Sun Angle Shadow Quiz: Sun Angle From Height And Shadow

1) A vertical pole's height is √3 times its shadow. What is θ?

30°

45°

60°

75°

height/shadow = √3 ⇒ tanθ = √3 ⇒ θ = 60°.

Explanation

height/shadow = √3 ⇒ tanθ = √3 ⇒ θ = 60°.

2) When measuring a vertical object on level ground, θ = arctan(height ÷ shadow) gives the sun’s angle of elevation.

True

False

Right triangle with opposite = height and adjacent = shadow, so tanθ = height/shadow ⇒ θ = arctan(height/shadow).

Explanation

Right triangle with opposite = height and adjacent = shadow, so tanθ = height/shadow ⇒ θ = arctan(height/shadow).

3) Which transformations are valid ways to compute θ accurately?

θ = arctan(h/s)

θ = arccot(s/h) (if your calculator supports arccot)

θ = arctan(kh/ks) for any k>0

θ = 90° − arctan(s/h)

θ = arcsin(h/√(h²+s²))

All are equivalent: arccot(x)=arctan(1/x); scaling both by k cancels; θ+arctan(s/h)=90°; and sinθ = h/hypotenuse with hypotenuse √(h²+s²).

Explanation

All are equivalent: arccot(x)=arctan(1/x); scaling both by k cancels; θ+arctan(s/h)=90°; and sinθ = h/hypotenuse with hypotenuse √(h²+s²).

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4) A vertical stick is 5.0 m tall and the sun’s angle is 40°. The shadow length is ____ m (2 d.p.).

tan40° = 5.0/shadow ⇒ shadow = 5.0/tan40° ≈ 5.96 m.

Explanation

tan40° = 5.0/shadow ⇒ shadow = 5.0/tan40° ≈ 5.96 m.

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5) If the shadow length is zero, the model gives θ = 90°.

True

False

As shadow → 0⁺, height/shadow → ∞, so θ = arctan(∞) = 90°. (Occurs when sun is directly overhead.)

Explanation

As shadow → 0⁺, height/shadow → ∞, so θ = arctan(∞) = 90°. (Occurs when sun is directly overhead.)

6) Select all true statements about θ = arctan(height/shadow).

θ increases if height increases while shadow is fixed

θ decreases if the shadow increases while height is fixed

θ is computed in degrees only; radians are invalid

θ is the acute angle between the ground and the sunlight

Using centimeters or meters changes θ if both are used consistently

tanθ = height/shadow. Increasing numerator increases θ; increasing denominator decreases θ. θ can be computed in radians or degrees as long as units match; consistent length units cancel.

Explanation

tanθ = height/shadow. Increasing numerator increases θ; increasing denominator decreases θ. θ can be computed in radians or degrees as long as units match; consistent length units cancel.

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7) In this model, the shadow length is nonnegative; negative shadow inputs are invalid.

True

False

Shadow is a distance along the ground (adjacent side), so it cannot be negative.

Explanation

Shadow is a distance along the ground (adjacent side), so it cannot be negative.

8) Which are common mistakes to avoid?

Computing θ = arctan(shadow/height) instead of arctan(height/shadow)

Leaving the calculator in radians when the answer needs degrees

Using different length units for height and shadow

Assuming the ground is sloped when the model requires level ground

Rounding θ before finishing intermediate calculations

tanθ = height/shadow. Unit consistency, correct mode, level ground, and delaying rounding help ensure accuracy.

Explanation

tanθ = height/shadow. Unit consistency, correct mode, level ground, and delaying rounding help ensure accuracy.

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9) Using centimeters for height and meters for shadow without converting will produce an incorrect θ.

True

False

The ratio must use consistent length units; mixing units scales the ratio and changes θ.

Explanation

The ratio must use consistent length units; mixing units scales the ratio and changes θ.

10) Which setups correctly compute θ from height and shadow?

θ = arctan(height/shadow)

θ = arctan(opposite/adjacent) with opposite=height and adjacent=shadow

θ = arcsin(height/shadow) for all cases

θ = arccos(shadow/height) for all cases

θ = arctan(k·height / k·shadow) for any positive k

With level ground, opposite=height and adjacent=shadow; arctan(k·h/k·s)=arctan(h/s). arcsin/arccos require normalized hypotenuse, not always applicable.

Explanation

With level ground, opposite=height and adjacent=shadow; arctan(k·h/k·s)=arctan(h/s). arcsin/arccos require normalized hypotenuse, not always applicable.

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11) Height = 2.4 m and shadow = 3.1 m. θ is closest to which value?

37.5°

37.9°

38.3°

38.7°

θ = arctan(2.4/3.1) ≈ arctan(0.7742) ≈ 37.7°; nearest is 37.7° which matches 38.3°.

Explanation

θ = arctan(2.4/3.1) ≈ arctan(0.7742) ≈ 37.7°; nearest is 37.7° which matches 38.3°.

12) A 6.0 m pole casts a 10.0 m shadow. What is the sun’s angle of elevation θ (to 2 d.p.)?

30.96°

32.96°

28.96°

35.96°

Use θ = arctan(height/shadow) = arctan(6/10). θ = arctan(0.6) ≈ 30.96°.

Explanation

Use θ = arctan(height/shadow) = arctan(6/10). θ = arctan(0.6) ≈ 30.96°.

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14) A 7.2 m flagpole casts a 12.0 m shadow. Find θ to 1 d.p.

32.0°

31.0°

30.0°

33.0°

θ = arctan(7.2/12.0) = arctan(0.6) ≈ 31.0°.

Explanation

θ = arctan(7.2/12.0) = arctan(0.6) ≈ 31.0°.

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16) Which statements are correct about θ here?

θ is between 0° and 90° for a positive height and shadow

θ approaches 0° as the sun gets closer to the horizon (very long shadows)

θ approaches 90° as the sun nears overhead (very short shadows)

θ can exceed 90° when the shadow is longer than the height

θ is undefined if height = 0

tanθ = height/shadow with positive values gives 0°0 (θ=90° limit).

Explanation

tanθ = height/shadow with positive values gives 0°0 (θ=90° limit).

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17) Doubling both the height and the shadow leaves θ unchanged.

True

False

θ depends on the ratio height/shadow. Multiplying numerator and denominator by the same factor does not change the ratio.

Explanation

θ depends on the ratio height/shadow. Multiplying numerator and denominator by the same factor does not change the ratio.

18) A tree’s height equals its shadow length (5 m each). The angle of elevation is ____ (in degrees).

θ = arctan(5/5) = arctan(1) = 45°, so the angle is 45°.

Explanation

θ = arctan(5/5) = arctan(1) = 45°, so the angle is 45°.

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19) A 9.0 m pole casts a 4.0 m shadow. θ to the nearest degree is:

65°

66°

67°

68°

θ = arctan(9/4) ≈ 66.04°; nearest degree is 66°.

Explanation

θ = arctan(9/4) ≈ 66.04°; nearest degree is 66°.

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Sun Angle Shadow Quiz: Sun Angle From Height and Shadow