Area of a Triangle

Summary

This math recipe will help you find the area of a triangle, coordinates of whose vertices are known.

Area of a Triangle

Skill Level

Easy

Time

Approx. 2 min

Ingredients

Coordinates of the three vertices: \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\)

Method

To find the area of \(\Delta ABC\), we’ll simply plug their coordinates into the formula below:

Area \( = \frac{1}{2}|x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3­(y_1 – y_2)|\)

Examples

Example 1 Find the area of the triangle, whose vertices are \(P(4, 6)\), \(Q(-2,2)\), and \(R(1, 1)\).

Solution The figure shows \(\Delta PQR\) on the XY plane.

Area of a Triangle

To find the area, we’ll use the above formula directly.

Area \(= \frac{1}{2}|4(2 – 1) –2(1 – 6) + 1(6 –2)|\)

\(= \frac{1}{2}|4(1) – 2(–5) + 1(4)|\)

\(= \frac{1}{2}|4 + 10 + 4|\)

\(= \frac{1}{2}|18|\)

\(= 9\ sq. units\)

Example 2 Find the area of the triangle, whose vertices are \(A(-1, 5)\), \(B(7, -3)\), and \(C(0, -2)\).

Solution The figure shows \(\Delta ABC\) on the XY plane.

Area of a Triangle

To find the area, we’ll again use the above formula.

Area \(= \frac{1}{2}|–1(–3 – (–2)) + 7(–2 – 5) + 0(5 – (–3))|\)

\(= \frac{1}{2}|(– 1)( –1) + 7(–7) + 0(8)|\)

\(= \frac{1}{2}|1 – 49 + 0|\)

\(= \frac{1}{2}|– 48|\)

\(= 24\ sq. units\)

That’s it for this recipe. Hope you found it helpful.

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