Areas, perimeters and composite plane shapes · Foundation
| English | 中文 | Pinyin |
|---|---|---|
| perpendicular height/ˌpɜːpənˈdɪkjʊlə haɪt/ | 垂直高度 | chuí zhí gāo dù |
A sloping garden fence can have a long side but a small perpendicular height. 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. The side length and the height serve different jobs in an area formula.
- This lesson studies perpendicular height 垂直高度: The distance between a base and its parallel opposite level, measured at right angles.
Choose the mathematical structure
- 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.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines perpendicular height?
The distance between a base and its parallel opposite level, measured at right angles.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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.
(6+10)×4/2=32.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The sloping side of a parallelogram is always its perpendicular height. This claim is false. Explain which definition or assumption it violates.
Find area of 8×6 rectangle with 3×2 corner removed.
8×6-3×2=42.
The sloping side of a parallelogram is always its perpendicular height.
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.
Interpret a new situation
- AQA G16/G17 includes triangle/parallelogram/trapezium area and composite perimeter/area. Mark the parallel bases, perpendicular height and counted outside edges before calculating.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the perimeter of that corner-cut shape.
Count the six outside edges; the notch replaces equal total lengths.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Foundation · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The distance between a base and its parallel opposite level, measured at right angles. Choose the relationship, show the method, check its assumptions and interpret the result.