The Area of a Triangle Revision Notes for VCE SSCE General Mathematics

The Area of a Triangle

Introduction

When calculating the area of a triangle, you need to choose the appropriate formula based on the information you have been given. There are three main methods:

Using base and height: When you know the base and perpendicular height
Using two sides and the included angle: When you know two sides and the angle between them
Using Heron's rule: When you know all three sides but no angles or heights

Understanding which method to use is essential for solving triangle area problems efficiently.

infoNote

The key to solving triangle area problems is identifying what information you have available. Each method is designed for a specific situation, so selecting the right formula from the start will save you time and effort.

Method 1: Area using base and height

The basic formula

The most fundamental formula for finding the area of a triangle is:

$\text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

Or more simply:

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

where:

$b$ represents the length of the base
$h$ represents the perpendicular height (the height measured at right angles to the base)

Understanding the formula

The diagram below illustrates why this formula works. A triangle fits inside a rectangle and occupies exactly half of its area. The rectangle has area $b \times h$ , so the triangle has area $\frac{1}{2} \times b \times h$ .

chatImportant

The height must always be perpendicular (at right angles) to the base. This perpendicular height can be drawn from any vertex to the opposite side, which becomes the base.

When to use this method

Use the formula $\text{Area} = \frac{1}{2} \times b \times h$ when:

You are given the base and the perpendicular height directly
You can calculate the perpendicular height from other given information

Worked example: Finding area using base and height

lightbulbExample

Worked Example: Finding Area Using Base and Height

Question: Find the area of the triangle shown to one decimal place.

Solution:

Step 1: Identify the given information

Base, $b = 7$ m
Height, $h = 3$ m

Step 2: Write the appropriate formula

Since we have the base and height, we use:

$\text{Area of triangle} = \frac{1}{2} \times b \times h$

Step 3: Substitute the values

$\text{Area} = \frac{1}{2} \times 7 \times 3$

Step 4: Calculate the result

$\text{Area} = \frac{1}{2} \times 21 = 10.5 \text{ m}^2$

Answer: The area of the triangle is 10.5 m²

Method 2: Area using two sides and an included angle

Deriving the trigonometric formula

When you know two sides of a triangle and the angle between them, you can use a trigonometric formula to find the area. This formula is derived from the basic base × height formula.

Consider triangle $ABD$ with sides $b$ and $c$ , and angle $A$ between them. If we draw the perpendicular height $h$ from vertex $B$ :

$\sin A = \frac{h}{c}$

Therefore:

$h = c \times \sin A$

Substituting this into the basic area formula:

$\text{Area} = \frac{1}{2} \times b \times h = \frac{1}{2} \times b \times c \times \sin A$

The three formula variations

Depending on which sides and angle you are given, there are three versions of this formula:

$\text{Area of a triangle} = \frac{1}{2}bc \sin A$

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

$\text{Area of a triangle} = \frac{1}{2}ab \sin C$

chatImportant

Important Pattern to Remember:

Each formula follows the same structure:

$\text{Area} = \frac{1}{2} \times (\text{product of two sides}) \times \sin(\text{angle between those two sides})$

The key is to identify which angle sits between which two sides. For example, angle A sits between sides b and c, so you use $\frac{1}{2}bc \sin A$ .

When to use this method

Use the formula $\text{Area} = \frac{1}{2}bc \sin A$ (or its variations) when:

You are given two sides and the angle between them (SAS - Side-Angle-Side)
The perpendicular height is not directly given
You need to use trigonometry to find the area

Worked example: Finding area using two sides and an angle

lightbulbExample

Worked Example: Finding Area Using Two Sides and an Angle

Question: Find the area of triangle $ABC$ where $b = 5$ cm, $c = 6$ cm, and angle $A = 135°$ . Give your answer to one decimal place.

Solution:

Step 1: Identify the given information

Side $b = 5$ cm
Side $c = 6$ cm
Angle $A = 135°$ (the angle between sides b and c)

Step 2: Choose the appropriate formula

Since we have two sides ( $b$ and $c$ ) and the angle between them ( $A$ ), we use:

$\text{Area of triangle} = \frac{1}{2}bc \sin A$

Step 3: Substitute the values

$\text{Area} = \frac{1}{2} \times 5 \times 6 \times \sin 135°$

Step 4: Calculate using a calculator

$\text{Area} = \frac{1}{2} \times 5 \times 6 \times 0.7071... = 10.606...$

Step 5: Round to one decimal place

$\text{Area} = 10.6 \text{ cm}^2$

Answer: The area of the triangle is 10.6 cm²

infoNote

Calculator Mode Reminder: Always make sure your calculator is in degree mode when working with angles measured in degrees, or in radian mode for angles in radians. Using the wrong mode is a common source of errors!

Method 3: Heron's rule

The formula

Heron's rule (also called Heron's formula) allows you to calculate the area of a triangle when you know the lengths of all three sides but don't have any angle measurements or the height.

$\text{Area of a triangle} = \sqrt{s(s-a)(s-b)(s-c)}$

where:

$s = \frac{1}{2}(a + b + c)$

Understanding the semi-perimeter

The variable $s$ is called the semi-perimeter because it equals half the perimeter of the triangle. It is calculated by adding all three side lengths together and dividing by 2:

$s = \frac{1}{2}(a + b + c)$

Once you have found $s$ , you subtract each side length from it to get $(s-a)$ , $(s-b)$ , and $(s-c)$ , then multiply these values together with $s$ and take the square root of the result.

infoNote

The semi-perimeter is a crucial intermediate step in Heron's formula. Always calculate and write down the value of $s$ before proceeding with the rest of the calculation. This helps prevent errors and makes your working easier to follow.

When to use this method

Use Heron's rule when:

You are given all three sides of the triangle (SSS - Side-Side-Side)
You don't know any angles
The perpendicular height is not given

Worked example: Finding area using Heron's formula

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Worked Example: Finding Area Using Heron's Formula

Question: The boundary fences of a farm form a triangle with sides of 6 km, 9 km, and 11 km. Find the area of the farm to the nearest square kilometre.

Solution:

Step 1: Identify the given information

Let:

$a = 6$ km
$b = 9$ km
$c = 11$ km

Step 2: Calculate the semi-perimeter

$s = \frac{1}{2}(a + b + c)$

$s = \frac{1}{2}(6 + 9 + 11) = \frac{1}{2}(26) = 13 \text{ km}$

Step 3: Write Heron's formula

$\text{Area of triangle} = \sqrt{s(s-a)(s-b)(s-c)}$

Step 4: Substitute the values

$\text{Area} = \sqrt{13(13-6)(13-9)(13-11)}$

$\text{Area} = \sqrt{13 \times 7 \times 4 \times 2}$

$\text{Area} = \sqrt{728}$

Step 5: Calculate using a calculator

$\text{Area} = 26.981... \text{ km}^2$

Step 6: Round to the nearest whole number

$\text{Area} = 27 \text{ km}^2$

Answer: The area of the farm is 27 km²

chatImportant

When using Heron's rule, always calculate s first and write it down before substituting into the main formula. This reduces the chance of making errors in your calculation and keeps your working organized.

Remember!

bookmarkSummary

Key Points to Remember:

Three methods for finding triangle area: Choose the method based on what information you have been given
Method 1 ( $\frac{1}{2} \times b \times h$ ): Use when you know the base and perpendicular height
Method 2 ( $\frac{1}{2}bc \sin A$ ): Use when you know two sides and the angle between them (SAS). Remember the pattern: half the product of two sides times the sine of the angle between them
Method 3 (Heron's rule): Use when you know all three sides (SSS) but no angles or heights. First calculate the semi-perimeter $s = \frac{1}{2}(a+b+c)$ , then use $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$
Always check your calculator mode: Make sure it's set to degrees when working with angles in degrees

The Area of a Triangle (VCE SSCE General Mathematics): Revision Notes