Parts of a circle

A circle is a set of all points equidistant from a fixed point called the Centre. It is one of the most fundamental shapes in geometry and is widely used in real-life applications like wheels, clocks, and orbits.

The parts of a circle include the center, radius, diameter, circumference, chord, arc, sector, segment, and tangent.

Key Components of a Circle

A circle has several important parts and properties that help describe its structure. Here are the main parts of a circle:

Radius

• The Radius (r) is the distance from the center to any point on the circle.
• It is always half of the diameter.

Formula:

• radius = d / 2 (from Diameter)
• r = \sqrt{(x-h)^2 + (y -k)^2 }

where,

• r = radius
• ( h, k) = Centre of circle
• ( x, y) = any point on the circle

Diameter

• The Diameter (d) of a circle is the longest straight-line segment that passes through its center and touches both sides of the boundary.
• It is denoted by d and is twice the radius.
• It is the longest chord of the circle.
• It is always twice of radius.

Formula: The diameter of a circle can be calculated through different methods:

• d = 2 r (from radius)
• d = C / 𝞹 (from circumference)
• d = 2 \sqrt{\frac{A}{\pi }} (from area of circle)

Real life-examples include: Wheel rotation, pipe flow rate.

Circumference

• The Circumference of the circle is the total distance around its boundary.
• It is the perimeter of the circle and is denoted by C. It plays an important role in geometry, physics, engineering, and real-world applications.

Formula:

C = 2 πr (using the radius)
C = πd (using the diameter)

Real-Life Examples: Measuring circular paths, rotational motion.

Chord

• A Chord is a line segment that connects two points on the circle.
• A circle has an infinite number of chords. The diameter is a chord of the circle.
• A chord divides the circle into two regions the minor segment and the major segment on the area covered by a circle.

Formula:

chord length = 2\sqrt{r^2-d^2 }

where d is the perpendicular distance from the center to the chord (not the diameter).

Tangent

• A Tangent of a circle is a straight line that touches the curve of the circle at exactly one point. (It does not intersect or enter the interior of the circle).
• It is also known as a non-intersecting line.
• The two important concepts of tangent are slope and point on the line.

Formula:

• For circle (x − h)² + (y − k)² = r², the tangent at (x1, y1) is: (x₁ - h) (x - h) + (y₁ - k) (y - k) = r²
• For circle x² + y² = r²_, the tangent at (x₁, y₁ ) is xx₁ + yy₁ = r²

Secant

• A Secant of the circle is a line that cuts across the circle intersecting the circle at two distinct points.
• The difference between a chord and a secant is that a chord is a line segment whose endpoints are on the circumference of a circle.

Formula:

(length of secant) × (its external segment) = (length of the tangent segment) ²

Arc

• An Arc of a circle is the curved part of a circle or part of the circumference of a circle.
• The curved portion of an object is mathematically known as an arc.
• There are two types of arc namely,
  • Minor arc,
  • Major arc.

Formula:

When θ is in radians:

• Arc length = θ × r (used in radians)

When θ\thetaθ is in degrees:

• Arc Length = \frac{\theta}{360} \times 2 \pi r

Sector

• A Sector is pie pie-shaped part of a circle made of an arc along with two radii dividing the circle into major and minor sectors.
• The larger portion is known as Major Sector and the smaller portion is known as the Minor Sector.

Formula:

Area of Sector=(θ / π360°) x r² (when the angle is given)
length of Sector=(θ πr❩ / 180 (when the length is given)
Perimeter of Sector= 2 r + ((θ/ 360) x 2 π r)

Solved Example of Parts of a Circles

Example 1: The radius of circle is 14 meter. Find the area of circle.

Here,
Radius of circle = 14 meter

Area of circle = πr²
Area = π(14)²
Area = 3.14 * 196
Area = 615.44 square meter

Example 2: The circumference of wheel is 600 cm. Find the radius and diameter.

Here,
Circumference of circle = 600 cm

Formula for circumference of circle = 2πr
Let us substiute the value of circumference
600 = 2πr
600/2 = 2*3.14*r
300 = 6.28r
r = 300 / 6.28
r = 95.54
Diameter = 2 * Radius
95.54*2
Diameter = 191.08

Radius = 95.54
Diameter = 191.08

Example 3: The diameter of sector is 30 cm, and the angle of sector is 45°. Find the area of the sector.

Here,
Diameter = 30cm and angle = 45 degree

Area of sector=𝛉 / 360 × 2πr²
=45 / 360 × 2 (15)²
=1/ 8 × 2 π(225)
=225 π / 8

A ≈ 225 × 3.1416 /8
A ≈ 706.86 / 8
A ≈ 88.36 cm²

Therefore, Area of sector is 88.36 cm²

Example 4: The radius of the arc is 50 cm and the angle substended by the arc is 90 . Find the length of arc.

Here,

Radius of arc= 50 meter

Angle subtend by the arc=90°

Length of arc = 𝛳 / 360 x 2πr
= 90/ 360 x 2π(50)
= 1 /4 x 2 π(50)
= 100π / 4
= 25π

Length of arc ≈ 25 × 3.1416 = 78.54 meter

Therefore, Length of arc is 78.54 meter

Related Articles

• Circle
• Arc of Circle
• Ways to calculate the area of a Circle

Unsolved Question on Parts of a Circle

Question 1:The circumference of wheel is 540 cm. Find the radius and diameter.

Question 2: The radius of circle is 21 meter. Find the area of circle.

Question 3: The radius of sector is 20 cm. The angle subtended by sector is 90°, find the area of the sector .

Question 4: A curved road sign is part of a circle with a radius of 6 meters. The arc of the sign subtends an angle of 75° at the center.
(a) Find the arc length of the sign.
(b) Find the area of the sector representing the curved sign.

Answer Sheet

1) radius = 85.9 , diameter = 171.8
2) 1384.74 m²
2) 3.1416 m²
3) 7.85 m, 23.56 m