The Area of a Triangle Revision Notes for Leaving Cert Mathematics

The Area of a Triangle

Understanding triangle notation

When working with triangles in trigonometry, we follow a standard labelling system that makes calculations easier to follow. This systematic approach is fundamental to solving trigonometric problems effectively.

In any triangle, we use capital letters (A, B, C) to represent the angles at each vertex, and lowercase letters (a, b, c) to represent the sides opposite to those angles.

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Triangle Labelling Convention:

Angle A is opposite side a
Angle B is opposite side b
Angle C is opposite side c

This consistent notation is essential for applying trigonometric formulas correctly in exam questions.

Deriving the trigonometric area formula

The basic formula for the area of any triangle is Area = ½ × base × height. However, when we know two sides and the included angle, we can use trigonometry to find the area more directly.

Starting with the basic area formula where the base is side $a$ and the height is $h$ :

$\text{Area} = \frac{1}{2} \times a \times h$

From the right triangle formed by the height, we know that:

$\frac{h}{c} = \sin B$
Therefore: $h = c \times \sin B$

Substituting this back into our area formula:

$\text{Area} = \frac{1}{2} \times a \times h \text{ becomes } \text{Area} = \frac{1}{2} \times a \times c \times \sin B$

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Trigonometric Area Formula:

$\text{Area} = \frac{1}{2}ac \sin B$

This formula allows us to find the area when we know two sides and the included angle.

The general trigonometric area formula

The formula can be written using any two sides and their included angle:

$\text{Area} = \frac{1}{2}ab \sin C = \frac{1}{2}ac \sin B = \frac{1}{2}bc \sin A$

In words: The area equals half the product of any two sides multiplied by the sine of the angle between them.

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Critical Point: The angle must be between the two sides you're using in the calculation. This is the most common error students make when applying this formula.

Worked examples

The following examples demonstrate how to apply the trigonometric area formula in different types of problems.

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Worked Example 1: Finding area when given two sides and included angle

Find the area of a triangle with sides 7 and 8, and an included angle of 46°.

Solution: Using the formula: $\text{Area} = \frac{1}{2} \times 7 \times 8 \times \sin 46°$

Step-by-step calculation:

$\text{Area} = \frac{1}{2} \times 7 \times 8 \times \sin 46°$
$\text{Area} = 0.5 \times 7 \times 8 \times \sin 46°$
Using a calculator: $0.5 \times 7 \times 8 \times 0.7193... = 20.14$

Answer: 20.1 square units (to 1 decimal place)

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Worked Example 2: Finding an angle when given area and two sides

If a triangle has an area of 40 cm², and two sides of lengths 10 cm and 14 cm, find the included angle A.

Solution: Start with the area formula and rearrange to solve for the angle:

Start with: $\text{Area} = \frac{1}{2} \times 10 \times 14 \times \sin A$
Substitute known values: $40 = \frac{1}{2} \times 10 \times 14 \times \sin A$
Simplify: $40 = 70 \times \sin A$
Rearrange: $\sin A = \frac{40}{70} = \frac{4}{7}$
Find the angle: $A = \sin^{-1}\left(\frac{4}{7}\right)$
Using a calculator: $A = \sin^{-1}(40 \div 70) = 34.85°$

Answer: A = 35° (to the nearest degree)

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Worked Example 3: Working with given sine values

If $\sin θ = 0.7$ , find the area of a triangle with sides 18 cm and 22 cm where θ is the included angle.

Solution:

$\text{Area} = \frac{1}{2} \times 18 \times 22 \times \sin θ$
$\text{Area} = \frac{1}{2} \times 18 \times 22 \times 0.7$
$\text{Area} = 9 \times 22 \times 0.7 = 138.6$

Answer: 139 cm² (to the nearest cm²)

Key exam techniques

Mastering these techniques will help you tackle trigonometric area problems with confidence in your exams.

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Calculator Usage Tips:

When finding an angle, use the inverse sine function: SHIFT + sin + (value) + =
Always check your calculator is in degree mode for Leaving Cert questions
Round final answers as requested in the question

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Common Problem Types:

Finding area: Use $\text{Area} = \frac{1}{2}ab \sin C$ directly
Finding an angle: Rearrange to $\sin C = \frac{2 \times \text{Area}}{ab}$ , then use $\sin^{-1}$
Finding a side: Rearrange the area formula to solve for the unknown side

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Exam Success Tips:

Always identify which two sides and which angle you're working with
Check that the angle is between the two given sides
Show your working step-by-step for full marks
Include units in your final answer
Round to the accuracy requested in the question

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Key Points to Remember:

The angle must be between the two sides you're using in the formula
Area = ½ab sin C where C is the angle between sides a and b
Capital letters represent angles, lowercase letters represent opposite sides
When finding an angle from area and sides, use sin⁻¹ on your calculator
Always check your calculator is in degree mode for exam questions

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