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Chapter 2: Problem 57

Compute the volume and lateral surface area of a regular hexagonal pyramid whose altitude has length \(h\) and makes the angle \(30^{\circ}\) with the apothem.

Short Answer

Expert verified
The volume is \(2\sqrt{3}h^3\) and the lateral surface area is \(12h^2\).

Step by step solution

01

Analyze the Structure of the Pyramid

A regular hexagonal pyramid has a hexagonal base and a single apex that forms the peak. The altitude of the pyramid is the perpendicular distance from the apex to the base. The apothem is the line drawn from the center of the base perpendicular to one of the base edges.
02

Use Trigonometry to Find the Slant Height

The altitude makes an angle of \(30^{\circ}\) with the apothem. Using trigonometry, particularly the sine function, we can find the slant height (\(l\)) of the pyramid using the relationship \[\sin(30^{\circ}) = \frac{h}{l} \quad \Rightarrow \quad l = \frac{h}{\sin(30^{\circ})} = 2h\]
03

Calculate the Apothem of the Base

Knowing the slant height, we use the cosine function for the angle to find the apothem (\(a\)) of the base,\[\cos(30^{\circ}) = \frac{a}{l} \quad \Rightarrow \quad a = l \cdot \cos(30^{\circ}) = 2h \cdot \frac{\sqrt{3}}{2} = h\sqrt{3}\]
04

Determine the Side Length of the Base

The apothem \(a = h\sqrt{3}\) of a regular hexagon relates to its side length \(s\) using the formula: \[ a = \frac{\sqrt{3}}{2}s \quad \Rightarrow \quad s = \frac{2a}{\sqrt{3}} = \frac{2h\sqrt{3}}{\sqrt{3}} = 2h.\]
05

Calculate the Area of the Hexagonal Base

The area of a regular hexagon with side length \(s\) is given by: \[A_{\text{base}} = \frac{3\sqrt{3}}{2}s^2 = \frac{3\sqrt{3}}{2}(2h)^2 = 6\sqrt{3}h^2\]
06

Compute the Volume of the Pyramid

The volume \(V\) is calculated using the formula:\[ V = \frac{1}{3} \cdot A_{\text{base}} \cdot h = \frac{1}{3} \cdot 6\sqrt{3}h^2 \cdot h = 2\sqrt{3}h^3\]
07

Calculate the Lateral Surface Area

The lateral surface area \(LSA\) is calculated as the total area of all the triangular faces, each having a base of \(s\) and height equal to the slant height \(l\):\[ LSA = \frac{1}{2} \cdot 6 \cdot s \cdot l = 3 \cdot (2h) \cdot (2h) = 12h^2\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume Calculation
Calculating the volume of a regular hexagonal pyramid involves understanding both its structure and the formula used in the calculation. This pyramid has a shape that includes a hexagonal base and an apex directly above the center. The key to finding its volume lies in the formula:\[ V = \frac{1}{3} \times A_{\text{base}} \times h \]- **Height (h):** This is the perpendicular distance from the apex to the center of the hexagon. - **Base Area \((A_{\text{base}})\):** For a regular hexagon with side length \(s\), the area can be calculated using: \[ A_{\text{base}} = \frac{3\sqrt{3}}{2}s^2 \]Using these, the volume of the pyramid becomes \(2\sqrt{3}h^3\), when substituting the base area and height accordingly. This demonstrates how the specific structure of the pyramid influences the volume calculation, making it necessary to first resolve the dimensions of the base and height.
Lateral Surface Area
The lateral surface area of a regular hexagonal pyramid represents the combined area of all its triangular side faces. This is an important aspect as it involves specifics about each triangular face's base and slant height.- **Triangular Face:** Each face is a triangle with a base equal to the side length \(s\) of the hexagonal base and a height equal to the slant height \(l\). To find the lateral surface area \(LSA\), use the following formula linking all six triangular faces: \[ LSA = \frac{1}{2} \times 6 \times s \times l \]Given \(s = 2h\) and \(l = 2h\), the calculation simplifies to \(12h^2\). This calculation reveals how geometric understanding of a structure translates into finding areas, emphasizing the role of both side length and slant height.
Trigonometry in Geometry
Understanding how trigonometry applies to geometry can greatly ease the determination of key measurements within a geometric figure. In analyzing the regular hexagonal pyramid, trigonometry helps in deriving the slant height and other necessary dimensions.- **Slant Height Derivation:** Knowing the angle of inclination and the altitude, trigonometric relationships such as the sine function are used: \[ \sin(30^\circ) = \frac{h}{l} \Rightarrow l = 2h \]- **Base Apothem:** Using cosine helps in finding the apothem \(a\) by considering the angle formed with the base and slant height: \[ \cos(30^\circ) = \frac{a}{l} \Rightarrow a = h\sqrt{3} \]Trigonometry provides the toolkit that translates angles and lengths into meaningful, measurable quantities for geometry. Whether looking for slant heights or base elements, trigonometric principles make complex geometric forms like pendants and pyramids manageable and their calculations achievable.

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Most popular questions from this chapter

Parallel cross sections of pyramids. Theorem. If a pyramid (Figure 49) is intersected by a plane parallel to the base, then: (1) lateral edges and the altitude \((S M)\) are divided by this plane into proportional parts; (2) the cross section itself is a polygon \(\left(A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\right)\) similar to the base; (3) the areas of the cross section and the base are proportional to the squares of the distances from them to the vertex.

In a rectangular parallelepiped with a square base and the altitude \(h\), a cross section through two opposite lateral edges is drawn. Compute the total surface area of the parallelepiped, if the area of the cross section equals \(S\). e.

Describe the cross section of a cube by the plane perpendicular to one of the diagonals at its midpoint.

How many planes of symmetry does a regular tetrahedron have?

Prisms. Take any polygon \(A B C D E\) (Figure 39 ), and through its vertices, draw parallel lines not lying in its plane. Then on one of the lines, take any point \(\left(A^{\prime}\right)\) and draw through it the plane parallel to the plane \(A B C D E\), and also draw a plane through each pair of adjacent parallel lines. All these planes will cut out a polyhedron \(A B C D E A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\) called a prism. The parallel planes \(A B C D E\) and \(A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\) are intersected by the lateral planes along parallel lines (\$13), and therefore the quadrilaterals \(A A^{\prime} B^{\prime} B, B B^{\prime} C^{\prime} C\), etc. are parallelograms. On the other hand, in the polygons \(A B C D E\) and \(A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\), corresponding sides are congruent (as opposite sides of parallelograms), and corresponding angles are congruent (as angles with respectively parallel and similarly directed sides). Therefore these polygons are congruent. Thus, a prism can be defined as a polyhedron two of whose faces are congruent polygons with respectively parallel sides, and all other faces are parallelograms connecting the parallel sides. The faces \(\left(A B C D E\right.\) and \(\left.A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}\right)\) lying in parallel planes are called bases of the prism. The perpendicular \(O O^{\prime}\) dropped from any point of one base to the plane of the other is called an altitude of the prism. The parallelograms \(A A^{\prime} B^{\prime} C, B B^{\prime} C^{\prime} C\), etc. are called lateral faces, and their sides \(A A^{\prime}, B B^{\prime}\), etc., connecting corresponding vertices of the bases, are called lateral edges of the prism. The segment \(A^{\prime} C\) shown in Figure 39 is one of the diagonals of the prism. A prism is called right if its lateral edges are perpendicular to the bases (and oblique if they are not). Lateral faces of a right prism are rectangles, and a lateral edge can be considered as the altitude. A right prism is called regular if its bases are regular polygons. Lateral faces of a regular prism are congruent rectangles. Prisms can be triangular, quadrangular, etc. depending on what the bases are: triangles, quadrilaterals, ete.
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