area of a triangle calculator

How to find the area of a triangle

The area of a triangle tells you how much space is inside its three sides. It is one of the most common geometry calculations used in school math, engineering, architecture, construction, surveying, and computer graphics.

This calculator lets you compute triangle area in three practical ways: using base and height, using all three sides with Heron's Formula, or using three coordinate points on a graph.

Triangle area formulas

1) Base and height formula

If you know the base and the perpendicular height:

Area = (base × height) / 2

• Fastest method when height is known.
• Works for right triangles, isosceles triangles, and scalene triangles.
• Make sure height is measured at a 90° angle to the base.

2) Heron's formula (three sides)

If you know side lengths a, b, and c, first calculate the semiperimeter:

s = (a + b + c) / 2

Then:

Area = √(s(s-a)(s-b)(s-c))

• Useful when no height is given.
• Requires valid triangle side lengths (triangle inequality must hold).
• Great for survey and measurement problems.

3) Coordinates formula

If triangle vertices are at points (x1, y1), (x2, y2), and (x3, y3):

Area = |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| / 2

• Best for coordinate geometry and graph-based problems.
• Works even with negative coordinates.
• If result is 0, points are collinear (not a triangle).

Step-by-step example calculations

Example A: base and height

Base = 12 cm, Height = 7 cm

Area = (12 × 7) / 2 = 84 / 2 = 42 cm²

Example B: three sides

a = 5 m, b = 6 m, c = 7 m

s = (5 + 6 + 7)/2 = 9

Area = √(9 × 4 × 3 × 2) = √216 ≈ 14.697 m²

Example C: coordinates

Points: (0,0), (6,0), (3,4)

Area = |0(0-4) + 6(4-0) + 3(0-0)|/2 = |24|/2 = 12 square units

Common mistakes to avoid

• Using slanted side length as “height” (height must be perpendicular).
• Mixing units (for example, base in meters and height in centimeters).
• Entering invalid side lengths that cannot form a triangle.
• For coordinates, accidentally swapping x and y values.

Quick FAQ

Can I use this for right triangles?

Yes. Use base-height, or use side lengths if you know all three sides.

What unit is the final answer in?

Area is always in square units. If inputs are in centimeters, output is in cm²; if inputs are in meters, output is in m².

Why does my Heron's formula result fail?

The three sides likely violate triangle inequality (for example, 2, 3, 10 cannot make a triangle).

Final thoughts

Whether you are solving homework, preparing for exams, or handling real-world design work, an area of a triangle calculator saves time and reduces errors. Use the method that matches your known values, and always double-check units.