Derivation of the Formula of a Circle

 

🎯 Goal

We want to derive the formula

$$A = \pi r^2$$

where $A$ is the area of the circle and $r$ is the radius.

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🧩 Step 1: Recall what a circle is

A circle is the set of all points that are at a fixed distance $r$ (the radius) from a central point.

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🧮 Step 2: Divide the circle into equal sectors

Imagine cutting the circle into many thin slices (like pizza slices).
If we rearrange these slices alternately (flipping every other one), they begin to resemble a parallelogram or rectangle shape.

As the number of slices increases, the shape more closely approximates a rectangle.

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📏 Step 3: Determine the dimensions of this "rectangle"

• Base (length): Half of the circumference of the circle

  $$\text{Base} = \frac{1}{2} \times \text{Circumference} = \frac{1}{2} \times 2\pi r = \pi r$$

• Height: Equal to the radius of the circle $r$

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📐 Step 4: Find the area of this “rectangle”

Area of rectangle = Base × Height

$$A = (\pi r) \times r = \pi r^2$$

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🌟 Step 5: The limiting argument (for precision)

As the number of sectors increases infinitely, the rearranged shape becomes exactly a rectangle — not just an approximation.
Thus, mathematically, the area of the circle is:

$$\boxed{A = \pi r^2}$$

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🔍 Step 6: Alternative Derivation (Using Integration)

For those familiar with calculus:

The area of a circle can also be found by integrating horizontal strips from $x = -r$ to $x = r$ under the curve:

$$x^2 + y^2 = r^2 \Rightarrow y = \sqrt{r^2 - x^2}$$

So,

$$A = 2 \int_{0}^{r} \sqrt{r^2 - x^2} \, dx$$

Using trigonometric substitution $x = r \sin\theta$, this becomes:

$$A = 2 \int_{0}^{\pi/2} r^2 \cos^2\theta \, d\theta = \pi r^2$$

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🌀 Step 7: Intuitive Meaning

• The π (pi) represents the constant ratio between a circle’s circumference and its diameter.

• The r² represents how the area scales — if you double the radius, the area quadruples.

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✅ Final Formula:

$$\boxed{A = \pi r^2}$$

Alternative derivation (step-by-step) using integration

We’ll compute the area of a circle of radius $r$ by integrating vertical slices.

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1. Equation of the circle and vertical slice

Circle: $x^2 + y^2 = r^2$.
Solve for the top half: $y = \sqrt{r^2 - x^2}$

For a given $x$ the vertical length of the circle is

$$\text{top} - \text{bottom} = \sqrt{r^2-x^2} - \big(-\sqrt{r^2-x^2}\big) = 2\sqrt{r^2-x^2}.$$

So the area is the integral of those vertical slices from $x=-r$ to $x=r$:

$$A=\int_{-r}^{r} 2\sqrt{r^2-x^2}\,dx .$$

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2. Use symmetry to simplify

The integrand is even, so

$$A = 2\int_{-r}^{r}\sqrt{r^2-x^2}\,dx = 4\int_{0}^{r}\sqrt{r^2-x^2}\,dx .$$

(We pulled a factor 2 outside and used $\int_{-r}^{r} = 2\int_{0}^{r}$.)

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3. Trigonometric substitution

Set $x = r\sin\theta$. Then

$$dx = r\cos\theta\,d\theta, \qquad \sqrt{r^2-x^2} = \sqrt{r^2 - r^2\sin^2\theta} = r\cos\theta.$$

Change the limits:

• $x=0 \Rightarrow \sin\theta=0 \Rightarrow \theta=0$,

• $x=r \Rightarrow \sin\theta=1 \Rightarrow \theta=\tfrac{\pi}{2}$.

Substitute into the integral:

$$\int_{0}^{r}\sqrt{r^2-x^2}\,dx = \int_{0}^{\pi/2} \big(r\cos\theta\big)\big(r\cos\theta\,d\theta\big) = r^2\int_{0}^{\pi/2}\cos^2\theta \,d\theta.$$

Thus

$$A = 4 \cdot r^2 \int_{0}^{\pi/2}\cos^2\theta \,d\theta.$$

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4. Evaluate $\displaystyle \int_{0}^{\pi/2}\cos^2\theta\,d\theta$

Use the identity $\cos^2\theta=\dfrac{1+\cos2\theta}{2}$:

$$\int_{0}^{\pi/2}\cos^2\theta\,d\theta = \int_{0}^{\pi/2}\frac{1+\cos2\theta}{2}\,d\theta = \frac{1}{2}\int_{0}^{\pi/2}1\,d\theta \;+\; \frac{1}{2}\int_{0}^{\pi/2}\cos2\theta\,d\theta.$$

Compute each term:

• $\displaystyle \frac{1}{2}\int_{0}^{\pi/2}1\,d\theta = \frac{1}{2}\cdot\frac{\pi}{2} = \frac{\pi}{4}$.

• $\displaystyle \frac{1}{2}\int_{0}^{\pi/2}\cos2\theta\,d\theta = \frac{1}{2}\left[\frac{\sin2\theta}{2}\right]_{0}^{\pi/2} = \frac{1}{4}\big(\sin\pi - \sin0\big) = 0.$

So

$$\int_{0}^{\pi/2}\cos^2\theta\,d\theta = \frac{\pi}{4}.$$

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5. Finish the computation

Plug that back into the area:

$$A = 4 r^2 \cdot \frac{\pi}{4} = \pi r^2.$$

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✅ Final result

$$\boxed{A=\pi r^2}$$

A=πr2​

​

Short derivation using polar coordinates

Use polar coordinates $(\rho,\theta)$ where $x=\rho\cos\theta,\; y=\rho\sin\theta$.
The area element in polar coordinates is $dA=\rho\,d\rho\,d\theta$ (the Jacobian).

The disk of radius $r$ is described by $0\le \rho\le r,\; 0\le\theta\le 2\pi$. So

$$A=\iint_{D} dA=\int_{\theta=0}^{2\pi}\int_{\rho=0}^{r} \rho\,d\rho\,d\theta.$$

Evaluate the inner integral:

$$\int_{0}^{r}\rho\,d\rho=\frac{\rho^{2}}{2}\Big|_{0}^{r}=\frac{r^{2}}{2}.$$

Now the outer integral:

THE CALL TO DIVINE RENEWAL

 

THE CALL TO DIVINE RENEWAL

 

“And God said, Let us make man in our image, after our likeness: and let them have dominion…”
— Genesis 1:26–27 (KJV)

In the beginning, the Eternal Father breathed into man His own Spirit and likeness. Man was not made a mortal creature struggling for survival, but a divine vessel carrying the breath of the Almighty. He was formed from the dust of the earth, yet quickened by the Spirit of God to live in perfect harmony — spirit, soul, and body — bearing the radiance of divine glory. The Hebrew phrase “b’tzelem Elohim” (in the image of God) describes not only resemblance but representation: man was designed to manifest the invisible God in visible form upon the earth.

When Adam walked in the Garden, he walked clothed with light. His consciousness was united with God’s presence; his mind was clear and his body immortal. There was no trace of decay, disease, or death, for he partook of the life of the Creator Himself. His thoughts, emotions, and actions flowed in seamless rhythm with the divine will. In this state, man fulfilled his original calling — to rule creation in righteousness and reflect the glory of his Maker.

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🌿 The Fall: When Corruption Entered and Mortality Began

“Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men…”
— Romans 5:12 (KJV)

But when man turned from God’s voice and followed self-will, something catastrophic occurred: the divine current of life that sustained him was broken. Separation from the Source of Life introduced a new condition — corruption. What once was immortal began to degenerate; what once was luminous became darkened. Man’s body, which had been the temple of divine energy, began to wither under the weight of sin. His mind, once aligned with heaven’s wisdom, was clouded by fear and confusion.

The glory departed, and mortality began its slow reign. The image was not erased, but it was marred. The Spirit that animated man withdrew, and flesh returned to dust. Yet even in the midst of judgment, the promise of redemption glimmered. The same God who formed man from the earth would one day breathe upon humanity again — this time through His Son.

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✝️ Redemption: The Return of Divine Life through Christ

“But if the Spirit of him that raised up Jesus from the dead dwell in you,
he that raised up Christ from the dead shall also quicken your mortal bodies
by his Spirit that dwelleth in you.”
— Romans 8:11 (KJV)

In the fullness of time, the Word became flesh. Jesus Christ, the second Adam, stepped into the fallen world to restore what was lost. He embodied the original pattern — man perfectly indwelt by God. He said, “I and my Father are one.” In Him, the image of God was fully manifested again.

Through His death and resurrection, Christ did not merely purchase forgiveness; He released the divine life that would renew man from the inside out. The Spirit that raised Jesus from the dead is the same Spirit that now indwells the believer, quickening mortal flesh and renewing corrupted nature. Salvation, therefore, is not only deliverance from sin but participation in divine vitality.

This is the mystery Paul declared: “Christ in you, the hope of glory.” The same Spirit who hovered over the chaos in Genesis now hovers within the believer, recreating, transforming, and energizing every cell with resurrection life.

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🌺 Partakers of the Divine Nature

“According as his divine power hath given unto us all things that pertain unto life and godliness,
through the knowledge of him that hath called us to glory and virtue:
Whereby are given unto us exceeding great and precious promises:
that by these ye might be partakers of the divine nature,
having escaped the corruption that is in the world through lust.”
— 2 Peter 1:3–4 (KJV)

This is the call to divine renewal — not to reform a corrupted life, but to awaken the divine seed implanted by grace. Through the knowledge of Christ and the indwelling Spirit, we become partakers of the divine nature. The Greek word koinōnos translated as “partakers” means sharers or partners — not distant recipients, but living participants of God’s very essence.

The believer’s journey, therefore, is one of transformation into the likeness of Christ — to think His thoughts, speak His words, and manifest His life. The same life that flowed through Jesus flows now in every regenerated believer. This life is not bound by decay, for it springs from eternity itself.

When Moses stood upon Mount Nebo at one hundred and twenty years old, Scripture records, “his eye was not dim, nor his natural force abated.” (Deuteronomy 34:7) Why? Because he lived in the constant presence of God. He reflected the light of divine communion — even his face shone with glory.

This is not myth; it is divine potential. When the Israelites came out of Egypt, “there was not one feeble person among their tribes” (Psalm 105:37). When they rested in obedience to the Sabbath — the covenant of renewal — their bodies and souls were restored, just as we partake of the Lord’s Table today, receiving the same life of the eternal Christ.

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🌞 Immortality Brought to Light

“But is now made manifest by the appearing of our Saviour Jesus Christ,
who hath abolished death, and hath brought life and immortality to light through the gospel.”
— 2 Timothy 1:10 (KJV)

Christ did not come merely to save the soul from damnation; He came to reveal the mystery of divine immortality — life that transcends decay. The gospel is the unveiling of immortality, the revelation that death has been defeated and eternal life is available now through union with Christ.

When Paul was bitten by a venomous serpent and suffered no harm (Acts 28:3–6), it was not an anomaly but a demonstration of divine life at work in a mortal body. When Peter’s shadow healed the sick, it was the overflow of the indwelling Spirit animating human form. When Jesus said, “He that believeth on me, the works that I do shall he do also; and greater works than these shall he do,” (John 14:12) He was not exaggerating but declaring the destiny of redeemed humanity.

The same Spirit that raised Christ from the dead is transforming the sons and daughters of God into living conduits of divine energy. This renewal begins in the spirit, reshapes the mind, and radiates through the body until every cell vibrates with resurrection life.

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🌿 The Vision of the Hebraic Path

The Hebraic Path of Divine Renewal calls every believer back to the root — to walk again in the ancient rhythm of creation: Teshuva (return), Tefilah (communion), Hitbodedut (reflection), Tzedakah (love in action), and Machashavah (sanctified imagination). These are not rituals, but living conduits that restore harmony between body, soul, and spirit under the lordship of the Messiah.

This path is not about escaping the physical world but transforming it — bringing heaven into flesh, eternity into time, and divinity into daily life. It is the restoration of the Edenic walk, the renewal of the body as the temple of the Spirit, and the fulfillment of Christ’s words: “That they all may be one; as thou, Father, art in me, and I in thee.” (John 17:21)

The call to divine renewal is, therefore, a call to live as Jesus lived on earth — full of the Holy Spirit, radiant in love, unbroken by death, and overflowing with life.