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The radius of a circle can be measured from the center to any point of the circumference. Radius is typically denoted by ‘r’. The radius of a circle is an obligatory quantity required in most circle-relevant formulae. In simple words, it is half the total distance of a diameter. The radius remains equidistant, regardless from where the line segment originates.
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Key takeaways: Radius of a circle, Circle, Radius, circumference, distance, center, diameter
Also read: Isosceles Triangle Theorems
Radius of a Circle
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The radius of a circle is usually measured from the center of any spherical object to a particular point on the circumference (the edge of the circular object). Radius, as is mentioned prior, is signified with the letter “r”.
Interestingly, a radius is not only half the diameter of a circular object, but also for semi-circular objects, cones that have circular bases or even cylindrical objects. Many define a circle as the locus point that typically moves in a plane, especially such that the measure remains constant from the center to the circumference either way.
In definition, the fixed center point in equal distance from the circumference, in a way that every plane remains in parity is known as radius. Most circle related formulas, like circumference and area, are determined mostly by first considering the radius.
The video below explains this:
Radius Formula Detailed Video Explanation:
Formula Related to Radius
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1. Radius From Diameter
Diameter of a circle is a plane that passes through the center point and eventually joins any point by the other end of the circle. In simple words, a diameter is twice the distance of a radius. A double radius can give rise to a total diameter. Dominantly, it is the largest chord ever determined and found in a circle. The formula of diameter is usually signified by:
Diameter = 2r (where, r is the radius)
2. Radius from Circumference
The circumference of a circle is the total length of the circle boundary, usually also defined as its perimeter. Mathematically, the circumference of a circle is one of the vital parts that decides many different areas of a spherical object. The formula of the circumference, in reference to radius, can be given as,
Circumference = 2πr units
where, r is radius, and π is a constant, often otherwise expressed as 3.141.
3. Radius from Area
The area of a circle is usually the total space it has occupied. The relationship between area and radius of a circle can gradually be established by the formula:
Area = πr2 square units (where, r is the radius).
4. Cartesian Plane
The radius of a circle can be also determined through the cartesian plane. By cartesian plane, the following formula can be obtained: (x-h)2 + (y-k)2 = r2
Read More: Conic Sections
Chord of a Circle
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The chord of a circle can be defined as a straight line segment that has endpoints lying over a given circular arc. In a nutshell, the seemingly infinite line segment of a chord, present in a circle, is known as a secant line. In majority, there are two formulae that help measure the chord length of a circle. The formulae are as expressed beneath,
Chord Length = 2 √(r2 – d2), this formula is generally used during the calculation of a perpendicular drawn out from the center point.
While, if speaking trigonometrically, the chord length can be expressed as = 2 r sin (c / 2).
Likewise, in reference to both area, diameter and circumference, the following formulae can be determined:
- Radius = C/2π (for circumference)
- Radius = √(A/π) (for area)
- Radius = D/2 (for diameter)
Chord of a Circle Theorem
Theorem 1: The chord is bisected by a perpendicular line drawn from the center of a circle to it.
Given: AB= Chord and OC⊥AB
To prove: AC=BC
Construction: Draw OA and OB
Proof: In ΔOAC and ΔOBC
| Statement | Reason |
|---|---|
| OA = OB | Radii of the same circle |
| OC = OC | Common |
| ∠OCA = ∠OCB | 90 degree Angle |
| ΔOAC ≅ ΔOBC | By RHS congruence rule |
| AC = CB | By CPCT (Corresponding parts of congruent triangles) |
Theorem 2: The line drawn through the centre of the circle to bisect a chord is perpendicular to the chord.
Given: The circle's chord AB has its midpoint at C, with the circle's centre at O.
To prove: OC⊥AB
Construction: Join OA, OB and OC
Proof: In ΔOAC and ΔOBC
| S. no | Statement | Reason |
|---|---|---|
| 1 | OA = OB | Radii of the same circle |
| 2 | OC = OC | Common |
| 3 | ∠OCA = ∠OCB | Corresponding parts of congruent triangle |
| 4 | ∠OCA + ∠OCB = 180 degree | Linear pair angle |
| 5 | ΔOAC ≅ ΔOBC | By SSS congruence rule |
| 6 | AC = CB | Given |
| 7 | ∠OCA = ∠OCB = 90 degree | From statement 3 and 4 |
| 8 | OC ⊥ AB | From statement 7 |
Radius of a Sphere
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As much as the radius of a circle, the radius of a sphere remains mostly similar. This is because the sphere is only a 3D version of a circle. It is a straight, equidistant line segment that eventually meets at the end of a sphere’s boundary.
However, the size of a sphere is mostly based on its radius. As like a circle, every radii drawn out from the focal point of the sphere has to remain the same, despite the size it attains. Luckily, we can estimate the volume and surface area by the radius of a sphere. The formulae for doing the same are,
- Radius of Sphere = 3√(3V)/4π units, (from volume)
- Radius of Sphere = √(A/4π) units (from surface area).
?Also read: Minors and Cofactors
Things to Remember
- The radius of a circle is half the length of its diameter, usually meeting at the boundary of the circle.
- Interestingly, every endpoint at which the center plane meets, always remains equidistant.
- The formula of radius of a circle is often denoted by “r”.
- There are different formulas for different events, much like: Radius = C/2π (for circumference), Radius = √(A/π) (for area), Radius = D/2 (for diameter).
- Most circle related formulas, like circumference and area, are determined by first considering the radius.
Also read: Area of a Triangle
Sample Questions
Ques. Determine the radius of a circle with the following points on the cartesian plane: O (2, 1), and point P (5, 5) that stands on the circumference. (3 marks)
Ans: In the given equation, the radius of a circle pointed in a cartesian plane is expressed by the following formula: (x − h)2 + (y − k)2 = r2
Now, after replacing the values, we get,
(5, 5) and (2, 1); we can get:
(5−2)2 + (5-1)2 = r2
As per the substitution,
= 32 + 42= r2
= 9 + 16 = r2
Therefore, r2 = 25
Hence, the radius of the given circle is 5 units.
Ques. Determine what the radius of a circle will be, considering that it has a diameter of 16 cm. (3 marks)
Ans: Given,
Diameter of the given circle= 16 cm
Now, going as per the formula we have learnt prior, it can be determined,
Radius of the circle = D / 2 (D = diameter)
Thus, Radius = 16 / 2
= 8 c.m.
Therefore, the radius of the given circle is 8 c.m.
Ques. Considering the circumference of a circle as 15 inches, determine what its radius will be? (3 marks)
Ans: As per the given equation, it can be said that the circumference is 15 inches.
Now, it gives an idea that the following formula needed to solve it will be,
R = C / 2π
Therefore, after replacing values,
R = 15 / 2 π
= (15×7) / (2×22) (obtained after cross replacing the values accordingly)
= 105 / 44 = 2.39
Hence, the circumference of the circle is 2.39 inches.
Ques. What is the circumference of a circle, presuming that the radius is given as 14 cm. (use the following: π = 22/7) (3 marks)
Ans: As per the given question, we need to determine the circumference considering the radius of the circle is given as = 14 c.m.
Now, after using the respective formula, we can say, Circumference = 2πr units
Substituting the values, we get,
= 2 * 22/7 * 14 (since we are using π = 22/7 in the equation)
= 88 c.m
Hence, the circumference of the circle is 88 c.m, considering if its radius is 14 c.m.
Ques. Determine a circle’s equation after considering its center as (1,2) alongside radius which is equal to 3 c.m? (3 marks)
Ans: As per the given equation, the center points of the given circle is (1, 2).
And, the radius is about 3 c.m.
Now, after using cartesian plane method, we get
Using the equation of the circle,
(x-1)2+ (y-2)2= 32
(x-1)2+ (y-2)2=9
(x2- 2x + 1)+ ( y2 -4y + 4) = 9
x2 + y2 - 2x -4y = 0
Therefore, the equation is as mentioned above.
Ques. Determine the approximate radius of a circle, after considering that it has an area of 36 m2. (3 marks)
Ans: In reference to the given equation, it says that the given area of the circle is 36 m2.
Now, to determine the approximate radius of the circle, we need to use the formula listed beneath,
r = √(A/π)
Now, after substituting the values, we get,
r = √36/ \(\prod\)
r = √36*7/22
r = 11.45
Hence, the radius of the given circle is roughly about 11.45 meters.
Ques. What is the radius of a circle whose chord length is about 8 c.m, besides the perpendicular distance from the focal center to the chord is 3 c.m? (3 marks)
Ans: As per the given equation, consider the chord length to be about = AB = 8 cm
Alongside, the perpendicular distance to be about = OP = 3 cm
Now, to find the radius, it is first important to consider its points = OA
Since we are already aware that a perpendicular line usually drawn from the center point to touch the chord often also bisects it.
Therefore, then it can be said, AP = PB = 4 c.m
In triangle OPA,
By Pythagoras theorem,
OA2 = OP2 +AP2
OA2 = 9 +16
OA2 = 25
OA = 5
Hence, the answer is = 5 cm
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