Triangle Area All Formulas & How to Use Them

This calculator works out the area of a triangle from the three side lengths you enter, using Heron’s formula, and returns the area in square units. It also gives you the semiperimeter so you can follow along with the calculation. Below you’ll find every standard triangle area formula, when to use each one, and common mistakes to avoid.

Formula

Heron’s formula gives the area of a triangle when you know all three side lengths:

Area = √[s(s − a)(s − b)(s − c)]

where a, b, and c are the side lengths, and s is the semiperimeter:

s = (a + b + c) / 2

All lengths are in the same unit (meters, feet, etc.), and the area comes out in that unit squared (m², ft², etc.). The semiperimeter is simply half the perimeter.

Worked Example

Using the default values from the calculator: a = 5 m, b = 6 m, c = 7 m.

1. Compute the semiperimeter: s = (5 + 6 + 7) / 2 = 18 / 2 = 9 m.
2. Subtract each side from s: (9 − 5) = 4, (9 − 6) = 3, (9 − 7) = 2.
3. Multiply: 9 × 4 × 3 × 2 = 216.
4. Take the square root: √216 ≈ 14.70 m².

The calculator shows the same result: area ≈ 14.70 m².

All the Standard Triangle Area Formulas

Depending on what information you already have, one of these formulas will be the easiest:

• Base and height: Area = ½ × base × height. This works for any triangle if you know the perpendicular height from the base to the opposite vertex.
• Two sides and the included angle (SAS): Area = ½ × a × b × sin(C), where C is the angle between sides a and b. Use this when you have two sides and the angle between them.
• Three sides (SSS): Heron’s formula above. Use this when you know all three sides but no height or angle.
• One side and two angles (ASA or AAS): Use the law of sines to find another side, then apply the SAS formula. For example, if you know side a and angles B and C, then b = a × sin(B) / sin(A), and area = ½ × a × b × sin(C).
• Equilateral triangle: Area = (√3 / 4) × side². This is a special case of both the base-height and SAS formulas.
• Coordinate geometry: If you have the vertices (x1,y1), (x2,y2), (x3,y3), use the shoelace formula: Area = ½ × |x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|.

How to Choose the Right Formula

Start by listing what you know about the triangle:

• If you have a base and the perpendicular height, use the simplest formula: ½ × base × height.
• If you have two sides and the angle between them, use the sine formula.
• If you have all three sides, use Heron’s formula.
• If you have two angles and one side, first find the missing side using the law of sines, then use the sine formula.
• If you have coordinates of the vertices, use the shoelace formula.

All formulas give the same area for the same triangle; the choice is about which measurements are easiest to obtain.

Common Mistakes and How to Avoid Them

• Mixing units: All side lengths must be in the same unit before calculating. Convert first, then compute.
• Using the wrong height: The height must be perpendicular to the base, not just any slanted side.
• Forgetting the half: The most common error is dropping the ½ in the base-height formula.
• Heron’s with invalid sides: If the sides don’t satisfy the triangle inequality (a + b > c for all combinations), the triangle doesn’t exist and the square root becomes negative. The calculator will show an error in that case.
• Rounding too early: Keep full precision during intermediate steps, then round the final answer.

Limits of Heron’s Formula

Heron’s formula is exact for any triangle in Euclidean geometry, but it has practical limits. It requires three side lengths that form a valid triangle. For very large or very small numbers, floating-point arithmetic can introduce tiny errors, but for everyday measurements the result is accurate. Also, Heron’s formula does not directly give the height or angles; if you need those, you must use other relations like the law of cosines.

Frequently Asked Questions

What is the simplest way to find the area of a triangle?

If you know the base and the perpendicular height, use Area = ½ × base × height. It is the most direct formula and works for any triangle, not just right triangles.

Can I use Heron’s formula for any triangle?

Yes, as long as you know all three side lengths and they form a valid triangle (each side is less than the sum of the other two). It works for scalene, isosceles, and equilateral triangles.

How do I find the area if I only know two sides and an angle?

Use the SAS formula: Area = ½ × a × b × sin(C), where C is the angle between the two known sides. Make sure your calculator is in the correct mode (degrees or radians) to match the angle unit.

What if I have the coordinates of the vertices?

Use the shoelace formula: Area = ½ × |x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|. This works for any triangle when you have the (x, y) coordinates.

Why does the calculator give an error for some side lengths?

Because those lengths don’t form a triangle. For example, sides of 1, 2, and 5 cannot meet to form a closed shape. The triangle inequality must hold for all three combinations.

Is there a formula for the area of a right triangle?

Yes, the base-height formula simplifies to Area = ½ × leg1 × leg2, since the two legs are perpendicular. You don’t need the hypotenuse for the area.

Related

• Chocolate Scent vs Caffeine for Workout Energy
• 7000mAh Batteries: What They Really Mean
• Best VPS With Full Root Access: Top Picks

in a | Side a | m | 5 | 0
in b | Side b | m | 6 | 0
in c | Side c | m | 7 | 0
out s | Semiperimeter | m | (a + b + c) / 2 | 2
out area | Area | m² | sqrt(s * (s - a) * (s - b) * (s - c)) | 2