Understanding the Geometry of Square Based Pyramids: Height, Slant Height, and Dihedral Angles

When dealing with geometric shapes, particularly square based pyramids, certain calculations are essential to fully understand their dimensions and properties. In this article, we explore the given problem and solve for height, slant height, and dihedral angles of a square based pyramid with a base length of 7 cm and a volume of 490 cmsup3;. We'll break down the steps and provide detailed calculations to ensure clarity for readers.

Revisiting the Given Problem

Given a square based pyramid:

Base length 7 cm Volume 490 cmsup3; The height (h) calculated is 30 cm

The volume formula for a pyramid is given by:

V 1/3 times; Base Area times; Height

Let's break down the formula and solve for the height.

Calculating the Height

First, calculate the base area using the base length:

Base Area Base Length times; Base Length 7 times; 7 49 cm2

Substitute the values into the volume formula:

490 1/3 times; 49 times; Height

Solving for the height:

Height 490 times; 3 / 49 1470 / 49 30 cm

Thus, the height of the pyramid is 30 cm.

Determining the Slant Height and Dihedral Angles

The slant height (s) and dihedral angles are other important dimensions in a square based pyramid. However, the slant angle cannot be determined without additional information. Here’s an exploration of the slant height and related angles.

Slant Height Calculation

The slant height is the distance from the midpoint of a base edge to the apex of the pyramid. To calculate the slant height, we can use the Pythagorean theorem:

slant height (s) radic;(h2 (base length/2)2)

Substitute the values:

s radic;(302 (7/2)2)

s radic;(900 12.25) radic;(912.25) ≈ 30.2 cm

Therefore, the slant height is approximately 30.2 cm.

Dihedral Angles and Trigonometric Relations

Dihedral angles are the angles between two adjacent faces of the pyramid. In a right square based pyramid, the dihedral angle can be calculated using trigonometric relations.

Base Dihedral Angle

The base dihedral angle (angle between a base edge and a slant height from the midpoint of the same base edge):

alpha; arctan(h / (base length / 2))

alpha; arctan(30 / 3.5) ≈ 81.37°

Edge Dihedral Angle

The edge dihedral angle (angle between an edge and the slant height at the apex):

beta; arctan(h / (base length / 2 times; radic;2))

beta; arctan(30 / (3.5 times; radic;2)) ≈ 81.22°

To find the slant length, use the Pythagorean theorem:

slant length (s) radic;((7 times; radic;2 / 2)2 302)

s ≈ 30.4 cm

Conclusion

This article has provided detailed calculations and explanations for determining the height, slant height, and dihedral angles of a square based pyramid. These calculations are fundamental for understanding the geometry and properties of pyramids and are crucial in various applications such as construction, engineering, and mathematics. By following the steps and formulas provided, you can confidently solve similar problems involving square based pyramids.

References

[1] Math Open Reference: Pyramids

[2] Khan Academy: Volume of a Pyramid