Concentric circles is a set of circles that have same center point but different radii. The term of “concentric” comes from the Latin word “concentricus,” which means “having a common center.” These circles are almost used in mathematics, geometry, art, and design.
Concentric Circles Meaning
Concentric circles are a set of circles that have the same center point, but different radii. The term of “concentric” comes from the Latin word “concentricus,” which means “having a common center.” These circles can be found in almost fields, including mathematics, geometry, art, design and engineering.The distance between of any two concentric circles is equal to the difference in their radii, and the area and perimeter of a concentric circle are proportional to its radius. Overall, the concentric circles are a useful and versatile geometric concept with more practical applications.
Concentric Circles – Theorem
In the two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
Proof :
Given that:
Consider two concentric circles C1 and C2, with centre of O and a chord AB of the larger circle C1, touching the smaller circle C2 at the point P as shown in the figure below.
Construction:
Join OP.
Concentric circles 2
To prove that: AP = BP
Since AB is the chord of larger circle C1, it becomes the tangent to C2 at P.
OP is the radius of circle C2.
We know that radius is perpendicular to the tangent at the point of contact.
So, OP ⊥ AB
Now AB is a chord of the circle C1 and OP ⊥ AB.
Then, OP is the bisector of the chord AB.
Thus, the perpendicular from the centre bisects the chord, i.e., AP = BP.
Applications of Concentric Circles
Geometry: Concentric circles are used in geometry to help define and understanding the circles, as well another geometric shapes.
Architecture: In architecture Concentric circles are used to create attractive patterns and designs.
Art: Concentric circles are used in art to create different visual effects and designs.
Engineering: Concentric circles are used in engineering to create gears, pulleys, and other mechanical systems.
Navigation: Concentric circles are used in navigation to plot courses and distances.
Concentric Circle Examples
The radius of the larger concentric circle is 10 cm and the radius of a smaller circle is 6 cm. than What is the area of the region between the two circles?
We can find the area of region between the two concentric circles by using below formula:
Area = πr1^2 – πr2^2
where r1 is a radius of the larger circle, and r2 is a radius of the smaller circle.
Substituting the values given in the problem, we get:
Area = π(10 cm)^2 – π(6 cm)^2
= π(100 cm^2) – π(36 cm^2)
= π(64 cm^2)
≈ 201.06 cm^2
Then, the area of region between the two concentric circles is approximately 201.06 square centimeters.
Find the area of lens? camera lens has three concentric circles with radii of 3 cm, 6 cm, and 9 cm.
To find the total area of the camera lens, we need to add up the areas of the three concentric circles. The area of a circle with radius r is given by the formula:
Area = πr^2
Using this formula, we can find the areas of each of the circles and add them together to get the total area:
Area of the first circle = π(3 cm)^2 = 9π cm^2
Area of the second circle = π(6 cm)^2 = 36π cm^2
Area of the third circle = π(9 cm)^2 = 81π cm^2
Total area = 9π + 36π + 81π = 126π
Therefore, the total area of the camera lens is 126π square centimeters (or approximately 395.85 square centimeters).
The Olympic Rings are five concentric circles with diameters of 6 cm, 5 cm, 4 cm, 3 cm, and 2 cm. Find the total area of the Olympic Rings logo?
To find the total area of the Olympic Rings logo, we need to add up the areas of the five concentric circles. We can use the formula for the area of a circle:
Area = πr^2
where r is the radius of the circle (which is half of the diameter given in the problem).
So, the radius of each circle is:
outermost circle, r = 6/2 = 3 cm
second circle, r = 5/2 = 2.5 cm
third circle, r = 4/2 = 2 cm
fourth circle, r = 3/2 = 1.5 cm
innermost circle, r = 2/2 = 1 cm
Using these radii, we can calculate the area of each circle and add them up:
Area of the outermost circle = π(3 cm)^2 ≈ 28.27 cm^2
Area of the second circle = π(2.5 cm)^2 ≈ 19.63 cm^2
Area of the third circle = π(2 cm)^2 ≈ 12.57 cm^2
Area of the fourth circle = π(1.5 cm)^2 ≈ 7.07 cm^2
Area of the innermost circle = π(1 cm)^2 ≈ 3.14 cm^2
Total area = 28.27 cm^2 + 19.63 cm^2 + 12.57 cm^2 + 7.07 cm^2 + 3.14 cm^2
= 70.68 cm^2
Therefore, the total area of the Olympic Rings logo is approximately 70.68 square centimeters.
