Circle lengths, areas and composite boundaries · Foundation
| English | 中文 | Pinyin |
|---|---|---|
| circumference/sɜːˈkʌmfrəns/ | 圆周长 | yuán zhōu cháng |
A semicircular window needs glass and a frame. Glass uses area; the frame includes a curved arc and the straight diameter.
- A semicircular window needs glass and a frame. Glass uses area; the frame includes a curved arc and the straight diameter.
- This lesson studies circumference 圆周长: The total length of a circle’s boundary.
Choose the mathematical structure
- Circle circumference is 2πr=πd and area is πr². A semicircle has half the circle area, but its complete perimeter includes the diameter as well as half the circumference. For composite shapes count every exposed boundary once, and separate straight edges from arcs.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines circumference?
The total length of a circle’s boundary.
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.
For radius 4 cm, circumference is 8π cm and area 16π cm². A semicircle of that radius has area 8π cm² and perimeter 4π+8 cm. A rectangular 8 by 3 cm window topped by this semicircle has area 24+8π cm². Its perimeter is the bottom 8, two vertical sides totalling 6 and the curved top 4π: 14+4π cm. The diameter across the join is internal and is not counted. An annulus with outer radius 5 and inner radius 3 has area π(25-9)=16π cm². Its two circular boundaries together have length 10π+6π=16π cm, despite this accidental equality of coefficients; length and area still have different units.
Circle lengths, areas and composite boundaries
Circle circumference is 2πr=πd and area is πr²
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find the coefficient of pi in area for radius 4.
r²=16.
Test a tempting shortcut
- Radius must be halved from a given diameter before squaring. Half a circle’s circumference is only its arc, not the complete semicircle perimeter. An internal join is not exposed boundary.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The complete perimeter of a semicircle is half a circle’s circumference. This claim is false. Explain which definition or assumption it violates.
Find the straight-edge contribution to the perimeter of an r=4 semicircle.
Its straight edge is diameter 2r=8.
The complete perimeter of a semicircle is half a circle’s circumference.
Radius must be halved from a given diameter before squaring. Half a circle’s circumference is only its arc, not the complete semicircle perimeter. An internal join is not exposed boundary.
Interpret a new situation
- AQA G17 includes exact pi answers and composite circle perimeters/areas. Give both the exact expression and the required final rounded value, keeping the meaning and units clear.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the coefficient of pi in annulus area for outer radius 5, inner radius 3.
Subtract squared radii: 25-9=16.
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 total length of a circle’s boundary. Choose the relationship, show the method, check its assumptions and interpret the result.