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<title>Adventure 1 — Measuring the World (Solution)</title>
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<h1>Adventure 1 — Measuring the World (Solution)</h1>
<div class="hero-right">
<img src="images/adventure_1_image_1_solution.png"
alt="Measure the World activity illustration with θ, D, and C">
</div>
<p>
Long before satellites and laser measuring tools, no one knew how big the Earth was. People guessed,
argued, and made wildly different claims. Some thought the planet was enormous; others believed it was
much smaller. What they all agreed on was this: the Earth was far too large to measure directly.
</p>
<p>
Then came Eratosthenes. He lived more than two thousand years ago and worked in the Great Library of
Alexandria — a place filled with scrolls, scholars, and a great deal of sunshine. He heard something
curious: in a southern city called Syene, the Sun shone straight down a well on the summer solstice.
At the very same moment in Alexandria, a vertical stick cast a small shadow.
</p>
<p>
This tiny difference — the angle of a single shadow — was the clue. Eratosthenes realized he could
“slice” the Earth using shadows the same way we slice area under a line: by taking a simple local
measurement and using it to understand something much bigger.
</p>
<p>
In this activity, you will follow the same idea. You start from the angle θ and the distance D between
the two cities, compute the ratio R = C / D, and then work your way to Earth’s circumference in meters
and kilometers. Finally, you compare your estimate with a present-day value and see how close you came.
The angle θ is the fraction of the full circle between the two cities. The ratio R = C / D tells you
how many times that distance D fits around the full Earth. From the geometry of the circle, you can use
the relationship R = 360° / θ.
</p>
<h2>1️⃣ Given angle and distance</h2>
<p>
Use the diagram to read the value of θ (in degrees). In Eratosthenes’ classic experiment, θ ≈ 7.2°.
The distance between Syene and Alexandria along the surface is D meters.
</p>
<div class="row-grid">
<div>
<label for="D_m">Distance between Syene and Alexandria D (meters):</label>
<input id="D_m" type="number" step="1" value="787500" readonly class="correct">
</div>
</div>
<h2>2️⃣ Ratio R = Earth’s circumference / D</h2>
<p>
Now use the idea above to compute the ratio R. Remember, R tells you how many times the distance D fits
around the entire Earth.
</p>
<p style="margin-left:0.7rem;">
<strong>R = 360° / θ</strong>
</p>
<div class="row-grid">
<div>
<label for="R_ratio">Compute R = 360 ÷ θ :</label>
<div style="font-size:0.8rem;color:#666;margin-top:0.2rem;"><em>(because θ tells you what fraction of the 360° circle separates the two cities)</em></div>
<input id="R_ratio" type="number" step="0.01" value="50" readonly class="correct">
</div>
</div>
<h2>3️⃣ Earth’s circumference in meters and kilometers</h2>
<p>
Once you know D (in meters) and R, you can find the Earth’s circumference in meters using:
</p>
<p style="margin-left:0.7rem;">
<strong>Circumference in meters = R × D</strong>
</p>
<p>
Then convert to kilometers by dividing by 1000:
</p>
<p style="margin-left:0.7rem;">
<strong>Circumference in kilometers = (circumference in meters) ÷ 1000</strong>
</p>
<div class="row-grid">
<div>
<label for="C_m">Your estimate of Earth’s circumference (meters):</label>
<input id="C_m" type="number" step="1" value="39375000" readonly class="correct">
</div>
<div>
<label for="C_km">Your estimate of Earth’s circumference (kilometers):</label>
<input id="C_km" type="number" step="0.1" value="39375" readonly class="correct">
</div>
</div>
<h2>4️⃣ Compare with a present-day value</h2>
<p>
A commonly used present-day value for Earth’s circumference is about
<strong>40,075 kilometers</strong>.
</p>
<p>
First, compute the <em>difference</em> between the present-day value and your estimate:
</p>
<p style="margin-left:0.7rem;">
<strong>difference = present-day value − your estimate</strong>
</p>
<p>
Then use this difference to find the <em>percent error</em> of your estimate:
</p>
<p style="margin-left:0.7rem;">
<strong>percent error = (difference ÷ present-day value) × 100%</strong>
</p>
<div class="row-grid">
<div>
<label for="abs_err">Difference (present value − your estimate) in km:</label>
<input id="abs_err" type="number" step="0.1" value="700" readonly class="correct">
</div>
<div>
<label for="pct_err">Percent error (%):</label>
<input id="pct_err" type="number" step="0.01" value="1.75" readonly class="correct">
</div>
</div>
<div class="solution" style="display:block;">
<strong>Filled answers (based on the diagram)</strong>
<ul>
<li><strong>θ = 7.2°</strong> (given in the diagram)</li>
<li><strong>D = 5000 stadia × 157.5 m/stadium = 787,500 m</strong></li>
<li><strong>R = 360 ÷ 7.2 = 50</strong></li>
<li><strong>Earth’s circumference (meters) = R × D = 50 × 787,500 = 39,375,000 m</strong></li>
<li><strong>Earth’s circumference (km) = 39,375,000 ÷ 1000 = 39,375 km</strong></li>
<li><strong>Difference = 40,075 − 39,375 = 700 km</strong></li>
<li><strong>Percent error = (700 ÷ 40,075) × 100% ≈ 1.75%</strong></li>
</ul>
</div>
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