# 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.