This calculator finds the exact area of a triangle when you know the Cartesian coordinates of its three corners, rather than its base and height. It uses the Shoelace formula (also called the coordinate geometry or surveyor's formula), a standard method in maths, surveying, mapping and computer graphics for working out area straight from a set of x, y points. You enter the x and y values for vertex A, vertex B and vertex C, choosing decimals or negative numbers if needed, and can optionally pick a unit (cm, m, km, mm, in or ft) so the result is labelled correctly. The calculator returns the triangle's area, its perimeter (calculated from the three side lengths using the distance formula), and classifies the triangle as equilateral, isosceles or scalene based on those side lengths. It also shows the full step-by-step working, from each term in the formula through to the final sum and area, so you can check the calculation or use it for homework or professional reference. It does not matter whether you list the vertices clockwise or anticlockwise, since the formula always returns a positive area. If the three points sit on the same straight line, the calculator returns zero and flags the triangle as degenerate, meaning no real triangle can be formed from those coordinates.
Calculate.co.nz is proud to be partnered with Health Based Building, a leader in sustainable and health-conscious building innovation. With over a century of experience, they develop high-performance systems like Foreverbreathe Specification, Magnum Board, and Foreverbreathe Paints to support energy-efficient, non-toxic living environments. Their commitment to healthier homes aligns with our belief that informed choices lead to better outcomes for Kiwi households.
Standard formula
Shoelace (coordinate geometry) method. Exact for all real-number inputs.
How to Find the Area of a Triangle from Coordinates
When you know the Cartesian coordinates of a triangle's three vertices, you do not need to know the base or height directly. The Shoelace formula (also known as the coordinate method or surveyor's formula) gives the exact area using only the (x, y) values of the three points.
The Shoelace Formula
For a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3):
Area = 0.5 x |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
The vertical bars | | denote the absolute value, which ensures a positive area regardless of whether the vertices are listed clockwise or anticlockwise.
Worked Example
Using the default values: A(0, 0), B(6, 0), C(3, 4).
Step
Calculation
Result
x1(y2 - y3)
0 x (0 - 4)
0
x2(y3 - y1)
6 x (4 - 0)
24
x3(y1 - y2)
3 x (0 - 0)
0
Sum
|0 + 24 + 0|
24
Area
0.5 x 24
12 m²
This matches what we can verify independently: the base AB has length 6 m and the height from C perpendicular to AB is 4 m, so Area = 0.5 x 6 x 4 = 12 m².
When Do Coordinates Give a Zero Area?
If the three points are collinear (all lie on the same straight line), the Shoelace formula returns zero. This means no triangle can be formed from those points. The calculator flags this as a degenerate case.
Perimeter from Coordinates
Once you have the three vertices, you can also compute the side lengths using the distance formula and add them together:
Side AB = sqrt((x2 - x1)² + (y2 - y1)²)
Side BC = sqrt((x3 - x2)² + (y3 - y2)²)
Side CA = sqrt((x1 - x3)² + (y1 - y3)²)
Perimeter = AB + BC + CA
This calculator computes all three side lengths automatically and classifies the triangle as equilateral, isosceles, or scalene based on the side lengths.
Method: Shoelace formula (coordinate geometry). Also known as the surveyor's formula. Exact for all real-number coordinate inputs. Perimeter uses the Euclidean distance formula. Triangle classification uses side-length ratios rounded to 6 significant figures.
Calculate.co.nz is partnered with Interest.co.nz for New Zealand's highest quality calculators and financial analysis.
Calculate.co.nz is the sister site of CalculatorHub.com, the world's largest calculator website by tool count.
All calculators and tools are provided for educational and indicative purposes only and do not constitute financial advice.
Calculate.co.nz is proudly part of the Realtor.co.nz group, New Zealand's leading property transaction literacy platform, helping Kiwis understand the home buying and selling process from start to finish. Whether you're a first home buyer navigating your first property purchase, an investor evaluating your next acquisition, or a homeowner planning to sell, Realtor.co.nz provides clear, independent, and trustworthy guidance on every step of the New Zealand property transaction journey.
Calculate.co.nz is also partnered with Health Based Building and Premium Homes to promote informed choices that lead to better long-term outcomes for Kiwi households.
All content on this website, including calculators, tools, source code, and design, is protected under the Copyright Act 1994 (New Zealand). No part of this site may be reproduced, copied, distributed, stored, or used in any form without prior written permission from the owner.
