Areas, perimeters and composite plane shapes · Higher
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| perpendicular height/ˌpɜːpənˈdɪkjʊlə haɪt/ | chiều cao vuông góc |
The side length and the height serve different jobs in an area formula.
- A sloping garden fence can have a long side but a small perpendicular height.
- This lesson studies perpendicular height 垂直高度: The distance between a base and its parallel opposite level, measured at right angles.
The method
- Triangle area is bh/2, parallelogram area bh and trapezium area (a+b)h/2 for parallel sides a,b.
- Heights are perpendicular to the selected base.
- Perimeter adds only the outside boundary.
- Split a composite shape into non-overlapping parts, or subtract a missing region from a containing shape.
Why must the height in A=bh/2 be perpendicular to the base?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The area formula measures how far the shape extends perpendicular to its base. Using a slant side inflates the answer - a very flat parallelogram has small area despite long sloping sides.
A worked case
A triangle with base 8 cm and height 5 cm has area 20 cm²; a parallelogram with the same base/height has area 40 cm². A trapezium with parallel sides 6 and 10 cm and height 4 cm has area (6+10)×4/2=32 cm². A rectangular 8 by 6 cm sheet with a 3 by 2 cm corner removed has area 48-6=42 cm². Its perimeter is still 28 cm: the two removed outside segments total 5 cm and the two new notch edges also total 5 cm. This perimeter equality depends on a corner rectangular cut; an internal hole adds a separate boundary. Rearranging triangle area gives h=2A/b.
Areas, perimeters and composite plane shapes
Triangle area is bh/2, parallelogram area bh and trapezium area (a+b)h/2 for parallel sides a,b
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find the trapezium area for parallel sides 6,10 and height 4.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
(6+10)×4/2=32.
Where it goes wrong
- Use perpendicular height rather than a sloping side.
- A shared internal division line is not part of the perimeter.
- An area answer has squared units; a perimeter answer has length units.
Not true: The sloping side of a parallelogram is its perpendicular height. The height must be perpendicular to the base; the sloping side is longer than the height.
Find area of 8×6 rectangle with 3×2 corner removed.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
8×6-3×2=42.
The height must be perpendicular to the base.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. The height must be perpendicular to the base; the sloping side is longer than the height.
Applying it
- AQA G16/G17 includes triangle/parallelogram/trapezium area and composite perimeter/area.
- Mark the parallel bases, perpendicular height and counted outside edges before calculating.
Find the perimeter of that corner-cut shape.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Count the six outside edges; the notch replaces equal total lengths.
Match each shape to its area formula.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
All three use the perpendicular height; the trapezium averages its two parallel sides before multiplying.
In your course
- AQA GCSE Mathematics (8300), Higher tier, specification section 3.4.
- Paper 1 is non-calculator: keep fractions, surds and π exact and show each step. On calculator papers, still write the expression before its value.
- Perpendicular height: The distance between a base and its parallel opposite level, measured at right angles.
- Triangle area is bh/2, parallelogram area bh and trapezium area (a+b)h/2 for parallel sides a,b.
- Watch for: use perpendicular height rather than a sloping side.