Area of Triangles and Parallelograms Revision Notes for Leaving Cert Mathematics

Area of Triangles and Parallelograms

Area of triangles

The area of a triangle measures the space enclosed within its three sides. Understanding how to calculate this area is fundamental in geometry and appears frequently in Leaving Cert exams.

Triangle area formula

The area of any triangle can be calculated using the formula:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{perpendicular height}$

Where:

Base is any side of the triangle
Perpendicular height is the distance from the base to the opposite vertex, measured at a 90° angle to the base

Key theorem about triangle area

chatImportant

An important principle to remember is that the area of a triangle remains the same regardless of which side you choose as the base. When you change the base, the corresponding perpendicular height changes accordingly, but the final area calculation stays constant.

This means you can use whichever base and height combination is most convenient for your calculation.

lightbulbExample

Worked Example: Triangle Area Calculation

Let's work through a comprehensive example to demonstrate the triangle area formula in practice.

Given: Triangle ABC where |BC| = 16 cm, |AB| = 12 cm, and |AD| = 10 cm Find: (i) Area of triangle ABC, (ii) Length |EC|

Solution:

(i) Finding the area:

Use the base |BC| = 16 cm and perpendicular height |AD| = 10 cm
Area = $\frac{1}{2} \times \text{base} \times \text{height}$
Area = $\frac{1}{2} \times 16 \times 10 = 80$ cm²

(ii) Finding |EC|:

We can also express the area using base |AB| and height |EC|
Area = $\frac{1}{2} \times |AB| \times |EC|$
$80 = \frac{1}{2} \times 12 \times |EC|$
$80 = 6|EC|$
$|EC| = 80 ÷ 6 = 13\frac{1}{3}$ cm

This example demonstrates how the same triangle area can be calculated using different base-height combinations.

Area of parallelograms

A parallelogram is a quadrilateral where opposite sides are parallel and equal in length. Understanding parallelogram area is crucial for many geometry problems.

Understanding parallelogram structure

When you draw a diagonal in a parallelogram, it creates two triangles that have a special relationship.

infoNote

The diagonal creates two triangles that are congruent (identical in size and shape). This means:

Both triangles have exactly the same area
The diagonal bisects (cuts in half) the total area of the parallelogram

Parallelogram area formula

Since a parallelogram consists of two identical triangles, we can derive its area formula:

$\text{Area of parallelogram} = \text{base} \times \text{perpendicular height}$

The perpendicular height is the shortest distance between the parallel sides, measured at a right angle to the base.

lightbulbExample

Worked Example: Parallelogram Area Calculation

Let's solve a practical parallelogram problem step by step.

Given: Parallelogram ABCD with base = 14 cm, side length = 8 cm, and we need to find the perpendicular height when |BC| = 9 cm

Find: (i) Area of parallelogram ABCD, (ii) Perpendicular height from A to |BC|

Solution:

(i) Finding the area:

Use base = 14 cm and perpendicular height = 8 cm
Area = base × height = $14 \times 8 = 112$ cm²

(ii) Finding the perpendicular height:

We can also calculate area using |BC| as the base
Area = $|BC| \times h = 9 \times h$
Since we know the area is 112 cm²:
$9h = 112$
$h = 112 ÷ 9 = 12\frac{2}{3}$ cm

Key formulas summary

Shape	Formula	Key Points
Triangle	$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$	Any side can be the base
Parallelogram	$\text{Area} = \text{base} \times \text{height}$	Height must be perpendicular to base

Common exam techniques and tips

Understanding the proper techniques for solving area problems will help you succeed in your exams.

infoNote

Essential Exam Tips

Always identify the perpendicular height - it's not necessarily the side length of the triangle or parallelogram
Use the most convenient base - choose the base that makes your calculation easiest
Check your units - ensure all measurements are in the same units before calculating
Show your working clearly - write down the formula, substitute values, then calculate
Double-check by using alternative base-height combinations when possible

bookmarkSummary

Key Points to Remember:

Triangle area = $\frac{1}{2} \times \text{base} \times \text{perpendicular height}$ - any side can serve as the base
Parallelogram area = $\text{base} \times \text{perpendicular height}$ - the diagonal divides it into two equal triangles
Perpendicular height is always measured at 90° to the chosen base, not along a slanted side
The same area can be calculated using different base-height combinations - use whichever is most convenient
Always include correct units (cm², m², etc.) in your final answer

Area of Triangles and Parallelograms (Leaving Cert Mathematics): Revision Notes