Acute Scalene Triangle: Properties & Examples

TL;DR
An acute scalene triangle has all three angles less than 90° and all three sides of different lengths, so no two angles and no two sides ever match. This article covers the definition, how a triangle can be acute and scalene at once, the properties, the area and perimeter formulas with derivation, six worked examples, and the common mistakes.

What Is an Acute Scalene Triangle?

An acute scalene triangle is a triangle that is both acute and scalene at once. Acute means all three angles are less than 90∘. Scalene means all three sides have different lengths — and unequal sides force unequal opposite angles, so all three angles differ too.

Put together: an acute scalene triangle has three different angles, each under 90∘, and three sides of three different lengths. There is no right angle, no obtuse angle, no equal sides, and no equal angles — nothing in it repeats.

Can a Triangle Be Both Acute and Scalene?

Students often ask this directly, so here it is: can a scalene triangle also be acute?

Yes. A scalene triangle — three unequal sides — can be acute, right, or obtuse, depending on its angles. It is acute when all three of its angles fall below 90∘. Since the three angles must add to 180∘, this happens whenever no single angle gets too large: angles like 80∘, 60∘, 40∘ work, and so do 70∘, 65∘, 45∘. As long as the three values are different and each stays under 90∘, the triangle is acute scalene.

Properties of the Acute Scalene Triangle

Everything about this triangle flows from "all angles acute, all sides unequal." The properties worth holding:

• Three different acute angles. Each is less than 90∘, no two are equal, and they sum to 180∘ like every triangle.
• Three sides of different lengths. No two sides match, so the longest side faces the largest angle and the shortest faces the smallest.
• No line of symmetry and no equal angles. It cannot be folded onto itself, because nothing in it repeats.
• The longest side rule still applies. The biggest angle (still under 90∘) sits opposite the longest side — useful for ordering sides without measuring.

Notice there is no right angle to lean on, so the Pythagorean shortcut is unavailable here: area comes from base and height, or from Heron's formula when only the three sides are known.

Area and Perimeter of an Acute Scalene Triangle

The formulas are the standard triangle formulas. What matters is knowing what each symbol stands for and why it holds.

Perimeter. The perimeter is the total distance around, so add the three sides a, b, and c:
P=a+b+c.

Area from base and height. Every triangle's area is half its base times its perpendicular height:
A=12×b×h,
where b is any side chosen as the base and h is the perpendicular height drawn to that base. In an acute triangle every height lands neatly inside the triangle.

Area from three sides (Heron's formula). When you know all three sides but no height, use Heron's formula:
A=s(s−a)(s−b)(s−c).
Heron's formula works for any triangle, which is exactly why it is handy for a scalene one.

Examples of Acute Scalene Triangle

Example 1. Two angles are 80∘ and 60∘. Find the third angle.
∠C=180∘−80∘−60∘=40∘.
Final answer: 40∘. All three angles (80∘, 60∘, 40∘) are different and each is under 90∘.

Example 2. Angles 40∘, 40∘, and 100∘. It is isosceles obtuse, not acute scalene.

Example 3. An acute scalene triangle has a base of 10 cm and a height of 12 cm.
A=12×10×12=60 cm².
Final answer: 60 cm².

Example 4. An acute scalene triangle has sides 7 cm, 9 cm, and 11 cm.
P=a+b+c=7+9+11=27 cm.
Final answer: 27 cm.

Example 5. A triangle has a perimeter of 68 inches, with two sides of 20 inches and 27 inches. Find the third side.
c=68−20−27=21 inches.
Final answer: 21 inches.

Example 6. Sides 6 cm, 7 cm, and 8 cm. Find its area using Heron's formula:
A=10.5(4.5)(3.5)(2.5)=20.33 cm².
Final answer: about 20.33 cm².

Why the Acute Scalene Triangle Matters

• It is what most real shapes actually are.
• It is the rigid building block.
• Acute triangles keep their heights inside.
• It anchors the longest-side-faces-largest-angle rule.

Common Errors When Working With Acute Scalene Triangles

Mistake 1: Reading the label off the triangle's overall look

Correct way: Run two separate tests.

Mistake 2: Using a side as the height for area

Correct way: The height is the perpendicular distance from a vertex to the opposite base.

Mistake 3: Assuming "scalene" tells you the angle type

Correct way: You need a word from each family — the side word and the angle word — to name the triangle fully.

Key Takeaways

• An acute scalene triangle has three different acute angles (each under 90∘) and three sides of different lengths.
• A scalene triangle can be acute, right, or obtuse; you need a word from each family to name it fully.
• Area is 12×b×h or Heron's formula when only the three sides are known.
• Acute triangles keep all three altitudes inside the triangle.