Area of a Triangle Revision Notes for Leaving Cert Mathematics

Area of a Triangle

Definition and formula

When you have a triangle with vertices at coordinates, you can calculate its area using a special coordinate geometry formula. This method is particularly useful when dealing with triangles plotted on coordinate axes.

Area of triangle = $\frac{1}{2}|x_1y_2 - x_2y_1|$

infoNote

The vertical lines around the expression indicate that you must take the absolute value of the result, ensuring the area is always positive.

Understanding the coordinate method

This formula works when one vertex of the triangle is at the origin (0,0) and the other two vertices are at points $(x_1, y_1)$ and $(x_2, y_2)$ . The method involves:

Cross multiplication: Multiply $x_1$ by $y_2$ and $x_2$ by $y_1$
Subtraction: Find the difference between these products
Absolute value: Take the positive value of the result
Half the result: Multiply by $\frac{1}{2}$ to get the final area

infoNote

This coordinate method is particularly powerful because it works directly with the position coordinates of the triangle's vertices, making it ideal for problems involving triangles on coordinate planes.

Step-by-step calculation method

Identify the coordinates of all three vertices
Check if one vertex is at the origin (0,0)
Label the other two vertices as $(x_1, y_1)$ and $(x_2, y_2)$
Substitute into the formula: $\frac{1}{2}|x_1y_2 - x_2y_1|$
Calculate the cross products and find their difference
Apply the absolute value to ensure a positive result
Multiply by $\frac{1}{2}$ to get the final area

Worked example 1

lightbulbExample

Worked Example: Finding Area with Origin Vertex

Find the area of the triangle with vertices (0,0), (-2,1) and (3,4).

Solution: One vertex is at the origin, so we can use the formula directly. Let $(x_1, y_1) = (-2, 1)$ and $(x_2, y_2) = (3, 4)$

Area = $\frac{1}{2}|x_1y_2 - x_2y_1|$ = $\frac{1}{2}|(-2)(4) - (3)(1)|$ = $\frac{1}{2}|-8 - 3|$ = $\frac{1}{2}|-11|$ = $\frac{1}{2} \times 11$ = 5.5 square units

Translation method for triangles not at the origin

When none of the vertices is at the origin, you must first translate the triangle so that one vertex moves to (0,0). This involves subtracting the same values from all coordinates.

infoNote

Translation is a powerful technique that allows us to apply the coordinate formula to any triangle, regardless of its position on the coordinate plane. The key insight is that translation doesn't change the area of the triangle.

Worked example 2

lightbulbExample

Worked Example: Triangle Translation Method

Find the area of the triangle with vertices (2,4), (-3,1) and (3,-5).

Solution: Since no vertex is at the origin, we translate by moving (2,4) to (0,0).

Translation: Subtract 2 from each x-coordinate and 4 from each y-coordinate.

$(2,4) \rightarrow (0,0)$
$(-3,1) \rightarrow (-5,-3)$
$(3,-5) \rightarrow (1,-9)$

Now apply the formula with $(x_1, y_1) = (-5, -3)$ and $(x_2, y_2) = (1, -9)$ :

Area = $\frac{1}{2}|x_1y_2 - x_2y_1|$ = $\frac{1}{2}|(-5)(-9) - (1)(-3)|$ = $\frac{1}{2}|45 + 3|$ = $\frac{1}{2}|48|$ = 24 square units

Key exam points

chatImportant

Essential Points for Exam Success:

Always check the absolute value - the area must be positive
Translation doesn't change the area - moving all vertices by the same amount preserves area
Choose any vertex to move to the origin - the result will be the same
Show all calculation steps clearly in your working
Double-check your arithmetic especially with negative numbers

Common exam traps

chatImportant

Avoid These Common Mistakes:

Forgetting to take the absolute value of the final calculation
Making sign errors when dealing with negative coordinates
Mixing up which coordinates to use as $(x_1, y_1)$ and $(x_2, y_2)$
Forgetting to multiply by $\frac{1}{2}$ at the end

Remember!

bookmarkSummary

Key Points to Remember:

The area formula is $\frac{1}{2}|x_1y_2 - x_2y_1|$ when one vertex is at the origin
Use translation to move any vertex to (0,0) if needed
Always take the absolute value to ensure a positive area result
Cross multiply coordinates: $x_1 \times y_2$ and $x_2 \times y_1$ , then subtract
The final step is always to multiply by $\frac{1}{2}$ to get the actual area

Area of a Triangle (Leaving Cert Mathematics): Revision Notes