If you wish to calculate a triangle area with vertices, then this area of a triangle with coordinates calculator is the right tool for you! You can also use this calculator to find the perimeter of the triangle with vertices. In this article, you shall learn:

  • What is the formula for the area of a triangle with vertices?
  • How do you find the area of a triangle with coordinates?
  • How do you calculate the perimeter of triangle using points?
  • How do you determine whether three given points are collinear?

Formula for area of triangle with vertices

Triangle formed by three points A, B and C.
Triangle formed by three points A(x1,y1)A(x_1, y_1), B(x2,y2)B(x_2, y_2) and C(x3,y3)C(x_3, y_3).

The formula for the area of a triangle from its three vertices is given by:

Area=12x1(y2y3)+x2(y3y1)+x3(y1y2)\footnotesize \begin{align*} \text{Area} = \frac{1}{2} &\big\lvert x_1(y_2-y_3) + x_2(y_3-y_1) \\ &+ x_3(y_1-y_2) \big\rvert \end{align*}

where:

  • Area\text{Area} is the area of the triangle ABCABC;
  • (x1,y1)(x_1,y_1) are the coordinates of the vertex AA;
  • (x2,y2)(x_2,y_2) are the coordinates of the vertex BB; and
  • (x3,y3)(x_3,y_3) are the coordinates of the vertex CC;

This simple formula is handy for calculating the area of a triangle given 3 coordinates. Another way to express this same formula is by using a determinant. To calculate the triangle area from 3 points:

Area=12111x1x2x3y1y2y3\text{Area} = \frac{1}{2} \begin{vmatrix} 1 & 1 & 1\\ x_1 & x_2 & x_3 \\ y_1& y_2 & y_3 \end{vmatrix}

How do you find the area of a triangle with coordinates?

To calculate the area of a triangle with its vertices A(x1, y1), B(x2, y2), and C(x3, y3), follow these simple steps:

  1. Evaluate the absolute value of the expression |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|.
  2. Divide this value by two to get the area of the triangle.
  3. Verify this result using our area of a triangle with the coordinates calculator.

How do you calculate the perimeter of triangle using points?

To calculate and find the perimeter of a triangle with its vertices A(x1, y1), B(x2, y2), and C(x3, y3), follow these simple steps:

  1. Calculate the length of the side AB using the distance formula AB = √(x2 − x1)2 + (y2 − y1)2.
  2. Similarly, find the lengths of the sides BC and AC using the distance formula.
  3. Add the lengths of the three sides to obtain the triangle ABC's perimeter.
  4. Verify this result using our area of a triangle with the coordinates calculator.

How to use this area of triangle with coordinates calculator

This calculator can do two functions simultaneously:

  1. Calculate the area of a triangle from 3 points.
  2. Calculate (or find) the perimeter of a triangle with points.

Simply enter the coordinates of the triangle vertices, and this calculator will do the rest.

FAQ

How do you determine whether three points are collinear?

To determine whether any three points A(x1, y1), B(x2, y2), and C(x3, y3) are collinear, follow these steps:

  1. Evaluate the value of the expression |x1(y2−y3) + x2(y3−y1) + x3(y1−y2)|.
  2. If this value equals zero, the points are collinear. If this value is non-zero, the points are non-collinear.

What is the area of the triangle formed by A(1,2), B(-1,1), and C(0,5)?

3.5 units. To calculate this value yourself, follow these steps:

  1. Evaluate the absolute value of the expression |(1)×(1−5)+(−1)×(5−2)+(0)×(2−1)| = |−4−3+0| = 7.
  2. Divide this value by 2 to get 7/2 = 3.5.
  3. Verify this result using our area of a triangle with the coordinates calculator.
Krishna Nelaturu
Area of triangle with vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃)
triangle with vertices A, B and C
x₁
y₁
x₂
y₂
x₃
y₃
Area
Perimeter
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