Acute Triangles: Properties and Formulas
An acute triangle has all three interior angles less than 90 degrees. It is one of three types classified by angles: acute, right, and obtuse.
The Pythagorean Inequality
Acute Triangle
a² + b² > c²
Sum of squares of shorter sides exceeds square of longest side. All angles < 90°.
Right Triangle
a² + b² = c²
Pythagorean theorem holds exactly. One angle = 90°.
Obtuse Triangle
a² + b² < c²
Sum of squares of shorter sides is less. One angle > 90°.
Key Formulas
Heron's Formula (Area)
s = (a+b+c)/2
A = sqrt(s(s-a)(s-b)(s-c))
Law of Cosines (Angles)
cos(A) = (b²+c²-a²)/(2bc)
Similarly for angles B and C.
Heights
h_a = 2·Area / a
Similarly for h_b and h_c.
Inradius & Circumradius
r = Area / s
R = abc / (4·Area)
Properties of Acute Triangles
- All altitudes lie inside the triangle.
- The orthocenter lies inside the triangle.
- The circumcenter lies inside the triangle.
- An equilateral triangle is always acute (all angles = 60°).
- Sum of any two angles > 90°.
Special Acute Triangles
Equilateral
All sides equal. All angles 60°.
Area = (sqrt(3)/4) · a²
Isosceles Acute
Two sides equal, all angles < 90°.
Example: 5, 5, 6
Scalene Acute
All sides different, all angles < 90°.
Example: 5, 6, 7
How to Verify Acute
- Check triangle inequality: sum of any two sides > third side.
- Identify the longest side (call it c).
- Check a² + b² > c².
- If true, it is acute. If equal, right. If less, obtuse.