medium difficulty6 min

Area of a Triangle Using Determinants

The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) can be found using a determinant formula.

In one line: Determinant gives double area; absolute value makes it area.

Understand it

In the coordinate formula, the determinant represents the signed double area of the triangle formed by the three points. The sign depends only on the order in which the vertices are taken, not on the actual size of the triangle, so the absolute value is necessary before halving. This method is especially useful in CBSE questions where points contain variables: if the determinant is zero, the three points are collinear; if an area is given, the same determinant equation can be used to find the unknown coordinate.

Example

For points (0,0), (4,0), and (0,3), area = 1/2 |0(0-3) + 4(3-0) + 0(0-0)| = 1/2 |12| = 6 square units.

Formula and rule

Area = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|.

Equivalent determinant form: Area = 1/2 |det([[x1,y1,1],[x2,y2,1],[x3,y3,1]])|.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1Find the area of triangle with vertices (1,2), (3,4), and (5,1).
2Area = 1/2 |1(4-1) + 3(1-2) + 5(2-4)| = 1/2 |3 - 3 - 10| = 1/2 × 10 = 5 square units.

Method you can reuse

1Write the three points in the formula in the same order.
2Substitute x and y values carefully.
3Simplify the bracket.
4Take absolute value.
5Multiply by 1/2 and add square units.

Visual tip

A coordinate-plane sketch helps show the three vertices and whether the triangle is non-degenerate or collinear.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What does the determinant value 0 imply in the area formula for three points?

Think first, then show answer

The area is 0, so the three points are collinear.

Common wrong answer

Omitting the factor 1/2 and giving double the correct area.

Don't mix this up

Forgetting the absolute value and reporting a negative area. For example, reversing the order of vertices may give -12 inside the modulus, but the area is still 6 square units, not -6.

Exam tip

If the final determinant is zero, write that the points are collinear instead of saying the triangle has a negative or invalid area.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Students may be asked to find the area, prove collinearity, or find a missing coordinate when area is given.

1
Enough for 1 mark
The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) can be found using a determinant formula.
2
Add for 2 marks
Example: For points (0,0), (4,0), and (0,3), area = 1/2 |0(0-3) + 4(3-0) + 0(0-0)| = 1/2 |12| = 6 square units.
3
Add for 3 marks
In the coordinate formula, the determinant represents the signed double area of the triangle formed by the three points.

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