Marble Mural

A marble wall in the form of regular 5-pointed star is in the lobby of the floor of a building.?

Each side of the issue is 3m. The mural is 50 cm thick, so the volume of marble is needed? The issue of volume uses a prism angle of the rules, trigonometry and the area of ​​a non-right triangles.

First, let me confirm that from the context we're talking 5-star tips generally known as a pentagram, and the length of 3 meters is applied to the emission lines from each vertex to the close bond with the inscription pentagon. They have been known for thousands of years and have been widely used as symbols of social groups, military and religious. They show many features interesting, one of which is that each section of single and continuous line that forms a tau is the golden ratio (φ) to another section. As result, the length of each section can be expressed in the form Lf (φ), where L is a length of baseline. To simplify the problem, we can break the staff based on five identical shapes on the sides of the inscribed pentagon. If we take the horizontal side AB, then the vertex V is symmetrically placed outside (top) and center Or the pentagon inscribed on the inside (below) of this line. The OV line is vertical and passes through the center C of AB, so that the angles at C are all the angles AVB straight and AOB are isosceles triangles. The fact that the lines between the vertices are co-linear with the sides of the pentagon inscribed allows different angles to be easily determined for example, CVA, CVB = π/10 (18 °), AVB, AOC, BOC = π / 5 (36 °), BAO, ABO = 3.π/10 (54 °) and VAB, VBA, AOB = 2.π / 5 (72 °). The prevalence of decimal multiples of π radians / 5 is the reason for the ubiquity of the golden ratio φ here as 2.cos φ = (π / 5) = (1 + √ 5) / 2 = 1.61803 and the various angles can be written as simple functions of φ or √ 5, for example sin (π/10) = (φ – 1) / 2: cos (π/10) = (1 / 2) √ (φ + 2) sin (π / 5) = (1 / 2) √ (3 – Φ). Cos (π / 5) = φ / 2 (an area) of the points: first note that the AB side of the pentagon inscribed in a golden ratio φ with the length L of the lines of VA and VB from the vertex, ie AB = L / φ. Ap area a "point" is only so Ap = (1 / 2). (L / φ). L.cos (π/10) = (L ² / 4.φ). √ (2 + φ) = 2.6450 m² and there are 5 points, the total area A1 = 5 x Ap = 13,225 m². (B) area of ​​the inscribed pentagon: the area of ​​isosceles triangle AOB that make up a fifth of the inscribed pentagon is Aq = (1 / 2). (L / φ). (L/2.φ). [1/tan (π / 5)] = [L ² / (4.φ ²).] cos (π / 5) / sin (π / 5). . .= [L ² / (4.φ)] / [√ (3 - φ)] giving inscribed pentagon area A2 as 5.Aq = 5.914 m². As a partial verification that calculates the area of ​​a regular pentagon with side (L / φ) the standard formula A2 = (L / φ) ². √ (25 + 10 √ 5) / 4, which gave the same response. Therefore the total surface the five-pointed star is 13 225 + 5914 = 19 139 m². As the mural was 0.50 m of uniform thickness, the volume contained marble was 9.57 m³.

Marble Mural II

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