# Obtuse Triangle - Definition, Properties, Formulas

An obtuse triangle is a triangle with exactly one angle greater than 90°, and that angle sits opposite the longest side. A triangle is obtuse when the square of its longest side beats the sum of the squares of the other two: c² > a² + b². This article gives the definition, the properties, the area and Heron's formula, the side test, and six worked examples.

## What Is an Obtuse Triangle?

An **obtuse triangle** (also called an obtuse-angled triangle) is a triangle in which one interior angle measures more than 90° but less than 180°. The other two angles are **acute** — each less than 90° — because the three interior angles of any triangle add to 180°. So if one angle is, say, 110°, the remaining two must share only 70° between them.

That single rule explains a lot. A triangle can hold **at most one** obtuse angle. Two angles above 90° would already exceed 180° on their own, which no triangle can do. By the same logic, an obtuse triangle can never also be a right triangle.

Obtuse triangles are one branch of the wider [types of triangle](/content/math/geometry/types-of-triangle/index.html), classified by their angles alongside acute and right triangles.

## How Do You Know If a Triangle Is Obtuse?

There are two ways to tell, depending on what you are given.

**By its angles.** Check the largest angle. If it is more than 90°, the triangle is obtuse. If exactly 90°, it is right; if all three are below 90°, it is acute.

**By its sides.** When you only know the three side lengths, square them. Take the longest side as c and the two shorter sides as a and b. Then compare:

a² + b² > c² ⇒ acute  
a² + b² = c² ⇒ right  
a² + b² < c² ⇒ obtuse

The obtuse case, written the other way round, is c² > a² + b². This is the [Pythagoras theorem](/content/math/geometry/pythagoras-theorem/index.html) used as an inequality — the longest side has "overshot" what a right triangle would allow, so the angle facing it has opened past 90°.

## Properties of an Obtuse Triangle

A short opener before the list: every property below follows from the one obtuse angle and the 180° angle sum.

- **One obtuse angle only.** Exactly one angle exceeds 90°; the other two are acute.
- **Longest side faces the obtuse angle.** The side opposite the obtuse vertex is the longest in the triangle — larger angle, longer opposite side.
- **The side test.** For the longest side c, the triangle is obtuse precisely when c² > a² + b².
- **Angle sum holds.** As in any triangle, the three interior angles total 180°, so the two acute angles add to less than 90°.
- **Circumcenter and orthocenter sit outside.** Unlike an acute triangle, both the circumcenter (centre of the circle through all three vertices) and the orthocenter (where the altitudes meet) fall _outside_ an obtuse triangle.
- **It can be scalene or isosceles, never equilateral.** An equilateral triangle has three 60° angles, so none can be obtuse.

Two named cases come up often. An [isosceles obtuse triangle](/content/math/geometry/isosceles-obtuse-triangle/index.html) has two equal sides with the obtuse angle wedged between them, while an [acute scalene triangle](/content/math/geometry/acute-scalene-triangle/index.html) is the opposite extreme — no equal sides and no obtuse angle at all. (A [right scalene triangle](/content/math/geometry/right-scalene-triangle/index.html) is the right-angled cousin.)

## Obtuse Triangle Formulas

The two everyday formulas are perimeter and area, and they read the same as for any triangle.

**Perimeter.** Add the three sides:

P = a + b + c

where a, b, c are the three side lengths (in the same unit).

**Area, base-and-height form.**  
Area = ½ × b × h

Here b is the chosen base and h is the perpendicular height to that base. This is where obtuse triangles surprise students: drop a perpendicular from the obtuse vertex and it lands _inside_ the base, but drop one from an acute vertex onto a short side and the foot can land _outside_ the triangle, on an extension of that side. The height still exists; it just sits beyond the edge.

**Area, Heron's formula** (when you know all three sides but no height). First compute the semi-perimeter s:

s = (a + b + c) / 2

then

Area = √(s(s−a)(s−b)(s−c))

Heron's formula, attributed to Heron of Alexandria, needs no angle and no height — only the three sides — which makes it the natural tool for an obtuse triangle whose height runs outside the figure.

## Examples of Obtuse Triangle

### Example 1

**Is a triangle with angles 30°, 40°, and 110° obtuse?**

Find the largest angle: 110°.

Since 110° is greater than 90° and less than 180°, the triangle has one obtuse angle.

Check the sum: 30° + 40° + 110° = 180°. Valid.

**Final answer:** Yes, it is an obtuse triangle.

### Example 2 (a tempting shortcut that fails)

**Do the sides 6 cm, 8 cm, and 9 cm form an obtuse triangle?**

_Wrong attempt._ A quick reflex is to add the two smaller squares and compare to the largest side rather than its square: 6² + 8² = 36 + 64 = 100, and "100 is bigger than 9," so the triangle "must be acute." That comparison is meaningless — it pits an area-sized number against a length.

_Correct._ Longest side is 9, so c² = 81. The other two give a² + b² = 36 + 64 = 100. Compare: 100 > 81.

**Final answer:** The triangle is acute, not obtuse — the side test only works once every term is squared.

### Example 3

**Do the sides 4 cm, 5 cm, and 7 cm form an obtuse triangle?**

Longest side c = 7, so c² = 49.
The other two: a² + b² = 4² + 5² = 16 + 25 = 41.
Compare: 49 > 41.

**Final answer:** Yes — the triangle is obtuse.

### Example 4

**An obtuse triangle has a base of 8 in and a perpendicular height of 5 in to that base. Find its area.**

Area = ½ × b × h  
Substitute:

Area = ½ × 8 × 5

Area = 20 in²

**Final answer:** 20 square inches.

### Example 5

**Find the area of an obtuse triangle with sides 5 cm, 6 cm, and 9 cm using Heron's formula.**

Semi-perimeter:

s = (5 + 6 + 9) / 2 = 10

Apply Heron's formula:

Area = √(10(10−5)(10−6)(10−9))

Area = √(10 × 5 × 4 × 1)  
Area = √{200} ≈ 14.14 cm²

**Final answer:** about 14.14 cm² (and the side test, 81 > 61, confirms it is obtuse).

### Example 6

**Two angles of an obtuse triangle are 25° and 35°. Find the third angle and confirm the triangle is obtuse.**

Subtract from 180°:

∠3 = 180° − 25° − 35° = 120°

Since 120° is greater than 90°, the triangle has one obtuse angle.

**Final answer:** The third angle is 120°, so the triangle is obtuse.

## Why the Obtuse Case Earns Its Own Name

That line is essentially **Proposition 12 of Book II of Euclid's _Elements_** — the obtuse-triangle companion to the Pythagorean theorem. The Greeks needed it because real triangles rarely arrive with a right angle, and surveyors, builders, and astronomers had to reason about the awkward, splayed-open ones too.

Where the obtuse case shows up:

- **Structural bracing.** A widely-splayed truss spreads load across a long span; engineers compute its area and centroid using exactly the Heron and base-height tools above. Because the orthocenter lies outside the figure, the geometry of the bracing forces behaves differently from a compact acute frame.
- **Navigation and surveying.** When a sightline between two landmarks subtends a wide angle from a third point, the triangle formed is obtuse, and the [triangle sum theorem](/content/math/geometry/triangle-sum-theorem/index.html) lets the surveyor recover the missing angle from the two measured ones.
- **The law of cosines.** For an obtuse angle, cos is negative, so c² = a² + b² − 2abcos C becomes c² = a² + b² + (a positive amount) — which is precisely why c² > a² + b². The side test isn't a separate rule; it is the law of cosines in disguise.

## Where Students Trip Up on Obtuse Triangles

### Mistake 1: Comparing a side to a sum of squares

**Where it slips in:** Using the side test with all three side lengths in hand.

Don't do this: Compare a² + b² to the longest side instead of to c².

**The correct way:** Square every side. Compare a² + b² with c² — squares against squares, never a square against a raw length.

### Mistake 2: Assuming the height is always inside the triangle

**Where it slips in:** Finding the area of an obtuse triangle from a base and height.

Don't do this: Insist the perpendicular height must land between the two base vertices.

**The correct way:** Allow the foot of the altitude to fall _outside_ the base, on its extension, when the height is dropped from an acute vertex. The height is still the perpendicular distance; it just lives outside the figure.

### Mistake 3: Thinking a triangle can have two obtuse (or an obtuse plus a right) angle

**Where it slips in:** Building or checking a triangle from its angles.

Don't do this: Allow two angles above 90°, or one obtuse and one right angle.

**The correct way:** Remember the 180° cap. One obtuse angle already leaves under 80° for the other two combined — a second 90°-plus angle is impossible.

## Key Takeaways

- An **obtuse triangle** has exactly one angle greater than 90°, and the longest side sits opposite it.
- The side test identifies one fast: a triangle is obtuse when c² > a² + b² for the longest side c.
- Area comes from ½bh or, when only the three sides are known, from Heron's formula.
- The most common error is comparing a side to a sum of squares — always square every term first.
- Unlike acute triangles, the circumcenter and orthocenter of an obtuse triangle lie outside the figure.