Acute Triangle
Key Formula
0°<A<90°,0°<B<90°,0°<C<90°,A+B+C=180°
Where:
- A = Measure of the first interior angle
- B = Measure of the second interior angle
- C = Measure of the third interior angle
Worked Example
Problem: A triangle has angles measuring 50°, 60°, and 70°. Determine whether it is an acute triangle.
Step 1: Check that the angles sum to 180°.
50°+60°+70°=180°✓
Step 2: Check each angle individually against 90°.
50°<90°,60°<90°,70°<90°
Step 3: Since every angle is less than 90°, all three angles are acute.
Answer: Yes, the triangle is an acute triangle because all three interior angles are less than 90°.
Another Example
Problem: A triangle has sides of length 5, 6, and 7. Use the side-length test to determine whether it is acute, right, or obtuse.
Step 1: Identify the longest side. Here the longest side is c = 7, with the other sides a = 5 and b = 6.
Step 2: Compute a² + b² and c².
a2+b2=25+36=61,c2=49
Step 3: Compare the two values. If a² + b² > c², the triangle is acute. If a² + b² = c², it is right. If a² + b² < c², it is obtuse.
61>49
Answer: Because a² + b² > c², the triangle with sides 5, 6, and 7 is an acute triangle.
Frequently Asked Questions
Can an equilateral triangle be an acute triangle?
Yes. An equilateral triangle has three 60° angles, and since 60° < 90°, every equilateral triangle is also an acute triangle. In fact, the equilateral triangle is the most 'balanced' acute triangle possible.
How do you tell if a triangle is acute from its side lengths?
Label the longest side c and the other two sides a and b. If a² + b² > c², the triangle is acute. This is an extension of the Pythagorean theorem: equality gives a right triangle, and if the sum is less than c² you get an obtuse triangle.
Acute Triangle vs. Obtuse Triangle
An acute triangle has all three angles less than 90°, while an obtuse triangle has exactly one angle greater than 90°. You can also distinguish them by side lengths: for the longest side c, an acute triangle satisfies a² + b² > c², whereas an obtuse triangle satisfies a² + b² < c². A right triangle sits exactly at the boundary, where a² + b² = c². Every triangle falls into exactly one of these three categories.
Why It Matters
Acute triangles appear frequently in geometry problems involving circumcenters and altitudes, because in an acute triangle the circumcenter and the orthocenter both lie inside the triangle. Engineers and architects favor acute triangles in structural trusses and mesh designs because they distribute forces more evenly and produce better numerical accuracy in computer simulations. Recognizing whether a triangle is acute also helps you apply the correct form of the law of cosines and avoid sign errors.
Common Mistakes
Mistake: Checking only one or two angles and concluding the triangle is acute.
Correction: All three angles must be less than 90° for the triangle to be acute. A triangle with angles 30°, 40°, and 110° has two acute angles but is actually obtuse because of the 110° angle.
Mistake: Using the side-length test with the wrong side as c.
Correction: The comparison a² + b² vs. c² only works when c is the longest side. Always identify the longest side first before applying the test.
Related Terms
- Triangle — General category that includes acute triangles
- Interior Angle — The angles tested to classify a triangle
- Acute Angle — An angle less than 90°; defines the type
- Obtuse Triangle — Triangle with one angle greater than 90°
- Right Triangle — Triangle with one angle exactly 90°
- Equilateral Triangle — Special case where all angles are 60°
- Pythagorean Theorem — Basis for the side-length classification test