The ratio of the perimeters of two circles is 5:4. What is the ratio of their radii?
Answer: 2πr1 : 2πr2 = r1 : r2, so the radii are also in the ratio 5 : 4.
Step-by-step solution
Idea: Perimeter = 2π × radius, so the perimeter is directly proportional to the radius; the common factor 2π cancels in the ratio.
- 2πr12πr2 = 54.1 mark
- Cancel 2π: r1r2 = 54, so r1 : r2 = 5 : 4.1 mark
The ratio of their radii is 5 : 4.
Check: Radii 5 cm and 4 cm give perimeters 10π and 8π, ratio 5 : 4 ✓.
Answer to write in the exam
2πr12πr2 = 54
r1r2 = 54
∴ r1 : r2 = 5 : 4
Common mistakes that cost marks
- Taking square roots and writing √5 : 2. That is for a ratio of areas, not perimeters.
- Squaring and writing 25 : 16, which would be the ratio of the areas.
How this can come in the exam
MCQ (1 mark)
The radii of two circles are in the ratio 3 : 2. The ratio of their areas is
- 3 : 2
- 9 : 4
- 6 : 4
- √3 : √2
Show answer
(B) 9 : 4
Area = πr2, so the ratio is 32 : 22 = 9 : 4 (perimeters would be 3 : 2).
Try one yourself
The circumferences of two circles are 66 cm and 44 cm. Find the ratio of their diameters.
Show answer
Diameter ∝ circumference: 66 : 44 = 3 : 2.
More questions like this
- What happens if the parallelogram is ‘thin’ (see the figure) and the foot of the perpendicular from C to AD does not lie on side AD? The construction then does not seem to work. How do we fix this ‘gap’?
- The area of a rectangle can be found when we know the lengths of its sides. Is the same true for a parallelogram? That is, can we find the area of a parallelogram when we know the lengths of its sides? Why or why not? (Hint: What happens to the area of a parallelogram if we decrease or increase the angle between the adjacent sides while keeping the lengths fixed?)
- You may wonder, like earlier, is there a gap in our argument? What would we do if angle EFG is obtuse and the triangle were shaped like triangle EFG in the figure? Please work out the answer to this question.
- Do you see why the two triangles fit together to make a parallelogram? (If you study the angles in the figure (e.g., ∠B’C’A’ and ∠BCA), you will see why this is so. Keep in mind the criterion by which we check whether two lines are parallel.)
- Since ΔABD and ΔACD have equal area, you may wonder — Can we divide ΔABD using straight cuts into two or more pieces that we can then rearrange to exactly cover ΔACD? What do you think? Is it possible?