Welcome back! In our last lesson, we focused on the rhombus and learned a specific formula to calculate its area using the diagonals. Today, we'll continue our journey through special quadrilaterals by examining the trapezium.
This lesson is all about mastering the area of a trapezium. You will learn to identify its key features—the parallel sides and the height—and understand and apply the formula to calculate its area. This is the final quadrilateral in this module, and it will complete your foundational toolkit for solving area problems involving four-sided shapes.
1. What is a Trapezium?
A trapezium (also called a trapezoid in some parts of the world) is a quadrilateral with one key feature: it has one pair of parallel sides. These parallel sides are called the bases of the trapezium. The other two sides, which are not parallel, are called the legs.
To find the area, we need to know the lengths of the two parallel sides (let's call them 'a' and 'b') and the perpendicular distance between them, which we call the height ('h').
The formula to calculate the area of a trapezium is:
In words, you add the lengths of the parallel sides, divide the sum by 2, and then multiply by the height.
Let's read a quick summary to solidify this definition and formula.
Area of Trapezium - Formula, Definition, Derivation, Examples
This article from Cuemath provides a clear and direct explanation of the trapezium area formula.
First, read the section What is Area of Trapezium? to understand the formula in words. Then, look at the "Area of Trapezium Formula" section, focusing on the formula and its variables.
2. Deriving the Formula: Why Does It Work?
Memorising a formula is useful, but understanding why it works makes it unforgettable. Just as in video editing where a final project is built by assembling smaller clips, we can understand the area of a trapezium by breaking it down into shapes we already know: triangles.
The following video by Eddie Woo provides a brilliant explanation of how the trapezium area formula is derived.
In this video, Eddie Woo demonstrates how to derive the area of a trapezium by dividing it into two triangles.
Please watch the segment from this timestamp. Observe how he draws a diagonal to split the trapezium into two triangles. He then calculates the area of each triangle and adds them together. Through algebraic factorization, he shows how this sum simplifies to the exact formula we use for the area of a trapezium.
This derivation beautifully connects back to your existing knowledge of the area of a triangle () and shows that you don't need to learn a "new" concept, but rather see how an existing one is applied in a new way.
3. Applying the Formula: Worked Examples
Now that you understand where the formula comes from, let's put it into practice. A crucial point to remember is to always use the perpendicular height, not the length of the slanted "leg" sides, unless that leg is also the height (as in a right-angled trapezium).
Let's watch a short video that walks through a calculation.
Area of a Trapezoid (Trapezium) | Math with Mr. J
This video from Math with Mr. J provides a clear, step-by-step example of calculating the area of a trapezium.
Watch the first example from the beginning of the problem. Notice how he identifies the bases (the parallel sides) and the height, plugs them into the formula, and follows the order of operations (parentheses first!) to get the final answer.
Here is another example in image format that you can study at your own pace.
Practice Problem:
Now, try one yourself. A trapezium has parallel sides of length 10m and 16m. Its perpendicular height is 7m. What is its area?
- Step 1: Add the parallel sides: m.
- Step 2: Divide the sum by 2: m.
- Step 3: Multiply by the height: m.
- Answer: The area is 91 m².
4. Your Formula Collection
Let's update your collection with today's formula.
| Shape | Formula | Variables |
|---|---|---|
| Parallelogram | Area = b × h |
b = base, h = perpendicular height |
| Rhombus | Area = (d₁ × d₂) / 2 |
d₁, d₂ = diagonals |
| Trapezium | Area = ((a + b) / 2) × h |
a, b = parallel sides, h = height |
Conclusion
In this lesson, you've learned how to calculate the area of a trapezium. By understanding how the shape can be decomposed into triangles, the formula becomes logical and easy to remember.
Here are the key takeaways:
- A trapezium is a quadrilateral with exactly one pair of parallel sides.
- The area is found by averaging the lengths of the parallel sides and multiplying by the perpendicular height.
- The formula is: Area = ½ × (sum of parallel sides) × height.
This lesson concludes our module on the area of special quadrilaterals. You now have the formulas and methods to find the area of parallelograms, rhombuses, and trapeziums.
In our next lesson, we will shift our focus from straight-sided shapes to curves as we learn how to calculate the area of a circle.