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's all about area! Area is just a fancy word for the amount of space inside a flat shape.

In this chapter, we're going on an adventure to learn how to measure the area of some cool shapes like parallelograms, triangles, and trapeziums. We'll even learn the secret to finding the area of almost ANY shape with straight sides. It's like being a shape detective!

Don't worry if some of these words are new. We'll break everything down into easy, fun steps. Let's get started!


Let's Remember: What is Area?

Before we learn about new shapes, let's do a quick review. Area is the measure of the surface inside a 2D (flat) shape. Think of it like the grass inside a football pitch or the chocolate spread on a slice of bread.

We measure area in square units, like square centimetres (cm²) or square metres (m²).

A Quick Look Back: Squares and Rectangles

You might remember these two shapes. They are the foundation for everything we will learn next!

Area of a Rectangle
To find the area of a rectangle, you just multiply its length by its width.
Example: A rectangle is 5 cm long and 3 cm wide. Its area is 5 cm × 3 cm = 15 cm². $$Area = Length \times Width$$

Area of a Square
A square is even easier because all its sides are the same length.
Example: A square has sides of 4 cm. Its area is 4 cm × 4 cm = 16 cm². $$Area = Side \times Side$$

Key Takeaway

Area is the space inside a shape. For rectangles and squares, you find it by multiplying the side lengths.


The Super Important Idea: Base and Height

For our new shapes, we need to understand two very important ideas: the base and the height.

The base is usually the side that the shape is "sitting" on. It's like the floor.

The height is how tall the shape is. But be careful! The height is always a straight line from the base to the very top. This special straight line is called perpendicular, which means it makes a perfect right angle (like the corner of a square) with the base.

Why is this so important?

Imagine measuring how tall you are. You stand up straight! You don't lean over. Shapes are the same. We must measure their height straight up.

Look at this triangle. The base is the bottom line. The red dotted line is the height. The slanted side is NOT the height!

Common Mistake to Avoid!

Many students accidentally use the slanted side of a shape as its height. Remember the rule: the height must be straight up and make a right angle with the base. Don't fall into the trap!

Key Takeaway

The base is the bottom side, and the height is the straight-up (perpendicular) distance from the base to the top. It is NOT the slanted side.


Shape #1: The Parallelogram

A parallelogram looks like a rectangle that has been pushed over. It has two pairs of parallel sides.

The Magic Trick to Find its Area

How do we find the area of this slanted shape? With a little magic trick! Imagine we cut a triangle off one side of the parallelogram...

...and move it over to the other side. What do we get? A perfect rectangle!

We already know how to find the area of a rectangle (Length × Width). For our new rectangle, the length is the base of the parallelogram, and the width is the height of the parallelogram.

So, the formula is super simple!

Formula for a Parallelogram:

$$Area = Base \times Height$$

Let's Try an Example!

Let's find the area of a parallelogram with a base of 10 cm and a height of 6 cm. (It also has a slanted side of 7 cm, but that's a trick! We don't need it.)

Step 1: Write down the formula.
Area = Base × Height

Step 2: Put in the numbers for the base and the REAL height.
Area = 10 cm × 6 cm

Step 3: Calculate the answer.
Area = 60 cm²

Easy! The area is 60 cm².

Did you know?

Rectangles and squares are actually special types of parallelograms! They are just parallelograms with perfect right angles.

Key Takeaway

The area of a parallelogram is found by multiplying its base by its perpendicular height. Just ignore the slanted sides!


Shape #2: The Triangle

Everyone knows what a triangle looks like! But finding its area has a secret. The secret is that every triangle is exactly half of a parallelogram!

If you take any triangle and put an identical, upside-down copy of it next to it, you will always make a parallelogram. Since the triangle is half of the parallelogram, its area must be half too!

Formula for a Triangle:

$$Area = (Base \times Height) \div 2$$

You might also see it written like this, which means the same thing:

$$Area = \frac{1}{2} \times Base \times Height$$

Let's Try an Example!

Let's find the area of a triangle with a base of 8 cm and a height of 5 cm.

Step 1: Write down the formula.
Area = (Base × Height) ÷ 2

Step 2: Put in the numbers for the base and height.
Area = (8 cm × 5 cm) ÷ 2

Step 3: Multiply the base and height first.
Area = 40 ÷ 2

Step 4: Now, divide by 2.
Area = 20 cm²

The area is 20 cm². You've got this!

Memory Aid

Just remember: "A triangle is half a pair, so divide by two with care!" This will help you remember the `÷ 2` part of the formula.

Key Takeaway

The area of a triangle is half of its base times its height. The most important step is to remember to divide by 2!


Shape #3: The Trapezium

A trapezium (sometimes called a trapezoid) is a four-sided shape that has one pair of parallel sides. Think of it as a triangle with its top chopped off!

The two parallel sides are both called bases. Let's call them 'a' (the top) and 'b' (the bottom).

The Teamwork Formula

This formula seems long, but it's all about teamwork. Just like we did with the triangle, we can see that two identical trapeziums fit together to make one big parallelogram.

The height of this big parallelogram is the same as the trapezium's height. But what is its base? The base is the two parallel sides of the trapezium added together (a + b)!

Since the trapezium is half of this giant parallelogram, we get our formula:

Formula for a Trapezium:

$$Area = (a + b) \times Height \div 2$$

In words, that's: (Add the parallel sides together) × Height, then ÷ 2.

Let's Try an Example!

Find the area of a trapezium where the parallel sides are 6 cm and 10 cm, and the height is 5 cm.

Step 1: Add the parallel sides (a + b).
6 cm + 10 cm = 16 cm

Step 2: Multiply by the height.
16 cm × 5 cm = 80 cm²

Step 3: Divide by 2.
80 cm² ÷ 2 = 40 cm²

The area of the trapezium is 40 cm². Great job!

Key Takeaway

To find the area of a trapezium, you add the two parallel sides first, then multiply by the height, and finally, divide by 2.


Super Challenge: Area of Any Polygon!

What if you get a really weird shape, like the one for a house floor plan? This is where your detective skills come in! A polygon is any shape with straight sides.

The secret to finding the area of a complex polygon is simple: Cut it up into shapes you already know!

You can split almost any polygon into a mix of squares, rectangles, triangles, and trapeziums.

Example: The L-Shape

Let's find the area of this L-shaped polygon.

Step 1: Cut it up! We can draw a dotted line to split it into two rectangles. Let's call them Shape A and Shape B.

Step 2: Find the area of Shape A. Let's say it's a rectangle that is 4 cm by 3 cm.
Area of A = 4 cm × 3 cm = 12 cm²

Step 3: Find the area of Shape B. Let's say it's a rectangle that is 5 cm by 2 cm.
Area of B = 5 cm × 2 cm = 10 cm²

Step 4: Add them together! The total area is the sum of the parts.
Total Area = Area of A + Area of B = 12 cm² + 10 cm² = 22 cm²

The total area is 22 cm². See? You just solved a complex problem by breaking it into smaller, easy ones!

Key Takeaway

To find the area of a complex polygon, divide it into simpler shapes you know (like rectangles and triangles), find the area of each part, and then add them all together for the final answer.