Triangle area (intro)

1Summary 重點摘要

Triangle area = base × height ÷ 2.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: Start from a rectangleDraw a rectangle with base 6 cm and height 4 cm; its area is 6 × 4 = 24 cm². Cut it along a diagonal and two identical triangles appear, each with exactly the same base and height as the rectangle. Why start from a rectangle? Because the rectangle's area is knowledge we already have, and using it to derive the triangle's formula is the mathematical way of turning new problems into old ones. For example, the two triangles cut along the diagonal cover each other exactly, which shows their areas are equal and each is half of the rectangle. A common mistake is to think the two cut-off triangles differ in size; cut along the diagonal of a rectangle and the two triangles are always exactly identical.
2. Part 2: A triangle is half the rectangleThe rectangle's 24 cm² is split into two equal shares by the diagonal, so each triangle has area 24 ÷ 2 = 12 cm². From this comes the formula: triangle area = base × height ÷ 2. Why divide by 2? Because two copies of the very same triangle fit together into a rectangle of the same base and height, so the triangle takes up exactly half of it. The formula holds for every triangle: acute, right or obtuse. For example, a triangle of base 10 cm and height 6 cm has area 10 × 6 ÷ 2 = 30 cm². A common mistake is to forget the ÷ 2 and use base times height directly, doubling the area; remember where the formula comes from — two identical triangles forming a rectangle — and the ÷ 2 will never be lost.
3. Part 3: Using the formulaThe order of calculation is to multiply the base and the height first and divide by 2 last: for a triangle of base 8 cm and height 5 cm, work out 8 × 5 = 40, then 40 ÷ 2 = 20, so the area is 20 cm². Write every step clearly for easy checking. Why multiply before dividing? Dividing first works too: 8 ÷ 2 × 5 = 20, the same result, because multiplication and division may swap their order, so choose whichever runs smoothest. The answer must carry the unit cm², because area is a square unit, different from the length unit cm. A common mistake is to multiply two sides at random without matching base and height; always use the base together with the height that corresponds to it. Another is to miss the square unit, writing cm instead of cm².
4. Part 4: The height must be perpendicularA triangle's height is the perpendicular segment drawn from the apex to the base, meeting the base at a right angle — this is the exact meaning of 'height' in the formula. Why must it be perpendicular? Because the area formula comes from the rectangle, where length and width meet at right angles; only when the triangle's height is perpendicular to its base does the formula hold. In an obtuse triangle the height sometimes has to be drawn down to the extended base line: the foot of the perpendicular falls outside the base, and the height is the perpendicular segment from the apex to that extension. To measure the height, brace one right-angle edge of a set square on the base and let the other pass through the apex. A common mistake is to measure along a slant and call it the height, or to draw a height that is not upright; check the angle with a set square before drawing.
5. Part 5: Right triangles are the easiestIn a right triangle the two legs are perpendicular to each other, so they themselves serve as the base and the height, with no extra drawing needed. For example, with legs 3 cm and 4 cm, area = 3 × 4 ÷ 2 = 6 cm². Why are the legs the base and the height? Because the height is defined as the perpendicular segment from the apex to the base, and one leg of a right triangle is exactly perpendicular to the other, so the definition is satisfied automatically; the hypotenuse is not the height unless specially drawn. Verify it: two identical right triangles fit together into a rectangle 4 cm by 3 cm, area 12 cm², each triangle 6 cm². A common mistake is to use the hypotenuse as the height, computing 3 × 5 ÷ 2 with the hypotenuse 5 cm; the area of a right triangle must be computed from the two legs.
6. Part 6: Word problemThe order for word problems is: find the base and the height in the question, confirm the units agree, substitute into the formula, and finish with the unit. For example, a triangular flag has base 30 cm and height 20 cm, so area = 30 × 20 ÷ 2 = 300 cm², and knowing the area tells how much cloth is needed. Why check the units first? Because if the base is in cm and the height in m, multiplying directly goes wrong; convert to one unit before calculating. For example, with base 2 m and height 50 cm, first turn 50 cm into 0.5 m, giving area = 2 × 0.5 ÷ 2 = 0.5 m². A common mistake is to start computing with a slant length, unclear which side is the height; sketch first, mark the base and the height, then calculate, and check the units at the end.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Triangle area = base × height ÷ 2
• Two identical triangles make a parallelogram — hence the half
• Height is perpendicular to the base, not the slant
• In right triangles the two legs are base and height
• Area answers use square units, cm²