Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:A circle is every point that sits exactly the radius away from one central point.
Common stuck point:The procedure for circles is the easy part; the trap is mixing up radius and diameter. Asking "Is every point on the boundary the same distance from one center?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Is every point on the boundary the same distance from one center?
Worked Examples
Example 1
easy
A circle has a radius of 7 cm. Find its diameter and state the relationship between the radius and the diameter.Radius = 7 cm — find the diameter
Answer
d=14 cm
First step
1
Key circle relationships: the diameter d spans the full width through the centre, so d=2r. The radius r is the distance from centre to any point on the circle, so r=2d.
Full solution
2
Substitute r=7 cm into d=2r: d=2(7)=14 cm.
3
Verify the relationship: r=2d=214=7 cm ✓. The diameter is always exactly twice the radius regardless of the circle's size.
The radius extends from the centre to any point on the circle, while the diameter passes through the centre connecting two opposite points. Understanding this relationship is fundamental to all circle calculations.
Example 2
medium
A chord of a circle is 24 cm long and is 5 cm from the centre. Find the radius of the circle.
Example 3
medium
A chord of length 16 is 6 from the center. Find the radius.Chord = 16, distance from center = 6 — find the radius
Example 4
medium
A square has side 12. Find the radius of the inscribed circle.
Example 5
hard
A chord of length 24 is in a circle of radius 13. Find the chord's distance from the center.Radius = 13, chord = 24 — find the distance from the center
Example 6
hard
Write the equation of a circle with center (0,0) and radius 7.
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
A circle has a diameter of 20 cm. What is its radius?Diameter
Example 2
easy
Two radii and one chord form a triangle inside a circle. If the radius is 10 cm and the triangle's perimeter is 34 cm, find the chord length.
Example 3
easy
A circle has radius 5. What is its diameter?Radius
Example 4
easy
A circle has diameter 14. What is its radius?Diameter
Example 5
easy
What is the name for the distance from the center of a circle to any point on the circle?
Example 6
easy
Which is longer in a given circle: a radius or a diameter?
Example 7
easy
A circle has radius 3. Write its circumference in terms of π.Radius
Example 8
easy
A circle has radius 4. Write its area in terms of π.Radius
Example 9
easy
A straight line segment from one point on a circle to another is called a what?
Example 10
easy
Using π≈3.14, estimate the circumference of a circle with diameter 10.Diameter
Example 11
medium
A circle has circumference 20π. Find its radius.Circumference
Example 12
medium
A circle has area 49π. Find its radius.Area
Example 13
medium
If you double a circle's radius, what happens to its circumference?
Example 14
medium
If you double a circle's radius, what happens to its area?
Example 15
medium
A square has side 10. A circle is inscribed inside it, touching all four sides. Find the circle's radius.
Example 16
medium
A bicycle wheel has radius 30 cm. About how far does it travel in one full turn? Use π≈3.14.
Example 17
medium
Two circles have radii 3 and 6. What is the ratio of their areas?
Example 18
medium
A semicircle has radius 6. Find its perimeter in terms of π.Semicircle with radius 6 — find the perimeter
Example 19
challenge
A circle is inscribed in a square of side 8, and another circle is circumscribed about the same square. Find the ratio of the larger circle's area to the smaller circle's area.
Example 20
challenge
Four identical coins of radius 1 are packed snugly in a square so each touches two neighbors. Find the area of the gap in the center between the four coins.
Example 21
challenge
A goat is tied by a 7-meter rope to a corner on the outside of a square barn with side 10 meters. Over what area can the goat graze?
Example 22
challenge
Explain why, if a circle's circumference equals its area numerically, the radius must be 2.
Example 23
easy
A circle has radius 9. What is its diameter?Radius
Example 24
easy
A circle has diameter 26. What is its radius?Diameter
Example 25
easy
A circle has radius 6. Find its area in terms of π.Radius
Example 26
easy
A circle has radius 8. Find its circumference in terms of π.Radius
Example 27
easy
A circular pizza has radius 7 inches. Use π≈3.14 to estimate its circumference.
Example 28
medium
A circle has area 25π. Find its radius.Area
Example 29
medium
A circle has circumference 30π. Find its diameter.
Example 30
medium
A semicircle has diameter 20. Find its perimeter in terms of π.Semicircle with diameter 20 — find the perimeter
Example 31
medium
Two circles have radii 4 and 10. Find the ratio of their circumferences.
Example 32
medium
Two circles have radii 3 and 9. Find the ratio of their areas.
Example 33
medium
A square has side 10. Find the radius of the circumscribed circle (passing through all 4 corners).
Example 34
hard
A circle has area A equal to circumference C numerically. Find r.
Example 35
hard
Three circles of radius 1 are arranged so each touches the other two. Find the side length of the equilateral triangle joining centers.
Example 36
hard
A circle is inscribed in an equilateral triangle of side 6. Find its radius.
Example 37
hard
A circle has center (2,3) and passes through (5,7). Find its radius.
Example 38
hard
A circle has equation (x−3)2+(y+2)2=16. State its center and radius.
Example 39
challenge
A circle of radius 5 has a chord of length 6. Find the distance from the center.Radius
Example 40
challenge
A circle has area A that is twice its circumference C numerically. Find r.
Example 41
challenge
A circle is inscribed in a regular hexagon of side 4. Find the circle's radius.