Obtuse Scalene Triangle: Properties & Examples

TL;DR

An obtuse scalene triangle is a triangle with one obtuse angle and all three sides of different lengths — combining the "obtuse" classification by angle with the "scalene" classification by side. This article covers the definition, the properties, how to find its area by base-and-height and by Heron's formula, real-world examples, six worked problems, and the common mistakes.

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° (but less than 180°). 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°:

∠A+∠B+∠C=180°.

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° — two would already exceed the 180° 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.

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

Area = 1/2 × b × 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.

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.

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 = (a+b+c)/2,

then

Area = √(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.

Examples of the Obtuse Scalene Triangle

Example 1

A triangle has angles 40°, 112°, and 28°, with sides 6 cm, 11 cm, and 5 cm. Classify it.

One angle (112°) exceeds 90°, 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° and 50°. A student concludes it can't be obtuse because neither given angle is over 90°. Are they right?

Find the third angle using the angle sum:

∠3 = 180° − 35° − 50° = 95°.

The third angle is obtuse. Final answer: it is an obtuse scalene triangle.

Example 3

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

By the angle sum:

∠3 = 180° − 105° − 32° = 43°.

The angles are obtuse scalene. Final answer: the third angle is 43°.

Example 4

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

Area = 1/2 × 14 × 6 = 42 square units.

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: P = 7 + 13 + 9 = 29 cm.

The semi-perimeter is s = 29/2 = 14.5 cm.

Area = √(14.5(14.5 - 7)(14.5 - 13)(14.5 - 9)) = √(896) ≈ 29.95 cm².

Example 6

An obtuse scalene triangle has area 84 cm² and a base of 24 cm. Find the height to that base.

Rearrange the area formula for the height:

h = 2 × Area / b = 2 × 84 / 24 = 7 cm.

Why the Obtuse Scalene Triangle Matters

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 have three different sides and one wide angle. The obtuse scalene triangle is what "no special structure" looks like.

Key Takeaways

• An obtuse scalene triangle has one obtuse angle (over 90°) and three unequal sides.
• Always compute the third angle before classifying — the obtuse angle is often the one not stated.

Practice These Problems to Solidify Your Understanding

1. A triangle has angles 120° and 25°. 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).