Volume and Surface Area of a Regular Quadrangular Pyramid with Specific Dimensions

When analyzing geometric shapes, a regular quadrangular pyramid is a fascinating object. A regular quadrangular pyramid features a square base with all pyramid edges of equal length. Let's explore the volume and surface area of such a pyramid, specifically with a base length of 6 cm and a side edge length of 9 cm. We will use geometric and algebraic methods to calculate the desired values.

Background

A regular quadrangular pyramid, otherwise known as a square pyramid, is a pyramid with a square base. The sides from the base to the apex are all equal in length. Our goal is to determine the volume and surface area of this specific pyramid. We start by identifying and defining the dimensions of the pyramid.

Dimensions and Calculations

Given that the base is a square with a side length of 6 cm, and each side edge is 9 cm, let's proceed with the necessary calculations.

Height of the Pyramid

Let OE h be the height of the pyramid, and a 6 cm be the base side length. The slant height of the pyramid, from the apex to the midpoint of the base, is given by

OE frac{a}{2} frac{6}{2} 3 cm

Right Triangle Analysis

Consider the right triangle EBC, where E is the apex, C is a vertex of the square base, and B is the midpoint of the base side BC. The height OE is the altitude of the pyramid.

Slant Height Calculation

The slant height, EF, of the isosceles triangle EBC can be found using the Pythagorean theorem. The equation for the slant height is

EF^2 l^2 - left(frac{a}{2}right)^2

Substituting the given values, we get

EF^2 9^2 - 3^2 81 - 9 72

EF sqrt{72} 3sqrt{8} approx 8.49 cm

Triangle Area Calculation

The area of the triangle EBC can be calculated using the formula for the area of a triangle, which is half the base times the height:

Area_{EBC} frac{1}{2} times 6 times 3sqrt{8} 9sqrt{8} text{ cm}^2

Total Slant Surface Area

The total slant surface area (As) is the sum of the areas of all four slant triangles:

A_s 4 times 9sqrt{8} 36sqrt{8} text{ cm}^2

Base Area Calculation

The area of the base, Ab, for a square with side length 6 cm is

A_b a^2 6^2 36 text{ cm}^2

Volume Calculation

The volume (V) of the pyramid can be calculated using the formula for the volume of a pyramid, which is one-third the base area times the height:

V frac{1}{3} times A_b times h

However, we need to find the height h first. Using the Pythagorean theorem in the right triangle formed by the height, half of the base, and the slant height, we get:

h^2 3^2 9^2

h^2 9 81

h^2 72

h sqrt{72} 6sqrt{2} approx 8.49 text{ cm}

Now, substituting the values into the volume formula:

V frac{1}{3} times 36 times 6sqrt{2} 72sqrt{2} approx 101.82 text{ cm}^3

Surface Area Calculation

The total surface area (A) of the pyramid is the sum of the base area and the slant surface area:

A A_b A_s 36 36sqrt{8} text{ cm}^2

A 36 36sqrt{8} approx 36 36 times 8.49 approx 354.64 text{ cm}^2

In summary, the volume and surface area of the given regular quadrangular pyramid are approximately 101.82 cm3 and 354.64 cm2, respectively.

By understanding the geometric properties and applying basic formulas, we can effectively calculate the volume and surface area of a regular quadrangular pyramid. These calculations are essential in various fields, including architecture, engineering, and mathematics.

Conclusion

In this article, we explored the volume and surface area of a regular quadrangular pyramid with specific dimensions. We applied geometric principles and formulas to derive accurate results. Armed with this knowledge, one can tackle similar problems involving other pyramid types or different dimensions.