Perimeter of a Triangle with Vertices Calculator
Enter the x and y coordinates of three points A, B, and C to get the full picture of your triangle: individual side lengths, perimeter, area, all three interior angles, and the triangle type by sides and by angles. Every step of the working is shown so you can follow the math from coordinates to final answer.
Formula
Worked example
For A(0,0), B(5,0), C(2.5,4): AB = sqrt(25) = 5, BC = sqrt(6.25+16) = sqrt(22.25) = 4.717, CA = sqrt(6.25+16) = 4.717. Perimeter = 5 + 4.717 + 4.717 = 14.434. Area = 0.5 |0(0-4) + 5(4-0) + 2.5(0-0)| = 0.5 |20| = 10. Triangle is isosceles and acute.
What this calculator does
Given the Cartesian coordinates of three vertices A, B, and C, this tool computes the full set of geometric properties of the triangle they define. It uses the distance formula to find each side length, the shoelace formula to find the area, and the law of cosines to derive all three interior angles. It then classifies the triangle by its sides (equilateral, isosceles, or scalene) and by its angles (acute, right, or obtuse). All six steps of the working are shown beneath the results so you can follow or verify each calculation.
How to find the perimeter from coordinates
The key tool is the distance formula, which is derived directly from the Pythagorean theorem. For two points (x1, y1) and (x2, y2), the distance between them is the square root of (x2 minus x1) squared plus (y2 minus y1) squared. You apply this three times, once for each side of the triangle (AB, BC, and CA), then add the three results together. That sum is the perimeter. For example, for A(0,0), B(5,0), and C(2.5,4): side AB = sqrt((5-0)^2 + (0-0)^2) = 5, side BC = sqrt((2.5-5)^2 + (4-0)^2) = sqrt(6.25 + 16) = 4.717, side CA = sqrt((0-2.5)^2 + (0-4)^2) = 4.717. Perimeter = 5 + 4.717 + 4.717 = 14.434 units.
Area and interior angles from coordinates
The area is found with the shoelace (or cross-product) formula: Area = 0.5 times the absolute value of x1(y2 minus y3) plus x2(y3 minus y1) plus x3(y1 minus y2). This works for any non-degenerate triangle regardless of orientation. The interior angles are found with the law of cosines: the angle at each vertex equals the inverse cosine of (a squared plus b squared minus c squared) divided by (2 times a times b), where a and b are the two sides meeting at that vertex and c is the opposite side. All three angles must sum to exactly 180 degrees.
Triangle types and what they tell you
Knowing whether a triangle is equilateral, isosceles, or scalene, and whether it is acute, right, or obtuse, gives you immediate insight into its geometry. An equilateral triangle has all sides equal and all angles 60 degrees, the most symmetrical case. An isosceles triangle has two equal sides and two equal base angles. A scalene triangle has no equal sides or angles. For angle classification, an acute triangle has all angles below 90 degrees, a right triangle has one angle exactly 90 degrees (and a hypotenuse which is the longest side), and an obtuse triangle has one angle above 90 degrees. In a right triangle, the Pythagorean theorem holds exactly: a squared plus b squared equals c squared, where c is the hypotenuse.
Triangle classification quick reference
| Category | Type | Criteria |
|---|---|---|
| By sides | Equilateral | All three sides equal; all angles 60° |
| By sides | Isosceles | Exactly two sides equal; two base angles equal |
| By sides | Scalene | No two sides equal; no two angles equal |
| By angles | Acute | All interior angles < 90° |
| By angles | Right | One interior angle = 90° (has a hypotenuse) |
| By angles | Obtuse | One interior angle > 90° |
Classification criteria by side lengths and by interior angles.
Frequently asked questions
What is the formula for the perimeter of a triangle from its vertices?
You use the distance formula on each pair of vertices to get the three side lengths, then add them. If the vertices are A(x1,y1), B(x2,y2), and C(x3,y3), then: AB = sqrt((x2-x1)^2 + (y2-y1)^2), BC = sqrt((x3-x2)^2 + (y3-y2)^2), CA = sqrt((x1-x3)^2 + (y1-y3)^2), and Perimeter = AB + BC + CA.
How do you find the area of a triangle given its vertices?
Use the shoelace (determinant) formula: Area = 0.5 x |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. The absolute value ensures a positive area regardless of the order the vertices are listed. This is equivalent to taking half the magnitude of the cross product of two edge vectors.
How do you calculate interior angles from vertex coordinates?
First compute the three side lengths using the distance formula. Then apply the law of cosines at each vertex: the angle at A = arccos((AB^2 + CA^2 - BC^2) / (2 x AB x CA)), and similarly for B and C by rotating which side is opposite. All three results must sum to 180 degrees.
How can I tell if a triangle formed by coordinates is a right triangle?
Check whether the square of the longest side equals the sum of squares of the other two sides (Pythagorean theorem). Alternatively, this calculator computes all three angles directly: if any angle is exactly 90 degrees (within floating-point precision), the triangle is a right triangle. The side opposite the 90-degree angle is the hypotenuse.
What does it mean if the calculator returns no result?
If the three points you entered are collinear (all on the same straight line), they do not form a triangle at all and the area is zero. Similarly, if any two vertices share the same coordinates, one side has length zero. In both cases the calculator returns a blank result because there is no valid triangle to measure.