The Three Types of Triangles: When to Use Which Formulas

Not all triangles are created equal. While right triangles dominate geometry classes with their Pythagorean theorem and tidy trigonometric ratios, they're just one member of a three-part family. Understanding all three types of triangles—right, acute, and obtuse—and knowing which mathematical tools apply to each is essential for solving real-world problems.

This guide will help you identify triangle types at a glance, understand why different formulas apply to different triangles, and choose the right mathematical approach for any triangle you encounter.

The Three-Way Classification

Every triangle falls into exactly one of three categories based on its largest angle:

• Acute Triangles: All three angles are less than 90°
• Right Triangles: Exactly one angle equals 90°
• Obtuse Triangles: Exactly one angle is greater than 90°

This classification is exhaustive and exclusive—every triangle fits one category, and no triangle fits more than one.

Why the Largest Angle Matters

Since the three angles of any triangle sum to 180°, the largest angle determines the type:

• If the largest angle < 90°, all angles must be < 90° → acute
• If the largest angle = 90°, the other two sum to 90° → right
• If the largest angle > 90°, the other two must be < 90° → obtuse

You can't have two angles ≥ 90° because they alone would sum to ≥ 180°, leaving no room for the third angle.

Acute Triangles: All Angles Sharp

Definition: All three interior angles are less than 90°

Visual Characteristics

• Appears "pointy" all around
• No angle approaches a right angle
• The triangle feels balanced, with no extremely sharp or flat angles

Examples of Acute Triangles

• Equilateral triangle (all angles = 60°)
• Isosceles triangle with angles 70°-70°-40°
• Scalene triangle with angles 50°-65°-65°

The Relationship Test for Acute Triangles

Here's a mathematical test: if you know all three sides (a, b, c where c is longest):

a² + b² > c² → The triangle is acute

Example: Sides 5, 7, 8

• Longest side: c = 8
• Check: 5² + 7² = 25 + 49 = 74
• Compare: 8² = 64
• 74 > 64 ✓
• This is an acute triangle

Why This Test Works

In a right triangle, a² + b² exactly equals c². In an acute triangle, the sides are "shorter" relative to the angles—the geometry is more compact. This manifests mathematically as the sum of squares exceeding the square of the longest side.

When to Use Acute Triangle Formulas

• Area calculation: Use Heron's formula or standard ½ base × height
• Finding angles from sides: Use Law of Cosines
• Finding sides from angles: Use Law of Sines
• You CANNOT use: Standard Pythagorean theorem or basic right triangle trig (SOH-CAH-TOA applies only to right triangles)

Right Triangles: The 90-Degree Sweet Spot

Definition: Exactly one interior angle equals 90°

Visual Characteristics

• One corner is a perfect square corner
• Two sides meet at a perpendicular angle
• The side opposite the right angle (hypotenuse) is visibly longest

Examples of Right Triangles

• 3-4-5 triangle (angles approximately 37°-53°-90°)
• 45-45-90 triangle (isosceles right triangle)
• 30-60-90 triangle

The Relationship Test for Right Triangles

a² + b² = c² → The triangle is right (where c is the longest side)

Example: Sides 6, 8, 10

• Longest side: c = 10
• Check: 6² + 8² = 36 + 64 = 100
• Compare: 10² = 100
• 100 = 100 ✓
• This is a right triangle (it's 3-4-5 scaled by 2)

Why Right Triangles Are Special

The 90° angle creates a unique geometric situation where:

• The two legs are perpendicular (independent dimensions)
• The hypotenuse relates to the legs via the Pythagorean theorem
• Trigonometric functions have simple definitions
• Many formulas simplify dramatically

This is why we spend so much time studying right triangles—they're both common and mathematically tractable. Certain special right triangles like 30-60-90 and 45-45-90 have particularly elegant properties.

When to Use Right Triangle Formulas

• Finding missing sides: Pythagorean theorem (a² + b² = c²)
• Finding angles from sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent
• Finding sides from angles: Rearrange the trig ratios
• Area calculation: ½ × leg₁ × leg₂ (the two perpendicular sides)
• You MUST use these tools: Right triangle formulas only work for right triangles!

Obtuse Triangles: One Wide Angle

Definition: Exactly one interior angle is greater than 90°

Visual Characteristics

• One angle appears "opened up" or flat
• The triangle looks lopsided or unbalanced
• One angle dominates the shape

Examples of Obtuse Triangles

• Triangle with angles 120°-40°-20°
• Triangle with angles 100°-50°-30°
• Any triangle where one angle exceeds 90°

The Relationship Test for Obtuse Triangles

a² + b² < c² → The triangle is obtuse (where c is longest side)

Example: Sides 5, 7, 10

• Longest side: c = 10
• Check: 5² + 7² = 25 + 49 = 74
• Compare: 10² = 100
• 74 < 100 ✓
• This is an obtuse triangle

Why This Test Works

In an obtuse triangle, the large angle "stretches out" the opposite side. The longest side becomes disproportionately long relative to the other sides. Mathematically, this causes c² to exceed a² + b².

When to Use Obtuse Triangle Formulas

• Area calculation: Heron's formula or ½ base × height (where height may fall outside the triangle)
• Finding angles from sides: Law of Cosines (essential for obtuse triangles)
• Finding sides from angles: Law of Sines
• Special note: The obtuse angle will have a negative cosine value, which is how the Law of Cosines handles it
• You CANNOT use: Pythagorean theorem or basic right triangle trig

Quick Identification Guide

When given three sides, identify the triangle type in seconds:

1. Step 1: Identify the longest side (call it c)
2. Step 2: Calculate a² + b² (using the two shorter sides)
3. Step 3: Calculate c²
4. Step 4: Compare:
   • If a² + b² > c² → Acute
   • If a² + b² = c² → Right
   • If a² + b² < c² → Obtuse

Example 1: Sides 7, 10, 12

• c = 12
• 7² + 10² = 49 + 100 = 149
• 12² = 144
• 149 > 144 → Acute triangle

Example 2: Sides 9, 12, 15

• c = 15
• 9² + 12² = 81 + 144 = 225
• 15² = 225
• 225 = 225 → Right triangle (3-4-5 triple scaled by 3)

Example 3: Sides 4, 5, 8

• c = 8
• 4² + 5² = 16 + 25 = 41
• 8² = 64
• 41 < 64 → Obtuse triangle

The Universal Formulas: Law of Sines and Law of Cosines

While right triangles have their special formulas, two laws work for ALL triangles:

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C)

This law relates sides to their opposite angles. It works for acute, right, and obtuse triangles equally.

Best used when:

• You know two angles and one side
• You know two sides and an angle opposite one of them

Example: Triangle with a = 7, A = 40°, B = 60°

• Find side b: b/sin(60°) = 7/sin(40°)
• b = 7 × sin(60°)/sin(40°)
• b = 7 × 0.866/0.643
• b ≈ 9.43

Works regardless of triangle type!

Law of Cosines

c² = a² + b² - 2ab·cos(C)

This law generalizes the Pythagorean theorem. For right triangles where C = 90°, cos(90°) = 0, and it reduces to a² + b² = c². The law of cosines is particularly essential for solving non-right triangles.

Best used when:

• You know all three sides and want angles
• You know two sides and the included angle, want the third side

Example: Triangle with a = 5, b = 7, C = 40°

• c² = 5² + 7² - 2(5)(7)cos(40°)
• c² = 25 + 49 - 70(0.766)
• c² = 74 - 53.62
• c² = 20.38
• c ≈ 4.52

Finding angles: Rearrange to solve for the cosine:

cos(C) = (a² + b² - c²)/(2ab)

If cos(C) is:

• Positive → angle C is acute
• Zero → angle C is 90° (right angle)
• Negative → angle C is obtuse

Decision Tree: Which Formula Should I Use?

Question 1: Is there a right angle?

Yes → Use right triangle formulas

• Pythagorean theorem for sides
• SOH-CAH-TOA for angles and sides
• Area = ½ × leg₁ × leg₂

No or Unknown → Continue to Question 2

Question 2: What information do I have?

Two sides and included angle:

• Use Law of Cosines to find third side
• Then use Law of Sines or Cosines for remaining angles

Two angles and one side:

• Use Law of Sines to find other sides
• Find third angle by subtracting from 180°

All three sides:

• Use comparison test to identify triangle type
• Use Law of Cosines to find angles

Three sides, need area:

• Use Heron's formula (works for all triangle types)

Area Formulas by Triangle Type

Calculating area varies by triangle type. For comprehensive coverage of all methods, see our guide on triangle area formulas.

For Right Triangles

Area = ½ × base × height = ½ × leg₁ × leg₂

Simple! The two perpendicular sides are your base and height.

For Acute and Obtuse Triangles

Option 1: Standard formula (if you can find the height)

Area = ½ × base × height

But finding the height may require trigonometry or geometry.

Option 2: Heron's Formula (when you know all three sides)

Area = √[s(s-a)(s-b)(s-c)]

Where s = (a+b+c)/2 (the semi-perimeter)

Example: Sides 5, 7, 9

• s = (5+7+9)/2 = 10.5
• Area = √[10.5(10.5-5)(10.5-7)(10.5-9)]
• Area = √[10.5 × 5.5 × 3.5 × 1.5]
• Area = √[303.1875]
• Area ≈ 17.41 square units

Option 3: Trigonometric formula (when you know two sides and included angle)

Area = ½ × a × b × sin(C)

Where C is the angle between sides a and b.

This works for ALL triangle types!

Special Cases and Considerations

Equilateral Triangles (All Acute)

• All angles = 60°
• All sides equal
• Special formula: Area = (s²√3)/4 where s is side length
• Highly symmetric

Isosceles Triangles (Can Be Any Type)

• Two sides equal
• Two angles equal (opposite the equal sides)
• Can be acute, right, or obtuse depending on the angle between equal sides

Right isosceles: 45-45-90 triangle

Acute isosceles: Equal angles > 45° (e.g., 70-70-40)

Obtuse isosceles: Equal angles < 45° (e.g., 30-30-120)

Degenerate Triangles (Not Really Triangles)

If a + b = c (the limiting case), the triangle "collapses" into a straight line. This violates the triangle inequality and isn't considered a valid triangle.

Common Mistakes in Triangle Type Identification

Mistake 1: Assuming All Isosceles Triangles Are Right

Reality: Isosceles just means two equal sides. The triangle could be acute, right, or obtuse.

Example: A triangle with sides 5-5-8 is obtuse isosceles (5² + 5² = 50 < 64 = 8²)

Mistake 2: Using Pythagorean Theorem on Non-Right Triangles

Problem: Applying a² + b² = c² to find the third side when the triangle isn't right.

Solution: First verify it's a right triangle. If not, use Law of Cosines instead.

Mistake 3: Thinking Obtuse Means "Not a Right Angle"

Confusion: Some students think "obtuse" means any angle that isn't 90°.

Reality: Obtuse specifically means > 90°. Acute means < 90°. All non-right angles are either acute or obtuse, not "obtuse" by default.

Mistake 4: Forgetting the Triangle Inequality

Problem: Creating "triangles" where a + b ≤ c.

Reality: This violates the triangle inequality. No triangle can exist.

Practical Applications by Type

Right Triangles

• Construction (squaring corners)
• Architecture (roof slopes, stairs)
• Navigation (simple distance calculations)
• Computer graphics (orthogonal coordinate systems)

Acute Triangles

• Structural engineering (trusses with all acute angles are often stiffer)
• Design (acute triangles appear more stable visually)
• Geodesic domes (mostly acute triangles)

Obtuse Triangles

• Irregular plots of land
• Artistic design (create visual interest with unusual angles)
• Real-world situations where right angles aren't possible

Conclusion: One Family, Three Members

Every triangle belongs to one of three types: acute, right, or obtuse. This simple classification has profound implications for which mathematical tools apply.

Right triangles get the spotlight because their formulas are simple and elegant. The Pythagorean theorem and basic trigonometry make them accessible and practical. But they're just one-third of the triangle family.

Acute triangles, with their "pointy" appearance and compact geometry, require more sophisticated tools like the Law of Cosines. Obtuse triangles, with their wide angles and stretched proportions, challenge our intuition but follow the same universal laws.

The beauty of mathematics is that while different tools apply to different types, underlying principles connect them all. The Law of Cosines reduces to the Pythagorean theorem for right triangles. The Law of Sines works equally well for all three types. Triangle area formulas generalize across the family.

Master the identification tests:

• a² + b² > c² → acute
• a² + b² = c² → right
• a² + b² < c² → obtuse

Remember which formulas apply when:

• Right triangles → Pythagorean theorem, SOH-CAH-TOA
• All triangles → Law of Sines, Law of Cosines, Heron's formula

With this knowledge, no triangle will intimidate you. Acute, right, or obtuse—you'll recognize it instantly and know exactly which mathematical tools to deploy. That's not just geometry knowledge; that's geometric wisdom.