Shadow of a Pole with a 30° Sun Angle

We are given:

• The sun's rays make an angle of $30^\circ$ with the ground.
• A vertical pole is $10\,\text{m}$ tall.

We want the length of the shadow cast on level ground.

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1. Geometric setup

Draw a right triangle:

• Vertical side = height of the pole = $h = 10$ m.
• Horizontal side = length of the shadow = $s$ (unknown).
• The hypotenuse = line of sunlight.
• The angle between the ground and the sun ray is $30^\circ$.

So the angle at the ground is $30^\circ$.

In that right triangle, the side opposite the angle is the height of the pole, and the side adjacent to the angle is the shadow length.

By definition of the tangent function:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{s}$

Here:

$\tan(30^\circ) = \frac{10}{s}$

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2. Solve for the shadow length

Rearrange:

$s = \frac{10}{\tan(30^\circ)}$

Recall the exact value:

$\tan(30^\circ) = \frac{1}{\sqrt{3}}$

Therefore:

$s = \frac{10}{1/\sqrt{3}} = 10\sqrt{3} \text{ m}$

Numerically:

$\sqrt{3} \approx 1.732 \Rightarrow s \approx 10 \times 1.732 = 17.32\,\text{m}$

So the shadow is:

$\boxed{10\sqrt{3} \text{ m} \approx 17.3\,\text{m}}$

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3. What the visualization shows

Use the sliders to change:

• Pole height (in meters).
• Sun angle (in degrees, measured from the ground).

The canvas shows:

• A horizontal ground line.
• A vertical pole at the origin.
• A glowing triangle of light: pole (vertical), shadow (horizontal), and the sun ray (hypotenuse).
• The shadow length dynamically computed by $s = \frac{\text{height}}{\tan(\text{angle})}$

When you set Height = 10 m and Sun Angle = 30°, the shadow length in the scene corresponds to $10\sqrt{3}$ m.

Note: As the angle gets smaller (sun lower in the sky), the shadow gets longer; as the angle approaches $90^\circ$ (sun directly overhead), the shadow shrinks toward zero.