Obtuse Scalene Triangle: Properties & Examples

What Is an Obtuse Scalene Triangle?

An obtuse scalene triangle is a triangle that is both:

• Obtuse — one of its interior angles measures more than $90^\circ$ (but less than $180^\circ$). The other two are necessarily acute.

• Scalene — all three sides have different lengths, and so all three angles have different measures.

Putting the two together: an obtuse scalene triangle has one obtuse angle, two unequal acute angles, and three unequal sides. As with every triangle, the three interior angles add to $180^\circ$:

$$\angle A + \angle B + \angle C = 180^\circ.$$

The two classifications are independent axes. "Obtuse" describes the angles; "scalene" describes the sides. A triangle can be obtuse and scalene, obtuse and isosceles, acute and scalene, and so on — this article is the one corner of that grid where a wide angle meets three mismatched sides.

Properties of an Obtuse Scalene Triangle

A handful of properties follow directly from the two-name definition.

• Exactly one obtuse angle. A triangle can have at most one angle above $90^\circ$ — two would already exceed the $180^\circ$ total on their own. So the obtuse angle is always alone, flanked by two acute angles.

• All three sides unequal, all three angles unequal. That is the scalene condition. No side equals another, so the triangle has no lines of symmetry.

• The longest side sits opposite the obtuse angle. In any triangle the largest angle faces the longest side, and here the obtuse angle is the largest, so the side across from it is the longest of the three.

• No equal angles, no axis of symmetry. Unlike an isosceles or equilateral triangle, you cannot fold it onto itself. It looks visibly lop-sided.

• The obtuse angle pushes a height outside the triangle. When you drop a perpendicular height to a side next to the obtuse angle, the foot of that height lands outside the base — a quirk that matters when computing area, and which we handle below.

How Do You Find the Area of an Obtuse Scalene Triangle?

The area uses the same formula as any triangle, but the obtuse angle adds one wrinkle worth seeing.

Base and height

The standard area formula is

$$\text{Area} = \tfrac{1}{2} \times b \times h,$$

where $b$ is the length of a chosen base and $h$ is the perpendicular height to that base. The formula comes from the fact that any triangle is exactly half of the parallelogram (or rectangle) built on the same base and height — cut a rectangle along a diagonal and each half is a triangle of area $\tfrac{1}{2} bh$.

The wrinkle: in an obtuse triangle, if you pick a base next to the obtuse angle, the perpendicular height falls outside the triangle, and you measure it to the extension of the base. The number you plug in is still the perpendicular distance from the opposite vertex to the base line.

Heron's formula (all three sides known)

When you know the three side lengths $a$, $b$, $c$ but not a height, use Heron's formula. First compute the semi-perimeter (half the perimeter):

$$s = \frac{a + b + c}{2},$$

then

$$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}.$$

Heron's formula works for any triangle, obtuse scalene included, and is the go-to when no height is given — which is common for scalene triangles, since their mismatched sides rarely hand you a clean height. The perimeter is simply the sum of the sides, $P = a + b + c$.

Examples of the Obtuse Scalene Triangle

With the definition, properties, and both area methods in hand, here are the calculations applied. The problems build from classification up to a Heron's-formula area.

Example 1

A triangle has angles $40^\circ$, $112^\circ$, and $28^\circ$, with sides $6$ cm, $11$ cm, and $5$ cm. Classify it.

One angle ($112^\circ$) exceeds $90^\circ$, so the triangle is obtuse. All three sides differ, so it is scalene. Final answer: it is an obtuse scalene triangle, with the longest side ($11$ cm) opposite the obtuse angle.

Example 2

An obtuse scalene triangle has two angles measuring $35^\circ$ and $50^\circ$. A student concludes it can't be obtuse because neither given angle is over $90^\circ$. Are they right?

A first instinct is to look only at the two given angles, see that both are acute, and decide the triangle isn't obtuse. But a triangle has three angles, and only the two smaller ones were given. Find the third using the angle sum:

$$\angle 3 = 180^\circ - 35^\circ - 50^\circ = 95^\circ.$$

The third angle is $95^\circ$ — obtuse. With all three angles different, the triangle is also scalene. Final answer: it is an obtuse scalene triangle; the obtuse angle was the one not handed to you, which is exactly why you check the third angle before classifying.

Example 3

Find the third angle of an obtuse scalene triangle whose other two angles are $105^\circ$ and $32^\circ$, and confirm the classification.

By the angle sum:

$$\angle 3 = 180^\circ - 105^\circ - 32^\circ = 43^\circ.$$

The angles are $105^\circ$, $32^\circ$, $43^\circ$ — one obtuse, all three different — so obtuse scalene is confirmed. Final answer: the third angle is $43^\circ$.

Example 4

An obtuse scalene triangle has base $14$ units and a height to that base of $6$ units. Find its area.

Apply the base-and-height formula:

$$\text{Area} = \tfrac{1}{2} \times 14 \times 6 = 42 \text{ square units}.$$

Final answer: $42$ square units. (The height is the perpendicular distance to the base line, whether or not its foot lands inside the triangle.)

Example 5

An obtuse scalene triangle has sides $a = 7$ cm, $b = 13$ cm, $c = 9$ cm. Find its perimeter and its area using Heron's formula.

The perimeter is the sum of the sides:

$$P = 7 + 13 + 9 = 29 \text{ cm}.$$

The semi-perimeter is

$$s = \frac{29}{2} = 14.5 \text{ cm}.$$

Now Heron's formula:

$$\text{Area} = \sqrt{14.5(14.5 - 7)(14.5 - 13)(14.5 - 9)} = \sqrt{14.5 \times 7.5 \times 1.5 \times 5.5}.$$

Multiplying inside: $14.5 \times 7.5 = 108.75$, and $1.5 \times 5.5 = 8.25$, so the product is $108.75 \times 8.25 = 897.19$ (to two decimals). Then

$$\text{Area} = \sqrt{897.19} \approx 29.95 \text{ cm}^2.$$

Final answer: perimeter $29$ cm, area $\approx 29.95$ cm$^2$.

Example 6

An obtuse scalene triangle has area $84$ cm$^2$ and a base of $24$ cm. Find the height to that base.

Rearrange the area formula for the height:

$$\text{Area} = \tfrac{1}{2} b h ;\Rightarrow; h = \frac{2 \times \text{Area}}{b} = \frac{2 \times 84}{24} = 7 \text{ cm}.$$

Final answer: the height is $7$ cm.

Why the Obtuse Scalene Triangle Matters

It might look like just one box in a classification chart, but the obtuse-plus-scalene combination is the most general triangle there is — and that generality is why it shows up everywhere irregular shapes do.

• It is the "default" triangle of the real world. Equilateral and isosceles triangles are special cases that need equal sides; most triangles you measure in the wild — a tilted roof gable, a wedge of land, a sail catching wind from the side — have three different sides and one wide angle. The obtuse scalene triangle is what "no special structure" looks like.

• It forces the general area tools. Because a scalene triangle rarely hands you a neat height, it is the case that makes Heron's formula genuinely useful — the formula a surveyor reaches for to find the area of an irregular three-cornered plot from side measurements alone.

• It appears in trusses, bracing, and design. Architects and engineers use obtuse scalene triangles wherever a span has to be bridged at an awkward angle — the diagonal of a ramp, a cantilevered support, a roof truss that isn't symmetric. The lop-sidedness is the point, not a flaw.

• It is where angle classification and side classification visibly disagree on symmetry. Studying it teaches that "obtuse" and "scalene" are independent descriptions — a lesson that carries into every later classification problem.

For a Grade 4 to 7 student first sorting triangles, the obtuse scalene triangle is the one that drives home the key idea: angles and sides are two separate questions, and a triangle has an answer to both.

Where Students Trip Up on the Obtuse Scalene Triangle

Mistake 1: Judging "obtuse" from only the angles you were given.

Where it slips in: A problem lists two acute angles and the student concludes the triangle is acute, never computing the third angle (as in Example 2).

Don't do this: Decide a triangle isn't obtuse just because the two given angles are both under $90^\circ$.

The correct way: Always find the third angle with $\angle 1 + \angle 2 + \angle 3 = 180^\circ$ before classifying. The obtuse angle is often the one not stated.

Mistake 2: Confusing the side classification with the angle classification.

Where it slips in: A student treats "scalene" and "obtuse" as the same kind of label, or thinks every scalene triangle is obtuse (or every obtuse triangle scalene).

Don't do this: Assume scalene implies obtuse, or that one name makes the other automatic.

The correct way: Keep the two axes separate — "scalene/isosceles/equilateral" classifies sides; "acute/right/obtuse" classifies angles. A triangle can be obtuse and isosceles, or scalene and acute. The memorizer who collapses the two systems into one list of triangle types loses the independence that makes classification work.

Mistake 3: Using a height that isn't perpendicular to the chosen base.

Where it slips in: Because an obtuse triangle's height can fall outside it, a student measures a slanted side as if it were the height, or uses the wrong base–height pairing.

Don't do this: Plug a side length in for $h$ when it isn't the perpendicular distance to the base.

The correct way: The height must be perpendicular to the base, even when its foot lands on the base's extension outside the triangle. If no perpendicular height is given, switch to Heron's formula from the three sides. The second-guesser who keeps swapping base–height pairs usually hasn't drawn the perpendicular to check where it lands.

Key Takeaways

• An obtuse scalene triangle has one obtuse angle (over $90^\circ$) and three unequal sides — obtuse by angle, scalene by side.

• The two acute angles plus the obtuse angle sum to $180^\circ$, and the longest side sits opposite the obtuse angle.

• Find the area with $\tfrac{1}{2} \times$ base $\times$ height, or with Heron's formula from the three sides when no height is given.

• Always compute the third angle before classifying — the obtuse angle is often the one not stated.

• "Obtuse" and "scalene" are independent labels: a triangle can be obtuse and isosceles, or scalene and acute.

Practice These Problems to Solidify Your Understanding

1. A triangle has angles $120^\circ$ and $25^\circ$. Find the third angle and classify the triangle by angle and side.

2. An obtuse scalene triangle has base $18$ cm and height $5$ cm. Find its area.

3. An obtuse scalene triangle has sides $8$ m, $15$ m, and $9$ m. Find its perimeter and area (Heron's formula).

Answer to Question 1: third angle $= 180^\circ - 120^\circ - 25^\circ = 35^\circ$; angles $120^\circ, 25^\circ, 35^\circ$ are all different with one obtuse, so it is obtuse scalene.

Answer to Question 2: Area $= \tfrac{1}{2} \times 18 \times 5 = 45$ cm$^2$.

Answer to Question 3: $P = 32$ m; $s = 16$; Area $= \sqrt{16 \times 8 \times 1 \times 7} = \sqrt{896} \approx 29.93$ m$^2$. If Question 1 came out as acute, check that you found the third angle before classifying (see Mistake 1).

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