A triangle has sides 3, 4, and 5. What is its area? Most people answer 10 — and most people are wrong. The answer is 6. The trap is picking the wrong side as the height, and it is the single most common area mistake in school geometry. Here is the trap, the fix, and the three formulas that solve any triangle.

The 3-4-5 trap: 6, not 10

The area of a triangle is half the base times the height: area = ½ × base × height. The height is the perpendicular distance from the base to the opposite vertex — the shortest distance, straight up at a right angle. In a 3-4-5 right triangle, the two legs (3 and 4) meet at the right angle, so each one is perpendicular to the other. Take base 3, height 4: area = ½ × 3 × 4 = 6.

The wrong answer, 10, comes from using the hypotenuse 5 as if it were the height: ½ × 4 × 5 = 10. But 5 is slanted — it is not perpendicular to the base 4, so it is not a height at all. The rule worth memorizing: the height must be perpendicular to the base you chose. If the side you picked is slanted, it is not the height.

The same triangle proves the point a second way. Heron's formula finds the area from the three sides alone, no height needed. First the semi-perimeter: s = (3 + 4 + 5) ÷ 2 = 6. Then area = √(s × (s − a) × (s − b) × (s − c)) = √(6 × 3 × 2 × 1) = √36 = 6. Two completely different routes, same answer — and the wrong route's 10 fails the check.

The three ways to find the area of a triangle

Every triangle-area problem is one of three cases, and each has its own formula.

1. Base and height — area = ½ × base × height. Use it when you know a side and its perpendicular height. The 3-4-5 triangle is the classic: ½ × 3 × 4 = 6. Another: base 6, height 8 gives ½ × 6 × 8 = 24 (the 6-8-10 triangle). The hard part is never the multiplication — it is confirming the height is genuinely perpendicular.

2. Two sides and the included angle (SAS) — area = ½ × a × b × sin(C). Use it when you know two sides and the angle between them. Example: sides 5 and 7 with a 60° angle between them give area = ½ × 5 × 7 × sin(60°) = 17.5 × 0.8660 = 15.16. The same case also finds the third side with the law of cosines: c² = 5² + 7² − 2 × 5 × 7 × cos(60°) = 25 + 49 − 35 = 39, so c = √39 ≈ 6.24.

3. Three sides (SSS) — Heron's formula. Use it when you know all three sides and no angles. Example: sides 5, 6, 7. Semi-perimeter s = (5 + 6 + 7) ÷ 2 = 9. Area = √(9 × 4 × 3 × 2) = √216 ≈ 14.70. Heron's formula also hands you the angles through the law of cosines: the angles opposite sides 5, 6, and 7 are 44.42°, 57.12°, and 78.46° — and they add to exactly 180°, which is the built-in check that the triangle is consistent.

If you know two angles and a side (ASA or AAS), the third angle is 180° minus the other two, and the law of sines finds the missing sides. Example: angles 50° and 60° with the side between them 10. The third angle is 70°. The other sides are a = 10 × sin(50°) ÷ sin(70°) ≈ 8.15 and b = 10 × sin(60°) ÷ sin(70°) ≈ 9.22, and the area is ½ × 8.15 × 9.22 × sin(70°) ≈ 35.30 — the same answer two different ways, which is how you know it is right.

The checks that catch every error

Four habits prevent nearly all triangle mistakes. First, the angles must add to 180° — if they don't, something is wrong before you go further. Second, the longest side is opposite the largest angle — in the 5-6-7 triangle, side 7 sits opposite 78.46°, the biggest angle. Third, the triangle inequality: any two sides must add to more than the third, so 2, 3, and 10 can never be a triangle (2 + 3 = 5, which is less than 10). Fourth, the height is perpendicular — if the number you used as the height is slanted, the answer is wrong, as the 3-4-5 trap shows.

Two more worked examples worth knowing by heart. The 5-12-13 right triangle: area = ½ × 5 × 12 = 30, perimeter = 5 + 12 + 13 = 30 — the same number twice, a coincidence students love to spot. And the equilateral triangle with side 6: area = (√3 ÷ 4) × 6² = 9√3 ≈ 15.59, with all three angles 60°.

Check it with code

The whole topic in one Python block:

import math # 3-4-5: base x height, and Heron — both give 6 print(0.5 * 3 * 4) # 6.0 s = (3 + 4 + 5) / 2 print(math.sqrt(s * (s-3) * (s-4) * (s-5))) # 6.0 # SSS 5-6-7: Heron s = (5 + 6 + 7) / 2 print(math.sqrt(s * (s-5) * (s-6) * (s-7))) # 14.6969... # SAS: sides 5, 7, included angle 60 degrees print(0.5 * 5 * 7 * math.sin(math.radians(60))) # 15.1554... print(math.sqrt(25 + 49 - 2*5*7*math.cos(math.radians(60)))) # 6.245...

A quick drill

  1. 3-4-5 right triangle: area = ½ × 3 × 4 = 6 — never 10, because 5 is the hypotenuse, not the height.
  2. Heron check: s = 6, area = √(6 × 3 × 2 × 1) = 6.
  3. Sides 5, 6, 7: area ≈ 14.70, angles 44.42° / 57.12° / 78.46°.
  4. Sides 5 and 7 with 60° between them: area ≈ 15.16, third side ≈ 6.24.
  5. Angles 50° and 60°, side 10 between them: the other sides are ≈ 8.15 and 9.22, area ≈ 35.30.

Want the steps shown instead of just the answers? A free triangle calculator solves all four modes — SSS, SAS, ASA/AAS — with the area, perimeter, and every angle worked out step by step. If the height still feels slippery, its guide to how to find the area of a triangle walks through the perpendicular-height rule with the 3-4-5 trap as the worked example.

One rule, three formulas: the height must be perpendicular to its base, the angles must add to 180°, and Heron's formula is the check that never lies. Master that, and any triangle — right, scalene, or equilateral — is just a few careful steps. Check your own triangles on a free triangle calculator and the trap stops working on you.