find the area of a triangle calculator

How to Find the Area of a Triangle

The area of a triangle tells you how much surface the triangle covers. It is always expressed in square units, such as square inches, square meters, or square feet. Whether you're solving geometry homework, working in construction, or building a design project, triangle area is one of the most useful formulas to know.

Most Common Formula: Base and Height

If you know the base and the perpendicular height, the area formula is:

Area = (1/2) × base × height

• The base can be any side of the triangle.
• The height must be measured at a right angle (90°) to that base.
• The result is always in square units.

When You Know All Three Sides

If you have side lengths a, b, and c, use Heron's formula:

• Compute the semi-perimeter: s = (a + b + c) / 2
• Then compute area: A = √(s(s-a)(s-b)(s-c))

This method is great when no height is given. Just make sure the three sides can actually form a triangle.

When You Know the Coordinates

For points (x1, y1), (x2, y2), and (x3, y3), the coordinate geometry formula is:

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

This is especially useful for graphing, CAD tools, and math problems involving the coordinate plane.

Worked Examples

Example 1: Base and Height

Base = 12, height = 5
Area = (1/2) × 12 × 5 = 30
Final answer: 30 square units

Example 2: Three Sides

a = 6, b = 8, c = 10
s = (6 + 8 + 10) / 2 = 12
Area = √(12 × 6 × 4 × 2) = √576 = 24
Final answer: 24 square units

Example 3: Coordinates

(0,0), (4,0), (0,3)
Area = |0(0-3) + 4(3-0) + 0(0-0)| / 2 = |12| / 2 = 6
Final answer: 6 square units

Common Mistakes to Avoid

• Using a slanted side as height without a 90° angle to the base.
• Forgetting to divide by 2 in the base-height formula.
• Using side lengths that fail the triangle inequality in Heron's formula.
• Mixing units (for example, meters and centimeters) without converting first.

Quick FAQ

Can the area of a triangle be zero?

Yes. If all three points are on the same line (collinear), the triangle has no enclosed surface, so area = 0.

What units should I use?

Use the same unit for every measurement, and report the answer in square units. For example, if sides are in cm, area is cm².

Which method should I pick in this calculator?

Choose the method that matches your known values:

• Base and Height: fastest when height is provided.
• Three Sides: best when only side lengths are known.
• Coordinates: ideal for graph and geometry-plane problems.

Use the calculator above anytime you need a quick and accurate triangle area result.