Triangle Area

1Summary 重點摘要

Learn a new formula to calculate triangle area without finding height. I'll teach you step by step.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The old area formulaFrom primary school: area of a triangle = base × height ÷ 2. This applies when the base and its corresponding height are known — for example a triangle with base 10 cm and height 6 cm has area 10 × 6 ÷ 2 = 30 square centimetres. The limitation: the height must be known, meaning the perpendicular distance from the vertex to the base. When a question gives only two sides and the angle between them — two sides 6 cm and 8 cm with included angle 30° — the height is unknown, and the old formula cannot be applied directly. That is exactly why a new formula is needed.
2. Step 2: The new formula: SAS areaThe new formula: Area = (12) × a × b × sin C, where a and b are the two known sides and C is their included angle — the angle between them, not some other angle. This is the SAS (Side-Angle-Side) area formula. The condition is strict: the angle must be the included one; any other angle does not qualify. Example: sides 6 cm and 8 cm with included angle 30° give Area = (12) × 6 × 8 × sin 30°. The formula's advantage is that no height is ever needed: two lengths and one angle go straight in, and the answer follows — especially useful where a height cannot be measured.
3. Step 3: Why it worksThe new formula follows from the old one — the two are the same idea. Given sides a and b with included angle C, take b as the base; the height is the perpendicular from the opposite vertex. In the right triangle so formed, sin C = height ÷ a (opposite over hypotenuse), so height = a × sin C. Substituting into the old formula: Area = base × height ÷ 2 = b × (a sin C) ÷ 2 = (12) ab sin C, exactly the new formula. So the formula is no invention from thin air: it merely performs the 'find the height' step at once with trigonometry. Knowing this, nothing needs memorising blindly — the formula can be re-derived on the spot.
4. Step 4: When to use the new formulaThe test is simple: when the question gives two sides and their included angle and asks for area, use the SAS formula. Three common shapes of question: a direct 'two sides with included angle', such as 7 cm and 10 cm with 45° between them; a list of two sides plus one of the triangle's angles, where you must confirm the angle is the included one; and diagram questions, where the triangle hides inside a larger figure and the two sides and included angle must be identified from the picture. By contrast, if the base and height are given, the old formula is quicker. The formulas are not better or worse — their conditions differ. Circle the known sides and angles as you read, and the choice makes itself.
5. Step 5: A worked calculationA full worked calculation: sides 6 cm and 8 cm with included angle 30°. Step one, write the formula: Area = (12) × a × b × sin C. Step two, substitute: = (12) × 6 × 8 × sin 30°. Step three, recall the special value: sin 30° = 0.5. Step four, compute: = 0.5 × 6 × 8 × 0.5 = 12 square centimetres, always quoting the unit cm². Practice again: sides 5 cm and 10 cm with included angle 90° give Area = 0.5 × 5 × 10 × sin 90° = 0.5 × 50 × 1 = 25 square centimetres. The special values such as sin 30° and sin 90° must be memorised — exams do not provide them.
6. Step 6: Comparing with the old formulaThe relation between old and new: the old formula is a special case of the new. When the included angle C = 90° (a right triangle), sin 90° = 1 and the new formula becomes Area = (12) × a × b × 1 = (12) ab — precisely base × height ÷ 2, since a right triangle's two legs serve as base and height for each other. Legs 5 and 12 give 5 × 12 ÷ 2 = 30 either way. Conversely, the smaller the included angle, the smaller sin C and the smaller the area: with the same sides 6 and 8, an angle of 90° gives area 24 while 30° gives only 12 — matching the intuition that the wider the two sides open, the bigger the triangle.
7. Step 7: Common mistakesThree traps surround the SAS formula. First, the wrong angle: the formula needs the included angle between the two known sides; substituting an opposite angle ruins everything. Mark the two sides and the angle on the diagram and confirm the angle sits between them. Second, calculator mode: before keying sin 30°, confirm the calculator is in Degree mode; in Radian mode sin 30° returns about −0.988, an absurd result. Third, losing the half or the unit: (12) × 6 × 8 × 0.5 = 12, but dropping the 0.5 doubles the answer to 24; the unit must be square centimetres (cm²). After each question verify three things: the angle is included, the mode is Degree, and the answer contains the division by 2.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• New formula: Area = (12)ab sin C
• Use when know two sides and included angle
• Calculate area without finding height
• Old formula is its special case (when right angle)
• Remember 12 in front