All formulas of circle

All formulas of circle are essential concepts in geometry.

Definition

A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius (r).

Here

• Diameter: The diameter is twice the radius: D = 2r.
• Circumference: The circumference (C) is the total distance around the circle: $$C=2\pi r$$
• Area: The area enclosed by the circle is

• Chord: A chord is a line segment joining two points in the circle.
• Arc: An arc is a part of  the circle’s circumference.
• Sector: A sector is a region enclosed by two radii and an arc.

Standard equation of a circle

The equation of circle depends on its center and radius

(a) Circle centered at the origin (0,0)

If the center is at the origin and the radius is r, then the equation is:

(b) Circle centered at (h,k)

If the center is at (h, k), and the radius is r, the equation is :

General equation of a circle

The expanded form of a circle’s equation is :

• The center is given by (-g,-f).
• The radius is

Important properties of circle

1. Tangents and normals to a circle

A tangent is a line that touches the circle at exactly one point.The tangent to the circle equation

In a general equation, x^2 +y^2 +2gx+2fy+c=0 , the equation of the tangent at (x_1​,y_1​) is:

2. Polar form of a circle: In polar coordinates (r,θ), the equation of a circle centered at the origin is

3. Position of any points with respect to a circle:

Distance formula: Calculate the distance d of the point (x_1,y_1) from the center (h,k):

Now, compare d with the radius r:

• If d<r: The point is inside the circle.
• If d=r : The point is on the circle.
• If d>r : The point is outside the circle.

4. Length of tangent from the point (x_1, y_1) to the circle: The length of the tangent from an external point (x_1, y_1) to a circle with equation

Example

Given circle

Find the tangent length from (7,1)

Using the formula

5. Angle of Intersection of two circles

The angle of intersections between two circles refers to the angle between their tangents at the points of intersection.It can be calculated using the formula

Example

Given two circles

(x-2)^2+(y-3)^2=25 – Center(2,3), Radius r_1=5

(x-7)^2+(y-6)^2=16 -Center(7,6), Radius r_2=4

Step 1: Find distance between centers

Step 2: Apply the formula

Step 3: Find θ

Special cases

1. Orthogonal circles ( θ=90∘)

• Two circles are orthogonal (intersect at right angles) if their centers and radii satisfy

• This means cos⁡θ=0 and θ=90∘

2. Concentric circles

• If two circles share the same center (d = 0), they do not intersect or  are the same circle .

3. Circles touching externally (θ=0∘)

• If d=r1+r2, then they touch externally.

4. Circles touching internally (θ=180∘)

• If d=∣r1​−r2​∣ they touch internally.

These are All formulas of circle.

How to make math fun through storytelling: The secret of the shadow pyramid

How to make math fun through storytelling: The math explorers cautiously stepped into a long hall, but something was off.The floor was covered in golden sand, and that moved in waves shifting unpredictably beneath their feet .The walls were lined with ancient measuring tools—rulers, scales, and hourglasses—glowing faintly with magical energy.

Chapter 3: The Shifting Sands of Measurement

Joey took a step forward, but the sand beneath him suddenly crumbled, nearly swallowing his foot.”Whoa! The ground isn’t stable, he gasped. Max pointed to an inscription on the wall.”Look ! A puzzle!Maybe it will help us cross.”

The inscription read: “To find the safe path, convert and compare:

• A rope is 2.5 meters long . How many centimeters is that?
• One stone weighs 3,000 grams. “How many kilograms is that?”

Ross grinned. “We just need to convert the units!”

“We know that 1 meter equals 100 centimeters.” Max said, writing on the sand with her finger .”So 2.5 meters is…”

2.5×100=250 cemtimeters.

“And for the weight ,”joey continued . “1 kilogram is 1000 grams.So we divide 3000 by 1000.”

3000÷1000=3 kilograms .

As soon as they said the answers aloud, the numbers curved into the walls began to glow . The sand shifted, revealing solid stepping stones that created a safe path forward .

“We did it!” Max cheered. “The numbers unlocked the way!”

One by one, the explorers carefully stepped onto the stones, making their way across the hallway.But as they reached the middle, another inscription appeared on a stone piller:

“To continue forward, solve these conversions:

• A river is 5 kilometers long.How many meters is that ?
• A bottle holds 2.5 liters of water. How many milliliters is that?
• A clock ticks every 15 minutes.How many ticks occur in 2 hours?

“We can do this !” said Ross, thinking quickly.

• 5 kilometers×1000=5000 meters.
• 2.5 liters×1000=2500 milliliters.
• 2 hours = 120 minutes.120÷15=8 ticks

Once they spoke the correct answers, the stepping stones ahead lit up, revealing the rest of the safe path .The shifting sand behind them swallowed the remaining unstable floor, making it impossible to turn back. “Looks like we solved it just in time,” Joey said with a relieved smile. “Let’s keep going !”

With their confidence growing, the team pressed forward, ready for the next challenge inside the shadow pyramid.

To be continued…

Stay tuned for Chapter 4: The room of ratios

Types of progression

A progression is a sequence of numbers following a specific pattern. The most common types of progression are:

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference (d).

1. General form of an AP: An arithmetic sequence is written as

a, a+d, a+2d, a+3d+…

Where

• a=first term
• d=common difference
• n=number of terms
• l=last term

2. The nth term of an AP is given by

3. Sum of first n terms

The sum of the first n terms is

Or,

Example 1: Find the 10th term of the AP: 3, 7, 11, 15,…

Here,

a=3, d=7-3=4, n=10

Using:

So, the 10th term is 39.

Example 2: Find the sum of the first 20 terms of the AP: 5, 8, 11, 14,…

Here,

a=5,d=8-5=3,n=20

Using

So the sum of the first 20 terms is 670.

Geometric Progression

A geometric progression is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed constant called the common ratio (r).

General form of a GP

Where

• a=first term
• r=common ratio
• n=number of terms

1.nth term of a GP:

2. Sum of first n terms of a GP

3. Sum of infinite GP(|r| < 1)

If ∣r∣<1, the infinite sum converges to:

Example 1: Find the 12th term of the GP: 5, 10, 20, 40, 80,…

Here,

a=5, r=10/5=2

Using

Example 2: Find the sum of the first 12 terms of the GP: 5, 10, 20, 40, 80,…

a=5,r=10/5=2

Using

Harmonic Progression

A harmonic progression is a sequence of numbers whose reciprocals form an Arithmetic progression (AP)

General form of an HP

1. nth term of HP

2. The sum of the first $n$ terms of the harmonic progression is:

Example 1: Find the 5th term of the HP: 3, 6, 9, 12,…

Here,

a=3, d=3

First find the 5th term of the HP

A5​=a+(n−1)d=3+(5−1)×3=3+12=15

Now, the 5th term of the HP is:

​Special algebraic series

i) Sum of First n natural numbers:

ii) Sum of First n squares

iii) Sum of first n cubes