Circle lengths, areas and composite boundaries · Foundation
Введено| English | Значение |
|---|---|
| circumference/sɜːˈkʌmfrəns/ | длина окружности |
Glass uses area; the frame includes a curved arc and the straight diameter.
- A semicircular window needs glass and a frame.
- This lesson studies circumference 圆周长: The total length of a circle’s boundary.
The method
- 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.
Why is the perimeter of a semicircle NOT half the circumference?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Half the circumference is only the curved arc. The semicircle region also includes its straight diameter edge, so its complete perimeter is πr+2r.
A worked case
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.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
r²=16.
Where it goes wrong
- 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.
Not true: The complete perimeter of a semicircle is half a circle’s circumference. The perimeter of a semicircle is half the circumference plus the diameter.
Find the straight-edge contribution to the perimeter of an r=4 semicircle.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Its straight edge is diameter 2r=8.
The perimeter of a semicircle is half the circumference plus the diameter.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
True. The perimeter of a semicircle is half the circumference plus the diameter.
Applying it
- 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.
Find the coefficient of pi in annulus area for outer radius 5, inner radius 3.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Subtract squared radii: 25-9=16.
Match each circle quantity to its formula.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Length answers stay in length units and area answers in square units; internal joins between parts are not exposed boundary.
In your course
- AQA GCSE Mathematics (8300), Foundation 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.
- Circumference: The total length of a circle’s boundary.
- Circle circumference is 2πr=πd and area is πr².
- Watch for: radius must be halved from a given diameter before squaring.