Consider a square-based straight pyramid. Suppose that the base is a square with sides 10 km long, and all other edges are 17 km long. Find an approximate value of the angle formed between the base and a triangular face. Present your answer in degrees, accurate up to four or more decimal places.

The approximate value of the angle formed between the base and a triangular face is 72.0775°.

Let be the distance between the center of the square base and the midpoint of the side of the square and let the height of the triangular face of the pyramid. By properties of the square and the Pythagorean theorem we derive expressions for and , respectively:

Square base

(1)

Triangular face

(2)

Where:

Finally, by trigonometry we can derive an expression for the angle between the base and a triangular face (), in sexagesimal degrees:

(3)

If we know that and , then the angle between the base and a triangular face is:

The approximate value of the angle formed between the base and a triangular face is 72.0775°.

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72.06

Step-by-step explanation:

Given that :

Side Lenght, a = 10

Edge length, b = 17

Using Pythagoras :

h = height of pyramid

h = sqrt (b² - 1/2a²)

h = sqrt(17^2 - 10^2/2)

h = sqrt(289 - 50)

h = 15.46

Slant height :

L = sqrt(h² + a²/4)

L = sqrt(15.46² + 25)

L = sqrt(239 + 25)

L = sqrt(264)

L = 16.25

Sinα = h/L

α = sin^-1 (15.46/16.25)

α = 72.060933

= 72.06

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