Area of a Triangle Formula: 5 Ways to Calculate

This calculator finds the area of a triangle from the inputs you choose. Enter the base and height, or pick a different method (three sides, two sides and an angle, or two angles and a side), and the calculator returns the area in square units. The first output is the area itself, and the same formula is explained below in words and symbols.

Formula

The most common formula is:

Area = ½ × base × height

In symbols: A = ½ × b × h

• A = area (square units, e.g., m², cm², ft²)
• b = base length (same linear unit as height)
• h = perpendicular height from the base to the opposite vertex (same linear unit)

The height must be measured at a right angle to the base. It is not the length of a slanted side unless the triangle is right-angled and that side is perpendicular to the base.

Worked Example

Using the default values from the calculator: base = 5 m, height = 3 m.

1. Multiply base by height: 5 × 3 = 15
2. Multiply by ½: 15 × 0.5 = 7.5

The area is 7.50 m². This matches the calculator output for the default inputs.

Other Formulas for Special Cases

Sometimes you don’t have the base and height. Here are other standard formulas:

• Heron’s formula (when you know all three sides a, b, c): first compute the semi-perimeter s = (a+b+c)/2, then Area = √(s(s-a)(s-b)(s-c)).
• Two sides and the included angle (SAS): Area = ½ × a × b × sin(C), where C is the angle between sides a and b.
• Two angles and a side (ASA or AAS): use the law of sines to find another side, then use the SAS formula. For example, if you know side a and angles B and C, then side b = a × sin(B)/sin(A), where A = 180° – B – C.
• Equilateral triangle with side s: Area = (√3/4) × s².

How to Choose the Right Formula

Your choice depends on what measurements you already have:

• If you have a base and its perpendicular height, use ½ × base × height.
• If you have all three side lengths, use Heron’s formula.
• If you have two sides and the angle between them, use the sine formula.
• If you have one side and two angles, use the law of sines first, then the sine formula.

For a right triangle, the two legs are the base and height, so the area is simply ½ × leg1 × leg2.

Common Mistakes and Misconceptions

• Using a slanted side as the height: The height must be perpendicular to the base. In an obtuse triangle, the perpendicular height may lie outside the triangle.
• Unit mix-ups: Convert all measurements to the same unit before calculating. If base is in meters and height in centimeters, convert one to the other first.
• Forgetting the ½: The area of a triangle is half the area of a rectangle with the same base and height. Skipping the ½ gives the rectangle’s area.
• Heron’s formula with invalid sides: The sum of any two sides must be greater than the third. If not, the triangle doesn’t exist and the formula gives an error.

Why the Formula Works

A triangle is exactly half of a parallelogram (or rectangle) that shares the same base and height. If you duplicate a triangle and rotate it 180°, the two copies form a parallelogram. The area of that parallelogram is base × height, so the triangle is half of that. This geometric relationship holds for all triangles, regardless of shape.

Limitations and Practical Notes

The formulas give exact area for a perfect mathematical triangle. In real-world measurements, accuracy depends on how precisely you measure the base, height, sides, or angles. For large areas like land plots, surveyors often use the sine formula because measuring angles is easier than measuring perpendicular heights. For very small triangles, rounding errors in side lengths can significantly affect the area, especially with Heron’s formula.

Frequently Asked Questions

How do I find the area if I only know the three sides?

Use Heron’s formula: compute the semi-perimeter s = (a+b+c)/2, then Area = √(s(s-a)(s-b)(s-c)). For example, sides 3, 4, 5 give s = 6, and area = √(6×3×2×1) = √36 = 6 square units.

What if the triangle is not right-angled?

The base-height formula works for any triangle, as long as the height is measured perpendicular to the base. For obtuse triangles, the height may fall outside the triangle, but the formula still works.

Can I use the formula if I only know two sides and the included angle?

Yes. Use Area = ½ × a × b × sin(C), where C is the angle between sides a and b. Make sure your calculator is in degree mode if the angle is in degrees.

What units is the area in?

The area is in square units of whatever linear unit you used. If base and height are in meters, area is in square meters (m²). If they are in feet, area is in square feet (ft²). Always use the same unit for both measurements.

How do I find the height if I know the area and the base?

Rearrange the formula: height = (2 × Area) / base. For example, if area is 10 m² and base is 4 m, height = (2×10)/4 = 5 m.

Why is the area half of base times height?

Because a triangle is half of a rectangle (or parallelogram) with the same base and height. Duplicate the triangle, rotate it 180°, and you get a parallelogram whose area is base × height.

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in b | Base | m | 5 | 0
in h | Height | m | 3 | 0
out area | Area | m² | 0.5 * b * h | 2