Inside, on or outside a circle at the origin
A circle has its centre at the origin O and radius 5. Is the dot (4, 4) inside it?
- A circle with centre O and radius 5.
- The dot (3, 4) is 5 from O. It is on the circle.
- The dot (1, −3) is nearer than 5. It is inside.
- The dot (4, 4) is further than 5. It is outside.
The rule
No: (4, 4) is outside the circle.
Rule: Compare x² + y² with r². Less is inside, equal is on, more is outside. Here x and y are the dot's numbers; r is the radius.
The dot is x across and y up from O. So its distance from O is √(x² + y²). A dot is on the circle when that distance equals r. So compare the squares. You do not need a square root.
Here r² = 5 × 5 = 25. Try (4, 4): 16 + 16 = 32. That is more than 25, so the dot is outside.
| Dot | x² + y² | Against 25 | Where |
|---|---|---|---|
| (3, 4) | 9 + 16 = 25 | equal | on |
| (1, −3) | 1 + 9 = 10 | less | inside |
| (4, 4) | 16 + 16 = 32 | more | outside |
| (−5, 0) | 25 + 0 = 25 | equal | on |
| (−5, 1) | 25 + 1 = 26 | more | outside |
This circle passes through twelve dots with whole-number coordinates. They are (±5, 0), (0, ±5), (±3, ±4) and (±4, ±3). The sign ± means plus or minus.
Watch out!
Adding x and y does not work. The dot (3, 4) is on the circle. But 3 + 4 = 7, not 5.
Slide the dot along the row. Find where it is on the circle.
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