Hello Future Maths Whizzes!
Welcome to your study notes on Area! Have you ever wondered how much wrapping paper you need for a present, or how much paint you need for a wall? That is all about area! Area is the amount of flat space inside a 2D shape.
In Primary 5, we are going on an adventure to master the Area of a Triangle and discover how to find the area of composite figures made up of squares, rectangles, and triangles. It is like being a shape detective!
Let's break everything down into easy, fun steps. Let's get started!
Let's Remember: What is Area?
Area is the measure of the surface inside a 2D (flat) shape. We measure area in square units, such as square centimetres (\(\text{cm}^2\)) or square metres (\(\text{m}^2\)).
A Quick Look Back: Squares and Rectangles
Area of a Rectangle
To find the area of a rectangle, multiply its length by its width.
Example: A rectangle is 5 cm long and 3 cm wide. Its area is \(5\text{ cm} \times 3\text{ cm} = 15\text{ cm}^2\).
\(\text{Area of Rectangle} = \text{Length} \times \text{Width}\)
Area of a Square
A square has four equal sides.
Example: A square has sides of 4 cm. Its area is \(4\text{ cm} \times 4\text{ cm} = 16\text{ cm}^2\).
\(\text{Area of Square} = \text{Side} \times \text{Side}\)
Key Takeaway
Area is the space inside a shape, measured in square units.
The Super Important Idea: Base and Perpendicular Height
To find the area of a triangle, we need two key measurements: the base and the height.
1. Any side can be the base: The base does not have to be the bottom side! You can choose any of the 3 sides of a triangle to be its base.
2. The height must be perpendicular: The height is the straight line drawn from the opposite corner (vertex) to meet the chosen base at a right angle (\(90^\circ\)). This is called the perpendicular height.
Three Types of Triangles and Their Heights
1. Right-angled Triangle: The two sides that form the right angle are already perpendicular to each other! One side is the base, and the other side is the height.
2. Acute-angled Triangle: The perpendicular height drops straight down from the opposite corner and falls inside the triangle onto the base.
3. Obtuse-angled Triangle (Important!): The perpendicular height often falls outside the triangle! We extend the base with a dotted line so the height can meet it at a right angle.
Common Mistake to Avoid!
Never use a slanted side as the height unless it makes a right angle with the base! The height must always be perpendicular to the chosen base.
Key Takeaway
Every base has one matching perpendicular height. Always look for the right-angle symbol (\(\llcorner\)) to find the correct height!
Finding the Area of a Triangle
Did you know that every triangle is exactly half of a rectangle or square with the same base and height?
Because a triangle takes up half the space of that rectangle, its area is also halved!
Formula for Area of a Triangle:
\(\text{Area of a Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height}\)
(You can also think of it as: \(\text{Base} \times \text{Height} \div 2\))
Let's Try an Example!
Find the area of a triangle with a base of 8 cm and a perpendicular height of 5 cm.
Step 1: Write down the formula.
\(\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}\)
Step 2: Put in the base and height.
\(\text{Area} = \frac{1}{2} \times 8\text{ cm} \times 5\text{ cm}\)
Step 3: Calculate.
\(\text{Area} = 4\text{ cm} \times 5\text{ cm} = 20\text{ cm}^2\)
The area of the triangle is \(20\text{ cm}^2\).
Memory Aid
Always remember: "Multiply base and height, then divide by 2 to get it right!"
Composite Figures: Adding and Subtracting Areas
In P5, you will often find the area of composite figures (shapes joined together or overlapping).
Method 1: Adding Parts Together
If a figure is made up of a rectangle and a triangle joined together:
Step 1: Find the area of the rectangle: \(\text{Length} \times \text{Width}\).
Step 2: Find the area of the triangle: \(\frac{1}{2} \times \text{Base} \times \text{Height}\).
Step 3: Add the two areas together to find the total area.
Method 2: Finding Shaded / Unshaded Regions (Subtracting)
When a triangle is cut out of a rectangle or square, find the shaded area by subtracting:
Example: A shaded triangle is inside a rectangle of 10 cm by 6 cm. The unshaded triangle has a base of 10 cm and a height of 4 cm.
Step 1: Total area of rectangle = \(10\text{ cm} \times 6\text{ cm} = 60\text{ cm}^2\).
Step 2: Area of unshaded triangle = \(\frac{1}{2} \times 10\text{ cm} \times 4\text{ cm} = 20\text{ cm}^2\).
Step 3: Shaded Area = \(60\text{ cm}^2 - 20\text{ cm}^2 = 40\text{ cm}^2\).
Key Takeaway
For composite figures, break the problem down into simple shapes (squares, rectangles, triangles), calculate each area, and then add or subtract as required.