Triangle Area: The 4 Formulas & When to Use Each

Q: How do I find the area of a triangle?

Default: A = ½ × base × height — the height must be PERPENDICULAR to the chosen base. Base 10, height 6: A = 30. Any side can serve as base, but the height must correspond. For slanted triangles, the height may fall outside the triangle — that is fine, use the extended base line.

Q: What is Heron's formula?

Area from three sides alone: s = (a+b+c)/2 (semi-perimeter), then A = √[s(s−a)(s−b)(s−c)]. Sides 5, 6, 7: s = 9, A = √(9×4×3×2) = √216 ≈ 14.7. No heights or angles needed — the surveyor's friend. Works for any triangle, including obtuse, as long as the three sides can form one.

Q: How do I use the sine area formula?

A = ½ × a × b × sin(C), where C is the angle BETWEEN sides a and b. Sides 8 and 5 with included angle 60°: A = ½ × 8 × 5 × sin60° = 20 × 0.866 = 17.32. Given a non-included angle, find the included one first (angles sum to 180°). sin90° = 1 collapses this to the right-triangle formula — they are the same family.

Q: What shortcuts exist for special triangles?

Right triangle: A = ½ × leg₁ × leg₂ (the legs are base and height). Equilateral side a: A = (√3/4)a² ≈ 0.433a² — side 6 → 15.59. Isosceles: split into two right triangles at the axis. 45-45-90 with leg a: A = a²/2. 30-60-90 with short leg a: A = (√3/2)a². Memorizing the equilateral coefficient saves real time.

Q: How do I check if three sides can form a triangle?

Triangle inequality: the sum of any two sides must exceed the third. Check only the two SHORTEST against the longest: 3+4 > 9? No — impossible triangle. 5+6 > 10? Yes (11 > 10), valid. Equality (3+4 = 7) gives a degenerate flat "triangle" with zero area. Heron's formula exposes invalid sets: a negative under the root means no such triangle.

Core answer: Triangle area = base × height ÷ 2. Any triangle's angles sum to exactly 180°. For right triangles, a² + b² = c² (Pythagoras). No base-height pair? Use Heron's formula from the three sides. A 3-4-5 triangle is the builder's best friend for squaring corners.

The area formulas (pick by what you know)

You know	Formula	Example
base & height	A = bh/2	b=10, h=6 → 30
three sides (Heron)	s=(a+b+c)/2; A=√(s(s−a)(s−b)(s−c))	5-6-7 → s=9, A=√216≈14.7
two sides + angle	A = ½ab·sinC	8,10,∠30° → 20
right triangle legs	A = ab/2	3,4 → 6

The laws that solve any triangle

Angle sum: A + B + C = 180°, always, on flat ground.
Pythagoras (right only): c = √(a²+b²). Classic triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
Law of sines: a/sinA = b/sinB = c/sinC — use with two angles + a side, or two sides + a non-included angle.
Law of cosines: c² = a² + b² − 2ab·cosC — the general Pythagoras; use with SAS or three sides.

Worked examples

Example 1 — Squaring a deck. Measure 3 m along one edge, 4 m along the other; adjust until the diagonal reads exactly 5 m. The corner is now 90°.

Example 2 — Roof pitch. A roof rising 2 m over a 6 m run: rafter length = √(4 + 36) = 6.32 m per side; pitch angle = arctan(2/6) ≈ 18.4°.

Example 3 — Land area. A triangular plot with sides 30, 40, 50 m: notice 30² + 40² = 50² — it's right! Area = 30×40/2 = 600 m², no Heron needed.

Common mistakes and myths

Using a slanted side as height — height must be perpendicular to the base; for obtuse triangles it may fall outside the triangle.
Applying Pythagoras to non-right triangles — 5-6-7 is not right (25 + 36 ≠ 49); check a² + b² = c² first.
Forgetting the triangle inequality — sides 2, 3, 6 cannot form a triangle (2 + 3 < 6); the two short sides must exceed the long one.
Believing AAA determines size — three angles fix the shape but not the scale; infinitely many similar triangles share them.
Mixing degrees and radians — sin(30°) = 0.5; sin(30 rad) = −0.988. Check the calculator mode.

Frequently Asked Questions

How do I find the area of a triangle?

Default: A = ½ × base × height — the height must be PERPENDICULAR to the chosen base. Base 10, height 6: A = 30. Any side can serve as base, but the height must correspond. For slanted triangles, the height may fall outside the triangle — that is fine, use the extended base line.

What is Heron's formula?

Area from three sides alone: s = (a+b+c)/2 (semi-perimeter), then A = √[s(s−a)(s−b)(s−c)]. Sides 5, 6, 7: s = 9, A = √(9×4×3×2) = √216 ≈ 14.7. No heights or angles needed — the surveyor's friend. Works for any triangle, including obtuse, as long as the three sides can form one.

How do I use the sine area formula?

A = ½ × a × b × sin(C), where C is the angle BETWEEN sides a and b. Sides 8 and 5 with included angle 60°: A = ½ × 8 × 5 × sin60° = 20 × 0.866 = 17.32. Given a non-included angle, find the included one first (angles sum to 180°). sin90° = 1 collapses this to the right-triangle formula — they are the same family.

What shortcuts exist for special triangles?

Right triangle: A = ½ × leg₁ × leg₂ (the legs are base and height). Equilateral side a: A = (√3/4)a² ≈ 0.433a² — side 6 → 15.59. Isosceles: split into two right triangles at the axis. 45-45-90 with leg a: A = a²/2. 30-60-90 with short leg a: A = (√3/2)a². Memorizing the equilateral coefficient saves real time.

How do I check if three sides can form a triangle?

Triangle inequality: the sum of any two sides must exceed the third. Check only the two SHORTEST against the longest: 3+4 > 9? No — impossible triangle. 5+6 > 10? Yes (11 > 10), valid. Equality (3+4 = 7) gives a degenerate flat "triangle" with zero area. Heron's formula exposes invalid sets: a negative under the root means no such triangle.

What mistakes happen with triangle area?

(1) Using a slant side as the height — height must be perpendicular to the base; (2) forgetting the ½ — bh gives the parallelogram, not the triangle; (3) sin formula with a non-included angle; (4) Heron arithmetic slips — compute s first, then each difference, check all positive; (5) not validating the triangle inequality before starting; (6) unit mixing: sides in cm and m produce nonsense cm·m areas.

Key Takeaways

The area formulas (pick by what you know)

The laws that solve any triangle

Worked examples

Common mistakes and myths

Frequently Asked Questions