Question

A regular hexagonal pyramid has the altitude \(h\) and the side of the base \(a\). Compute the lateral edge, apothem, lateral surface area, and total surface area.

Short answer

Lateral edge: \(l_s = \sqrt{h^2 + \left(\frac{a \sqrt{3}}{2}\right)^2}\); Lateral area: \(3a l_s\); Total area: \(3a l_s + \frac{3 \sqrt{3}}{2} a^2\).

Step-by-step solution

Understanding the Hexagonal Pyramid

The given pyramid is a regular hexagonal pyramid, meaning its base is a regular hexagon and all lateral sides (triangles) are identical. The side length of the hexagon is given as \(a\) and the pyramid's altitude (height from the base to the apex) is \(h\).

Calculate the Apothem of the Base

The apothem \(a_p\) of a regular hexagon with side length \(a\) is the distance from the center of the hexagon to the midpoint of one of its sides. It can be calculated using the formula:\[ a_p = \frac{a \sqrt{3}}{2} \]This formula derives from dividing the hexagon into 6 equilateral triangles and using trigonometry in one such triangle.

Calculate the Slant Height (Lateral Edge)

The lateral edge is the slant height \(l_s\) of one of the triangular faces of the pyramid. It can be computed using the Pythagorean theorem in the triangle formed by the pyramid's altitude, the apothem of the base, and the slant height:\[ l_s = \sqrt{h^2 + a_p^2} \]Substitute \(a_p\) from the previous step to get:\[ l_s = \sqrt{h^2 + \left(\frac{a \sqrt{3}}{2}\right)^2} \]

Calculate the Lateral Surface Area

The lateral surface area \(A_L\) is the sum of the areas of the 6 identical triangular faces. The area of one face is:\[ \text{Area of one triangle} = \frac{1}{2} \times a \times l_s \]Thus, the lateral surface area is:\[ A_L = 6 \times \frac{1}{2} \times a \times l_s = 3a l_s \]

Calculate the Total Surface Area

The total surface area \(A_T\) is the sum of the lateral surface area and the base area. The area of the base (a regular hexagon) is:\[ A_B = \frac{3 \sqrt{3}}{2} a^2 \]So, the total surface area is:\[ A_T = A_L + A_B = 3a l_s + \frac{3 \sqrt{3}}{2} a^2 \]

Key concepts

Lateral Edge

In geometry, the lateral edge of a pyramid refers to the slant height of one of its triangular faces. For a regular hexagonal pyramid, this is particularly important because each triangular face is identical. The lateral edge connects the apex of the pyramid to a vertex of the base.
To calculate the lateral edge, we consider the right triangle formed between the pyramid's altitude (the vertical height), the apothem of the base, and the lateral edge itself. By applying the Pythagorean theorem, we can derive the length of the lateral edge as follows:

• Given the altitude, \( h \), and the apothem of the base, \( a_p = \frac{a \sqrt{3}}{2} \), the lateral edge \( l_s \) is calculated by: \[ l_s = \sqrt{h^2 + a_p^2} \]

This equation helps visualize how the pyramid extends spatially from its base to its top, showing the relationship between its vertical height and the sloping sides.

Apothem

The apothem of the base is a fundamental measurement in calculating various aspects of a hexagonal pyramid. It is the line from the center of the hexagonal base to the midpoint of one of its sides. The apothem is crucial because it helps in determining both the lateral edge and the surface areas of the pyramid.
In a regular hexagon, every side and angle are equal, which simplifies calculations significantly. The formula for finding the apothem of a regular hexagon is:

• \[ a_p = \frac{a \sqrt{3}}{2} \]

Here, \( a \) represents the length of the base's sides. The apothem essentially divides the hexagon into six smaller right triangles, each with one apex at the center of the hexagon, helping bridge formulas for the overall surface area calculations.

Lateral Surface Area

The lateral surface area of a pyramid is the sum of the areas of all its triangular faces, excluding the base. For a hexagonal pyramid, having six identical triangular faces makes this calculation straightforward. Knowing the base side, \( a \), and the lateral edge or slant height, \( l_s \), is essential.
Each triangular face has an area given by half the product of its base and height, the latter being the lateral edge:

• \[ \text{Area of one triangle} = \frac{1}{2} \times a \times l_s \]

Because there are six such triangles, the total lateral surface area \( A_L \) is:

• \[ A_L = 6 \times \frac{1}{2} \times a \times l_s = 3a l_s \]

This formula emphasizes how the regularity and symmetry of the hexagon aid in simplifying these calculations.

Total Surface Area

The total surface area of a hexagonal pyramid is the combination of its lateral surface area and its base area. This total provides a complete measure of the surface exposed on the entire pyramid.
The base, being a regular hexagon, has its area calculated using:

• \[ A_B = \frac{3 \sqrt{3}}{2} a^2 \]

This formula stems from dividing the hexagon into six equilateral triangles. Combining this with the lateral surface area, you get the total surface area \( A_T \):

• \[ A_T = A_L + A_B = 3a l_s + \frac{3 \sqrt{3}}{2} a^2 \]

Understanding these components allows for a full grasp of the pyramid's spatial properties, providing insights into both geometry and algebraic manipulations via symmetry and regularity.