Triangle Area (Coordinates) Calculator

About Triangle Area (Coordinates) Calculator

The shoelace formula calculates area from coordinates. Also known as the surveyor's formula.

How to use this calculator

1. Enter the x and y coordinates for all three triangle vertices.
2. Check that each point is written as an ordered pair, with x first and y second.
3. Click calculate to get the triangle area instantly.
4. If needed, compare the step-by-step explanation to see how the result was found.

The formula explained

• \(A\) = area of the triangle
• \(x_1, y_1\) = coordinates of the first vertex
• \(x_2, y_2\) = coordinates of the second vertex
• \(x_3, y_3\) = coordinates of the third vertex

Step by step method

1. List the three vertices as ordered pairs, such as (x1, y1), (x2, y2), and (x3, y3).
2. Use the coordinate area formula to combine the x and y values in the correct order.
3. Take the absolute value, then divide by 2 to get the area.
4. Read the final answer in square units.

Worked example

Suppose you want the area of a triangle with vertices at (0, 0), (4, 0), and (0, 3).

1. Substitute the points into the formula: \(A = \frac{1}{2}|0(0-3) + 4(3-0) + 0(0-0)|\).
2. Simplify inside the bars: \(A = \frac{1}{2}|0 + 12 + 0|\).
3. Finish the calculation: \(A = \frac{1}{2}\cdot 12 = 6\).

Tips and common mistakes

• Write the coordinates in the correct order. Swapping x and y changes the result.
• Use all three vertices, even if the triangle looks simple on the graph.
• The answer is always a positive area, so the absolute value matters.
• If two points are the same or all three points lie on one line, the area will be 0.

Frequently asked questions