 90°
Unlabeled sketch only	Find widest interior angle visually	Tentative obtuse call if angle looks wider than 90°
Side-length comparison question	Locate side opposite the largest angle	That opposite side is longest in an obtuse triangle too
Area problem with a “height” line	Check whether altitude lands on base extension	Common in obtuse triangles; still valid for area
Triangle type sorting worksheet	Separate angle type from side type	Can be obtuse-scalene or obtuse-isosceles

Examples That Make The Idea Stick

Let’s run through the kinds of questions students get most often. The aim is to build a quick habit: spot the largest angle or longest side, then test from there.

Example 1: Classify By Angles

Triangle A has angles 112°, 38°, and 30°. One angle is above 90°, and the total is 180°. So Triangle A is an obtuse-angled triangle.

Notice the two smaller angles add to 68°, which is less than 90°. That pattern shows up every time in an obtuse triangle.

Example 2: Find A Missing Angle

Triangle B has angles 27° and 41°. What is the third angle, and is the triangle obtuse?

Third angle = 180° − (27° + 41°) = 112°. Since 112° is greater than 90°, Triangle B is obtuse.

Britannica’s geometry entry gives a reliable general definition of triangles and their geometric terms, which fits the standard school treatment used in this article: triangle definition and facts.

Example 3: Classify By Side Lengths

Triangle C has side lengths 5, 7, and 9. The longest side is 9.

Now compare squares:

• 9² = 81
• 5² + 7² = 25 + 49 = 74

Since 81 > 74, the triangle is obtuse.

Example 4: A Non-Obtuse Case That Looks Wide

Triangle D is sketched with a wide-looking top angle, but the labels say 88°, 52°, and 40°. The largest angle is 88°, so the triangle is not obtuse. It is acute. This is why labels beat appearance.

Area, Perimeter, And Height In Obtuse Triangles

Many students can identify an obtuse triangle, then get stuck when the problem shifts to area. The formulas are the same triangle formulas you already know. The part that changes is where the height is drawn.

Perimeter

Perimeter is the sum of all three sides.

Perimeter = a + b + c

The angle type does not change this formula. An obtuse triangle and an acute triangle use the same perimeter rule.

Area With Base And Height

Area still follows:

Area = 1/2 × base × height

In an obtuse triangle, the perpendicular height from a chosen vertex may fall outside the triangle on an extended base line. That is normal. The height is still the perpendicular distance to the base line, even if the foot of the perpendicular lies outside the segment.

Area With Side Lengths Only

If the question gives only side lengths, you can use Heron’s formula. That works for any triangle type, including obtuse ones. Many school tasks stay with base-height, yet Heron’s formula is useful when no height is shown.

Obtuse Triangle Facts Students Use Most

Property Or Formula	What It Means	Classroom Use
One angle > 90°	Defines an obtuse-angled triangle	Basic classification
Only one obtuse angle	Angle sum in a triangle is 180°	Error check in angle problems
Other two angles are acute	They add to less than 90°	Quick verification
Longest side opposite largest angle	Angle-side relationship in any triangle	Side comparison questions
c² > a² + b² (c longest)	Side-length test for obtuse type	No-angle classification
Area = 1/2 × b × h	Same area rule as any triangle	Area with drawn height
Height may be outside	Use perpendicular to base line extension	Avoid wrong altitude placement

Common Mistakes And How To Avoid Them

Most wrong answers on this topic come from a small set of habits. If you fix these, your accuracy jumps fast.

Mixing Up “Obtuse Angle” And “Obtuse Triangle”

An obtuse angle is a single angle measure greater than 90°. An obtuse triangle is a full triangle that contains one obtuse angle. Students sometimes mark one angle and stop there without checking whether the shape is actually a triangle with valid total angle sum.

Forgetting The 180° Sum Check

A set like 100°, 50°, 40° is valid and obtuse. A set like 100°, 60°, 40° is not a triangle because the total is 200°. Always run the sum check.

Calling The Longest Side A Hypotenuse

The word “hypotenuse” belongs only to right triangles. In an obtuse triangle, say “longest side” or name the side directly.

Using The Wrong Side As c In The Side-Length Test

In the rule c² > a² + b², c must be the longest side. If you pick a shorter side as c, your test can fail even when your arithmetic is correct.

Drawing The Height Inside Every Time

That works for acute triangles but not always for obtuse ones. If your area sketch looks awkward, extend the base line and drop the perpendicular there.

Study Tips For Classwork And Exams

When a triangle question lands on the page, start with classification before doing anything else. That one step often tells you which rule to use next. A short routine helps:

1. Check what data is given: angles, sides, or diagram.
2. Find the largest angle or longest side.
3. Run one test only (angle test or side-square test).
4. Write the triangle type clearly.
5. Then move to area, perimeter, or missing values.

Also, practice with mixed sets. If every practice question is already labeled “obtuse triangle,” your brain gets a free hint. Mixed practice builds the real skill: spotting the type before the worksheet tells you.

Final Takeaway On Obtuse Triangles

An obtuse-angled triangle has one angle greater than 90°, two acute angles, and a longest side opposite the obtuse angle. You can identify it by angle measures or by side lengths using the square comparison test. Once you spot the type, the rest of the problem usually becomes much cleaner.