Circle and Sector of a Circle

Circle

For any circle, = a constant.

This constant is represented by π (pi). For many school-level calculations, π = is used.

Circle

Let the diameter be AB = d and the radius be OB = r.

🔹 Radius of a circle = × diameter of the circle, so

🔹 Circumference of a circle = π × diameter = πd = 2πr (∵ d = 2r)

🔹 Area of a circle = π × (radius)² = πr²

Semicircle

Semicircle

🔹 Area of a semicircle = × area of the circle = πr²

🔹 Length of the semicircular arc = πr

🔹 Perimeter of a semicircle = semicircular arc + diameter = πr + 2r = r(π + 2)

Circular Ring

Circular Ring

Let the outer radius be R and the inner radius be r.

🔹 Width of the ring = R − r

🔹 Area of the ring = area of the outer circle − area of the inner circle = πR² − πr² = π(R + r)(R − r)

Semi-Circular Ring

Semi-Circular Ring

Let the outer radius be R and the inner radius be r.

🔹 Width = R − r

🔹 Area of the semi-circular ring = = (R + r)(R − r)

🔹 Perimeter of the semi-circular ring = outer semicircular arc + inner semicircular arc + 2 × width

🔹 Therefore, perimeter = πR + πr + 2(R − r) = π(R + r) + 2(R − r)

Sector of a Circle

Sector of a Circle

In a circle with centre O and radius r, sector AOB subtends a central angle of θ°.

The length of arc AB can be calculated using the following formula.

🔹 Arc length, AB = × 2πr

Or, AB = rφ, where φ = radians

🔹 Area of the sector = × πr²

Or, area of the sector = × arc length × radius

Square Inscribed in a Circle

Square Inscribed in a Circle

🔹 Diagonal of the inscribed square = diameter of the circle = 2r.

🔹 Area of the square = = = 2r²

🔹 Perimeter of the square = 4 × √2 × r = 4√2r

Square Circumscribed About a Circle

Square Circumscribed About a Circle

For a square that exactly surrounds a circle, the side of the square equals the diameter of the circle, so side = 2r.

🔹 Area of the circumscribed square = (2r)² = 4r²

🔹 Perimeter of the circumscribed square = 4 × 2r = 8r

Incircle of a Square

Let the side of the square be a.

Incircle of a Square

🔹 Radius of the incircle r =

🔹 Circumference of the incircle = πa

🔹 Area of the incircle =

Circumcircle of a Square

Circumcircle of a Square

🔹 Diameter of the circumcircle = diagonal of the square = a × √2

🔹 Area of the circumcircle = =

🔹 Circumference of the circumcircle = 2π × = πa × √2 = √2πa

Incircle of an Equilateral Triangle

Let the inradius of equilateral triangle ABC be r.

Incircle of an Equilateral Triangle

🔹 Height of the equilateral triangle AD = 3r

🔹 Side of the equilateral triangle = a =

🔹 Area of the triangle =

🔹 Perimeter of the triangle

Circumcircle of an Equilateral Triangle

🔹 Circumradius of an equilateral triangle R = 2r.

🔹 Area of the circumcircle = πR² = 4πr²

🎯 Quick Revision

🔸 Circumference of a circle = 2πr

🔸 Area of a circle = πr²

🔸 Perimeter of a semicircle = πr + 2r

🔸 Area of a circular ring = π(R + r)(R − r)

🔸 Arc length of a sector = × 2πr

🔸 Area of a sector = × πr²

🔸 Area of a square inscribed in a circle = 2r²

🔸 Area of a square circumscribed about a circle = 4r²