Trapezium area

1Summary 重點摘要

Area = (a+b)×h÷2.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Know the trapeziumA trapezium has one pair of parallel sides: the shorter one is the upper base, say 3 cm; the longer one is the lower base, say 5 cm; the two slanted sides are the legs. The perpendicular distance between the two bases is the height, say 4 cm. A trapezium looks like the side of a flower pot, narrow on top and wide below, though some are wide on top; either way, the parallel pair are the bases. Why identify the two bases and the height? Because the area formula uses exactly these three quantities, and misidentifying any of them breaks the substitution. A common error is taking a leg for the height; the height must be perpendicular to the bases.
2. Step 2: The formulaTrapezium area = (upper base + lower base) × height ÷ 2, written A = (a + b) × h ÷ 2. The order is: add the two bases, multiply by the height, then divide by 2. The brackets mean the bases are added first, and this step must come first in the working. Why do both bases appear? Because the formula is derived from a parallelogram, and the two bases together decide the length of the combined base; neither can be missing. For bases 3 and 5 with height 4, it is (3 + 5) × 4 ÷ 2. A common error is dropping one base and multiplying by a single one, which is far off; count both bases before substituting.
3. Step 3: Why divide by 2Why divide by 2? Take two identical trapeziums and flip one over: together they form a parallelogram with base 3 + 5 = 8 cm and height 4 cm, so its area is 8 × 4 = 32 cm². Since the parallelogram is made of two trapeziums, one trapezium is 32 ÷ 2 = 16 cm². That is where the division by 2 comes from; it is not an arbitrary rule. With this understanding the formula need not be memorised blindly, and if forgotten, the derivation can be rebuilt by fitting two copies together. The usual doubt is whether to divide or to multiply by 2; the combined whole is two pieces while a single trapezium is one piece, so dividing is certain.
4. Step 4: SubstituteSubstitute the numbers: (3 + 5) × 4 ÷ 2 = 8 × 4 ÷ 2 = 32 ÷ 2 = 16, so the answer is 16 cm². Write one step at a time, each step doing a single thing, with the middle results set out clearly so checking is instant. Why write in steps? Because finishing in one breath invites errors, and a mistake cannot then be located; step-by-step writing lets every line be checked on its own, first 3 + 5 = 8, then 8 × 4 = 32, finally 32 ÷ 2 = 16. A common fault is mental skipping of steps, which hides mistakes; in tests especially, write every line.
5. Step 5: Keep the order of operationsThe formula mixes brackets, multiplication and division, and the order cannot be shuffled: work out the brackets (3 + 5) first, then multiply by the height 4, and finally divide by 2. Ignoring the brackets and computing 3 + 5 × 4 ÷ 2 gives 5 × 4 ÷ 2 = 10 and then 10 + 3 = 13, completely different from the correct 16. Why do brackets come first? Because the formula is designed with the two bases merged into one whole first, which is exactly what brackets express. The common error is overlooking the brackets, and one wrong step ruins the result; circle the brackets before starting as a reminder of what to compute first.
6. Step 6: Find the heightWith the area and both bases, the height can be found in reverse: area 16 cm², upper base 3 cm, lower base 5 cm, so height = 16 × 2 ÷ (3 + 5) = 32 ÷ 8 = 4 cm. Use the formula backwards: the sum of the bases = area × 2 ÷ height, find the sum first, then the height. Why multiply the area by 2? Because the forward direction divides by 2, so the reverse must multiply by 2; multiplication and division cancel each other. Reverse questions are common in tests, and a formula must be mastered both ways. A common error is forgetting the × 2 and computing 16 ÷ 8 = 2; substituting back into the original formula exposes it at once.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Trapezium area = (a + b) × h ÷ 2
• Two identical trapeziums form a parallelogram
• Order: brackets → × height → ÷ 2
• Area units are square units
• Missing height: area × 2 ÷ (a + b)