What is the area of a triangle?
The area of a triangle is the 2D space it covers, measured in square units (cm², m², sq inches, etc.). The most familiar formula is one of the first things taught in geometry:
A = ½ × base × height
Where the height is the perpendicular distance from the base to the opposite vertex — NOT the slant side length. For a right triangle, the two legs are conveniently the base and height. For other triangles, you may need to drop a perpendicular line first to find the height.
This calculator handles three different methods so you can solve for area no matter what you know:
- Base × height — the everyday classroom formula
- Heron's formula — when you know all three side lengths but not the height
- Two sides + included angle (SAS) — when you know two sides and the angle between them
Pick whichever matches the inputs you have.
Method 1: Base × Height (the classic)
A = ½ × b × h
Worked example: a triangle with base 10 and height 6.
- A = 0.5 × 10 × 6 = 30 square units
Why the ½? Because a triangle is exactly half a parallelogram (or rectangle, in the right-angle case). Imagine duplicating your triangle and flipping the copy to fit alongside the original — together they form a parallelogram with area = base × height. So the original triangle is half of that.
The trickiest part is identifying the height correctly: it's perpendicular to the base, not the slant. For an obtuse triangle (one angle > 90°), the height might fall outside the triangle itself — in that case you'd extend the base mentally to find where the perpendicular lands.
Method 2: Heron's formula (3 sides only)
If you have all three side lengths but no height, Heron's formula is the answer. It's beautiful in how it doesn't need any angles or perpendiculars:
First compute semi-perimeter: s = (a + b + c) / 2
Then area: A = √(s × (s − a) × (s − b) × (s − c))
Worked example: a 3-4-5 right triangle.
- s = (3 + 4 + 5) / 2 = 6
- A = √(6 × 3 × 2 × 1) = √36 = 6
Cross-check with method 1: for the 3-4-5 right triangle, the legs ARE the base (3) and height (4). A = 0.5 × 3 × 4 = 6. Same answer.
Heron's formula is named after Hero of Alexandria (~10–70 AD), but evidence suggests Archimedes knew it earlier. It's particularly useful for surveying — measuring the perimeter of a triangular plot of land is much easier than measuring perpendicular heights to interior points.
Validity check: not every set of three numbers forms a valid triangle. The triangle inequality says the sum of any two sides must exceed the third. So 3-4-5 works, 1-2-5 doesn't (1 + 2 < 5 — you couldn't even close the triangle with sides those lengths). The calculator checks this and warns you if your input is invalid.
Method 3: Two sides and the included angle (SAS)
A = ½ × a × b × sin(C)
Where a and b are two sides, and C is the angle BETWEEN them (the "included angle"). Critical: C must be the angle between a and b, not opposite either of them.
Worked example: a triangle with sides 5 and 6, angle 60° between them.
- A = 0.5 × 5 × 6 × sin(60°)
- = 15 × 0.866
- ≈ 12.99
This formula is useful in trigonometry-heavy contexts (physics, navigation, surveying). The intuition: a × sin(C) gives you the height of the triangle when b is treated as the base. So you're back to ½ × base × height, just with the height computed via trigonometry.
Special cases worth knowing
Equilateral triangle (all three sides equal to s):
A = (√3 / 4) × s² ≈ 0.433 × s²
For s = 6: A ≈ 0.433 × 36 ≈ 15.59. This is a derived shortcut from Heron's (try plugging in s_perim = 9 and a = b = c = 6 — same answer).
Right triangle (one angle is 90°): the two legs are the base and height. Area = ½ × leg₁ × leg₂. No need to compute perpendicular height separately. For a right triangle with legs 5 and 12, area = 30.
Isosceles triangle (two sides equal): if the two equal sides are length s and the base is b, height = √(s² − (b/2)²) by the Pythagorean theorem, and area = ½ × b × height.
Common mistakes
Using slant length as height. The "height" in the basic formula is the perpendicular distance from base to opposite vertex. Using the slant edge instead gives the wrong (larger) answer.
Wrong angle in SAS. The angle must be BETWEEN the two sides you're using. If you use the angle opposite one of the two sides, the formula gives nonsense. Make sure you're using the included angle.
Forgetting the ½. A common mistake is computing base × height without halving. That gives you the area of the surrounding rectangle, not the triangle.
Mismatched units. If base is in cm and height is in mm, the area is in cm × mm — not a real square unit. Convert to a single unit first.
Real-world uses
Roofing. Roof slopes are usually triangular; calculating shingle area requires triangle geometry.
Surveying. Plot of land described by three distance measurements (boundary corners) — Heron's formula gives the area without surveying any perpendicular.
Sailing. Sail areas are often computed via two sides and the included angle (SAS) since sails attach at corners.
Computer graphics. 3D models are tessellated into triangles; rendering and physics engines compute triangle areas constantly.
Construction. Triangular reinforcement panels, gables, dormers — triangular areas come up in many building contexts.
Related calculators
- For right-triangle math (finding the hypotenuse or a leg), use the Pythagorean Theorem Calculator.
- For straight-line distance between points (the building block of side lengths from coordinates), the Distance Formula Calculator.
- For circle math, the Circumference Calculator.
- For 3D volume calculations, the Volume of a Cylinder Calculator.
Frequently asked questions
What's the formula for the area of a triangle?
The most common is A = ½ × base × height, where height is perpendicular to the base. For other inputs: Heron's formula uses three sides; SAS uses two sides and the included angle.
How do I find area when I know all three sides?
Use Heron's formula. Compute the semi-perimeter s = (a + b + c) / 2, then A = √(s × (s−a) × (s−b) × (s−c)). For a 3-4-5 triangle: s = 6, A = √(6 × 3 × 2 × 1) = 6.
What's the area of an equilateral triangle?
A = (√3 / 4) × s², where s is the side length. For s = 6: A ≈ 15.59. Derived from Heron's formula but easier to remember as a shortcut.
How do I find the height if I know area and base?
Rearrange: height = (2 × area) / base. So a triangle with area 24 and base 8 has height 6.
What's the triangle inequality?
The sum of any two sides must exceed the third. So 3-4-5 works (3+4 > 5; 3+5 > 4; 4+5 > 3). 1-2-5 doesn't (1+2 < 5). The Heron's mode of this calculator validates the inequality and warns you if your input can't form a real triangle.
Is my input saved or sent anywhere?
No. The calculation runs entirely in your browser using JavaScript — nothing is sent to a server, logged, or stored.