Area of a Trapezium

🔮 Definition of A trapezium

A trapezium (also called a trapezoid) is a four-sided shape with only one pair of parallel sides.

• We call these parallel sides the bases.
• The other two sides are not parallel.

🌟 Note: If both pairs of opposite sides were parallel, the shape would be a parallelogram instead.

🔮 Height of a Trapezium

The height of a trapezium is the perpendicular distance between the two bases.

The height must meet both bases at a right angle (90°).

🌟 Important: The height is not the length of the slanted sides. A common mistake is to use one of the non-parallel sides as the height. Don’t do this! 😵‍💫

🔮 Trapezoid Area Formula:

$$\large \text{Area} = \frac{\text{Base}_1 + \text{Base}_2}{2} \times \text{Height}$$

Steps:

1. Add the lengths of the two bases.
2. Divide the result by 2 (this gives the average base length).
3. Multiply by the height.

🔮 Example 1: Finding the Area of a Regular Trapezium

A trapezium has:

• • Base 1 = 8 cm
  • Base 2 = 12 cm
  • Height = 6 cm

$$\text{Area} = \frac{8 + 12}{2} \times 6 = 10 \times 6 = 60 \text{ cm}^2$$

So the area of this trapezium is 60 cm² ✅ 

🌟 What is a Right-Angled Trapezium?

A right-angled trapezium has two right angles. Good news! 🥳 The height is easy to spot: it’s the short side that meets both bases at right angles.

🔮 Example 2: Finding the Area of a Right-Angled Trapezium

• • Base 1 = 6 cm
  • Base 2 = 9 cm
  • Height = 4 cm

$$\text{Area} = \frac{6 + 9}{2} \times 4 = 15 \times 2 = 30 \text{ cm}^2$$

So the area of this right-angled trapezium is 30 cm² ✅ 

🌟 What is a Isosceles Trapezium?

An isosceles trapezium has two equal non-parallel sides.

If you drop perpendicular lines from the ends of the top base to the bottom base, you split it into:

• Two identical right-angled triangles (on each side)
• One rectangle (in the middle)

🔮 Example 3: Finding the Height of a Isosceles Trapezium

• • Top base = 5m
  • Side parts = 2m each
  • Bottom base = 2m $9$m
  • Area of the trapezium = 21 m²

Calculation:

$$
\begin{aligned}
\frac{5 + 9}{2} \times h &= 21 \\
7 \times h &= 21 \\
h &= 3 \text{ cm}
\end{aligned}
$$

So the height of this isosceles trapezium is 3 cm ✅