Polygon Triangle Area Formula

This calculator works out the area of a triangle from its base and height, and it also computes the area using Heron’s formula when you enter the three side lengths. You get the area in square units matching your input, with a step-by-step breakdown of the calculation. The formula is straightforward: Area = ½ × base × height, and for side lengths, Heron’s formula gives the area without needing the height.

Formula

The most common formula for the area of a triangle is:

Area = ½ × base × height

where base is the length of one side of the triangle (any side can be chosen), and height is the perpendicular distance from that base to the opposite vertex, measured at a right angle to the base. The result is in square units, such as square meters (m²), square feet (ft²), or square centimeters (cm²), depending on the units used for base and height.

For the three-side version, Heron’s formula is:

Area = √(s(s − a)(s − b)(s − c))

where a, b, and c are the lengths of the three sides, and s is the semiperimeter, calculated as s = (a + b + c) / 2. Heron’s formula works for any triangle, as long as the side lengths satisfy the triangle inequality (each side is shorter than the sum of the other two).

Worked Example

Using the default values from the calculator: base = 6 m, height = 4 m, side a = 5 m, side b = 6 m, side c = 5 m.

For the base-height formula:

1. Multiply base by height: 6 × 4 = 24
2. Divide by 2: 24 ÷ 2 = 12
3. Area = 12 m²

For Heron’s formula:

1. Calculate semiperimeter: s = (5 + 6 + 5) / 2 = 16 / 2 = 8 m
2. Compute s − a = 8 − 5 = 3, s − b = 8 − 6 = 2, s − c = 8 − 5 = 3
3. Multiply: 8 × 3 × 2 × 3 = 144
4. Take square root: √144 = 12
5. Area = 12 m²

Both methods give the same result, 12 m², because the triangle with sides 5, 6, 5 has a height of 4 m relative to the base of 6 m (it is an isosceles triangle).

When to Use Each Formula

Choose the base-height formula when you know or can easily measure the perpendicular height. This is common in construction, geometry problems, and when working with right triangles or triangles drawn on a grid. If you only know the three side lengths, use Heron’s formula. It is especially useful for irregular triangles where measuring the height is impractical, such as land surveys or when the triangle is not drawn to scale.

For right triangles, you can also use the two legs as base and height directly. For equilateral triangles, you can derive the height from the side length using the Pythagorean theorem, but Heron’s formula works just as well without extra steps.

How to Read the Result

The area is always expressed in square units. If you enter base and height in meters, the area is in square meters; if you use feet, the area is in square feet. The calculator shows two decimal places by default, which is precise enough for most practical purposes. For very large or very small areas, you may want to round to a sensible number of significant figures. For example, a triangle with base 2.3 m and height 1.7 m gives an area of 1.955 m², which you might round to 1.96 m² or even 2.0 m² depending on the precision of your measurements.

Common Mistakes and Misconceptions

• Using a slanted height instead of perpendicular height: The height must be measured at a right angle to the base. Using the length of a slanted side will give a wrong area.
• Forgetting to divide by 2: The formula is ½ × base × height, not base × height. This is a frequent error.
• Unit mix-ups: If base is in meters and height is in centimeters, convert one to the other first. The area unit will be the square of the unit you use for both.
• Heron’s formula with invalid sides: If the side lengths do not satisfy the triangle inequality, the expression under the square root becomes negative, and the area is undefined. The calculator will show an error or a non-real number.
• Confusing base and height: Any side can be the base, but the height must correspond to that base. The area is the same regardless of which side you choose, as long as you use the correct height.

Limits of the Formulas

Both formulas work for any triangle in Euclidean geometry, but they have limitations. The base-height formula requires you to know the perpendicular height, which may be difficult to measure directly for obtuse triangles (where the height falls outside the triangle). Heron’s formula avoids that issue but requires precise side measurements; small errors in side lengths can lead to larger errors in the area, especially for nearly degenerate triangles (where one side is almost equal to the sum of the other two). For very large triangles on the Earth’s surface, spherical geometry applies, and these planar formulas are not accurate.

Also, the formulas assume a flat, two-dimensional triangle. If you are working with three-dimensional shapes, you may need other methods, such as cross products of vectors.

Alternatives to These Formulas

If you know two sides and the included angle, you can use the formula Area = ½ × a × b × sin(C), where C is the angle between sides a and b. This is useful in trigonometry and when angles are easier to measure than heights. Another alternative is the shoelace formula (or surveyor’s formula) for a triangle given by coordinates of its vertices in a plane: Area = ½ × |xâ‚�(yâ‚‚ − y₃) + xâ‚‚(y₃ − yâ‚�) + x₃(yâ‚� − yâ‚‚)|. For a triangle on a grid, you can also count the number of unit squares and half-squares inside it, though this is less precise.

Frequently Asked Questions

What if I only know the three side lengths?

Use Heron’s formula. Enter the three side lengths in the calculator, and it will compute the semiperimeter and then the area using the square root of s(s−a)(s−b)(s−c). Make sure the sides form a valid triangle (each side less than the sum of the other two).

Can I use any side as the base?

Yes, you can choose any side as the base, but you must use the height that is perpendicular to that side. The area will be the same regardless of which side you pick, as long as you use the correct corresponding height.

What is the difference between base and height?

The base is any side of the triangle you choose to measure from. The height is the perpendicular distance from that base to the opposite vertex (the corner not on the base). It is always measured at a 90-degree angle to the base, not along a slanted side.

How do I find the height if I only know the sides?

You can use Heron’s formula to find the area, then rearrange the base-height formula to solve for height: height = (2 × area) / base. For a specific base, you can also use trigonometry or the Pythagorean theorem if the triangle is right or isosceles.

Why does the calculator show the same area for both methods?

Because both formulas calculate the same true area of the triangle. The base-height method uses the base and perpendicular height, while Heron’s formula uses only the side lengths. If the inputs are consistent (the height matches the base and the sides form the same triangle), the results will match. In the default example, the triangle with sides 5, 6, 5 has a height of 4 m relative to the base of 6 m, so both give 12 m².

Can I use this formula for any type of triangle?

Yes, the base-height formula works for all triangles—acute, obtuse, right, equilateral, isosceles, or scalene—as long as you correctly identify the perpendicular height. Heron’s formula also works for any triangle with valid side lengths, without needing to know the height.

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in b | Base | m | 6 | 0
in h | Height | m | 4 | 0
in a | Side a | m | 5 | 0
in c | Side c | m | 5 | 0
out area_bh | Area (base×height) | m² | 0.5 * b * h | 2
out semiperim | Semiperimeter s | m | (a + b + c) / 2 | 2
out area_heron | Area (Heron) | m² | sqrt(semiperim * (semiperim - a) * (semiperim - b) * (semiperim - c)) | 2