Welcome to Area: Parallelograms, Trapeziums, and Polygons!
Hello mathematicians! In Primary 4, you learned how to find the area of simple flat shapes like squares and rectangles by counting squares or multiplying lengths. Now in Primary 5, we are taking our measurement skills to the next level. We will discover how to find the area of parallelograms, triangles, trapeziums, and even irregular polygons!
Don't worry if this seems a bit tricky at first. All of these shapes are closely related to the simple rectangle you already know. Once you see how they connect, finding their areas is like solving a fun geometric puzzle!
Quick Review from P4:
• Area is the amount of flat surface enclosed inside a 2-D boundary.
• Area is measured in square units, such as square centimetres (\(\text{cm}^2\)) and square metres (\(\text{m}^2\)).
• Area of a rectangle = \(\text{length} \times \text{width}\)
• Area of a square = \(\text{side} \times \text{side}\)
The Secret Key: Base and Corresponding Height
Before using any new formula, there is one super important concept you must master: the relationship between a base and its corresponding height.
What is a Base?
A base can be any chosen side of a shape that we use as our baseline for measurement.
What is a Corresponding Height?
The height (often written simply as \(h\)) is the perpendicular distance from the opposite vertex (or opposite parallel side) down to the base line. "Perpendicular" means it meets the base at a right angle (\(90^\circ\)).
The Golden Rule of Height: The height is NEVER a slanted side! It must always be straight and form a square corner (\(90^\circ\)) with the chosen base.
Key Takeaway: Whenever you see a shape, look for the right-angle symbol. The two lines meeting at the right angle give you the base and its matching height!
1. Area of a Parallelogram
How Does It Work? (The Cut-and-Slide Trick)
Imagine holding a paper parallelogram. If you cut off a right-angled triangle from one side along its vertical height and slide it across to the opposite side, what shape do you get? A perfect rectangle!
Because no paper was added or thrown away, the area of the parallelogram is exactly the same as the area of that rectangle:
• The length of the rectangle is equal to the base of the parallelogram.
• The width of the rectangle is equal to the height of the parallelogram.
The Formula
Area of a parallelogram = \(\text{base} \times \text{height}\)
In symbols: \(\text{Area} = b \times h\)
Worked Example
A parallelogram has a base of \(8\text{ cm}\). The perpendicular height drawn to this base is \(5\text{ cm}\). The slanted side is \(6\text{ cm}\). Find its area.
Step-by-step Solution:
Step 1: Identify the correct numbers. Base = \(8\text{ cm}\), Height = \(5\text{ cm}\). (Note: The slanted side of \(6\text{ cm}\) is extra information—do not use it!)
Step 2: Write the expression and calculate:
\(\text{Area} = 8 \times 5 = 40\)
Step 3: Write down the final statement with the correct unit:
The area of the parallelogram is \(40\text{ cm}^2\).
Common Trap to Avoid!
Never multiply the two adjacent side lengths together (e.g., \(8 \times 6\)). A parallelogram is "tilted", so multiplying two outer sides gives an answer that is too large. Always use the perpendicular height!
2. Area of a Triangle
Why Do We Divide by 2?
Take any triangle. Now make an exact twin of it (a congruent triangle), flip it upside down, and join their matching slanted edges. Together, they form a parallelogram!
Since two identical triangles make one whole parallelogram, just one triangle must be exactly half of that parallelogram. That is why we divide by \(2\)!
The Formula
Area of a triangle = \(\text{base} \times \text{height} \div 2\)
Or written as a fraction: \(\text{Area} = \frac{\text{base} \times \text{height}}{2}\)
Height in Different Types of Triangles
• Acute-angled triangle: The height drops straight down inside the triangle to the base.
• Right-angled triangle: The two sides forming the right angle are already perpendicular! One side is the base, and the other side is the height. You do not need to draw a separate height line.
• Obtuse-angled triangle: The height drops from the top vertex straight down to an extended base line outside the triangle. Even though the height sits outside, the base length is still just the actual side of the triangle (do not include the extended dotted line in your base length!).
Worked Example
A triangle has a base of \(10\text{ m}\) and a corresponding height of \(7\text{ m}\). Find its area.
Step-by-step Solution:
Step 1: Identify base = \(10\text{ m}\) and height = \(7\text{ m}\).
Step 2: Apply the formula:
\(\text{Area} = 10 \times 7 \div 2\)
\(\text{Area} = 70 \div 2 = 35\)
Step 3: State the answer:
The area of the triangle is \(35\text{ m}^2\).
Key Takeaway: Always remember the magical step for triangles: divide by 2!
3. Area of a Trapezium
A trapezium is a quadrilateral with exactly one pair of opposite parallel sides. We call these parallel sides the upper base and the lower base.
How the Formula is Discovered
Just like we did with triangles, take two identical trapeziums. Turn one upside down and place it next to the other so their slanted sides touch. Together, they form one large parallelogram!
Look closely at this large parallelogram:
• Its long base is made by combining the upper base and the lower base: \((\text{upper base} + \text{lower base})\).
• Its height is the perpendicular distance between the parallel lines: \(\text{height}\).
• The area of this big parallelogram is: \((\text{upper base} + \text{lower base}) \times \text{height}\).
• Because our original trapezium is only half of this big parallelogram, we divide by \(2\)!
The Formula
Area of a trapezium = \((\text{upper base} + \text{lower base}) \times \text{height} \div 2\)
Or written as: \(\text{Area} = \frac{(\text{upper base} + \text{lower base}) \times \text{height}}{2}\)
Memory Trick: Remember to put the bases in a "hug" (brackets) first! Add the parallel bases together before you multiply by the height.
Worked Example
A trapezium has parallel sides of length \(6\text{ cm}\) and \(10\text{ cm}\). The perpendicular distance between them is \(4\text{ cm}\). Find its area.
Step-by-step Solution:
Step 1: Identify upper base = \(6\text{ cm}\), lower base = \(10\text{ cm}\), height = \(4\text{ cm}\).
Step 2: Set up the calculation (do the brackets first!):
\(\text{Area} = (6 + 10) \times 4 \div 2\)
\(\text{Area} = 16 \times 4 \div 2\)
\(\text{Area} = 64 \div 2 = 32\)
Step 3: State the conclusion:
The area of the trapezium is \(32\text{ cm}^2\).
Common Mistake Alert!
If you forget the brackets and write \(6 + 10 \times 4 \div 2\), the order of operations would make you multiply \(10 \times 4\) first! That would give an incorrect answer of \(26\). Always calculate \((6 + 10)\) first!
4. Area of Polygons (Splitting and Filling)
In everyday life and examination questions, you will often meet composite shapes or irregular polygons. These are shapes made by putting together several straight-sided figures. There is no single formula for an irregular polygon, but you can easily find its area using one of two powerful strategies:
Strategy A: The Splitting Method (Dissection)
Cut the complicated polygon into simpler, friendly shapes that you already know how to calculate (rectangles, triangles, parallelograms, or trapeziums). Then, add their individual areas together.
Steps for the Splitting Method:
1. Draw dotted lines to split the polygon into non-overlapping shapes.
2. Find the missing side lengths for each smaller shape.
3. Calculate the area of each smaller shape separately.
4. Add all the areas together: \(\text{Total Area} = \text{Area}_1 + \text{Area}_2 + \dots\)
Strategy B: The Filling Method (Subtraction)
Sometimes it is much easier to draw a "box" (a large rectangle) around the whole polygon. Then, calculate the area of the large outer rectangle and subtract the empty corner pieces that do not belong to the shape.
Formula: \(\text{Area of Polygon} = \text{Area of Big Rectangle} - \text{Area of Extra Pieces}\)
Worked Example: Splitting an L-shaped Garden
A school garden plot is shaped like a pentagon that can be split into a rectangle of length \(12\text{ m}\) and width \(5\text{ m}\), and a triangle on top with a base of \(12\text{ m}\) and a height of \(4\text{ m}\). What is the total area of the garden?
Step-by-step Solution:
Step 1: Calculate the area of the bottom rectangle:
\(\text{Area of rectangle} = 12 \times 5 = 60\text{ m}^2\)
Step 2: Calculate the area of the top triangle:
\(\text{Area of triangle} = 12 \times 4 \div 2 = 48 \div 2 = 24\text{ m}^2\)
Step 3: Add the two areas together:
\(\text{Total area} = 60 + 24 = 84\)
Step 4: Write the conclusion:
The total area of the garden is \(84\text{ m}^2\).
Summary of Area Formulas
Keep this quick reference guide handy whenever you practice:
• Rectangle: \(\text{Area} = \text{length} \times \text{width}\)
• Square: \(\text{Area} = \text{side} \times \text{side}\)
• Parallelogram: \(\text{Area} = \text{base} \times \text{height}\)
• Triangle: \(\text{Area} = \text{base} \times \text{height} \div 2\)
• Trapezium: \(\text{Area} = (\text{upper base} + \text{lower base}) \times \text{height} \div 2\)
• Polygons: Split into familiar shapes and add, or box into a rectangle and subtract.
TSA Examination Tips: How to Score Full Marks
In Hong Kong primary assessments (including school exams and TSA application problems), questions are marked in three distinct parts:
1. Method Mark: You earn this by writing down the correct mathematical expression showing how the numbers are used with the formula.
2. Answer Mark: You earn this by correctly calculating the numerical value.
3. Presentation Mark: You earn this by showing neat, logical working and ending with a clear conclusion sentence that includes the correct unit (e.g., \(\text{cm}^2\) or \(\text{m}^2\)).
Your Pre-Submission Checklist:
• Did I choose the perpendicular height instead of a slanted side?
• Did I remember to divide by 2 for triangles and trapeziums?
• Did I add the two parallel sides in brackets first for a trapezium?
• Is my final unit written as an area unit with a small superscript 2 (\(\text{cm}^2\) or \(\text{m}^2\)) rather than a length unit (\(\text{cm}\) or \(\text{m}\))?
Keep practicing these steps, and you will master area calculations with confidence and ease!