Circles and composite boundaries
Use the radius for circle area and trace every part of a composite boundary.
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Radius, diameter and circumference
The radius joins the centre to a point on a circle. The diameter passes through the centre from one side to the other and is twice the radius: d = 2r.
The circumference is C = π d = 2π r, where π is the constant ratio of circumference to diameter. The area enclosed is A = π r². One useful model cuts a circle into many thin sectors and alternates them into a shape approaching a rectangle: its height approaches r and its base approaches half the circumference, π r. Hence the area approaches π r × r.
Read the given dimension before choosing a formula. A diameter of 14 cm gives radius 7 cm; putting 14 into the radius formula would double the circumference and make the area four times as large.
For radius 7 cm, circumference is 14π cm and area is 49π cm². If told to use π = 22/7, these are 44 cm and 154 cm². Use the value of π or accuracy requested; area and circumference have different units. The substitution π = 22/7 is a question’s approximation instruction, not an exact identity for the mathematical constant. If an exact answer in terms of π is requested, keep π in the answer instead. When rounding is requested, keep the intermediate values and round only the final measure.
A fraction of an area is not the whole perimeter
A semicircle of radius 7 cm has area (1/2)π × 7² = 77 cm² when π = 22/7. Its curved edge is half the full circumference, 22 cm. Its complete perimeter also includes the 14 cm diameter: 22 + 14 = 36 cm.
Follow both kinds of edge
The solid outline follows the semicircular arc and returns along the diameter.
View figure data
| Given | Information |
|---|---|
| Region | Semicircle |
| Full circle diameter | 14 cm |
| Pi instruction | Use pi = 22/7. Do not round the result. |
A quarter-circle’s perimeter contains one quarter of the circumference plus two radii. Trace the actual boundary with a finger before writing the expression.
Shared edges are not outside edges
A semicircle is joined to a rectangle along its 14 cm side. The rectangle’s other dimension is 10 cm. The area is 14 × 10 + 77 = 217 cm². The outside perimeter is 14 + 10 + 10 + 22 = 56 cm; their shared diameter is inside and is not counted.
Check your understanding 1 Area or boundary?
A quarter-circle has radius 14 cm. Using π = 22/7, find its area and its complete perimeter. State why the same fraction of a full circle cannot give both answers.
Separate the quarter arc from the two radii
The arc is one quarter of the full circumference. Both straight sides are radii.
View figure data
| Given | Information |
|---|---|
| Region | Quarter circle |
| Circle radius | 14 cm |
| Pi instruction | Use pi = 22/7. Do not round the result. |
Show answer
Area is 1/4 × 22/7 × 14² = 154 cm². The curved edge is 22 cm; add two radii of 14 cm to get perimeter 50 cm. A quarter of the full circumference alone omits both straight edges.
Try another choice of dimension
Check your understanding 2 Choose the radius before using the area formula
A circle has diameter 10 m. Give its area and circumference in terms of π. Explain how the diameter enters each formula.
One circle, two different measures
The full diameter is 10 m; it is not the radius.
View figure data
| Given | Information |
|---|---|
| Region | Circle |
| Full circle diameter | 10 m |
| Pi instruction | Keep pi exact; type pi or π. Do not substitute a decimal approximation. |
Show answer
The radius is 5 m, so area is π × 5² = 25π m². The circumference is π × 10 = 10π m. Using 100π m² would treat the diameter as the radius. Area requires the squared radius, whereas circumference can use the full diameter directly.
Check your understanding 3 Keep the given radius until the final answer
A circular patch has radius 1.25 m. Use π = 3.14 and find its area to two decimal places. Explain why you do not first round the radius to 1.3 m.
Show answer
The given radius is 1.25 m, so A = 3.14 × 1.25² = 4.90625 m². Rounding only this final value gives 4.91 m². Changing the radius first changes the circle being measured. The two-decimal-place instruction applies to the requested final area.
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