Marble Mural
A marble mural in the shape of a regular 5 pointed star is set into a building’s lobby floor.?
Each side of the point is 3m. The mural is 50cm thick what volume of marble is needed?
The question uses Volume of a prism, angle rules, trigonometry and area of a non right angled triangles.
First, let me confirm that, from the context, we are talking about the 5-pointed star generally known as a pentagram, and that the length of 3m applies to the lines issuing from each vertex to the nearest junction with the inscribed pentagon.
They have been known for thousands of years, and have been widely used as symbols by social, military and religious groups. They show many interesting features, one of which is that each section of the single continuous line that forms them is in a golden ratio tau (φ) with another section. As a result, the length of every section may be expressed in the form L.f(φ), where L is a baseline length.
To simplify the problem, we may break the pentagram into five identical shapes based upon the sides of the inscribed pentagon. If we draw the side AB horizontally, then the vertex V is symmetrically placed outside (above) and the centre O of the inscribed pentagon inside (below) this line. The line OV is vertical and passes through the centre C of AB, so that the angles at C are all right angles and AVB and AOB are isosceles triangles. The fact that the lines between the vertices are co-linear with the sides of the inscribed pentagon enables the various angles to be determined easily eg 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°).
This prevalence of multiples and sub-multiples of π/5 radians is the reason for the ubiquity of the golden ratio φ here, since
φ = 2.cos(π/5) = (1 + √5)/2 = 1.61803
and the various angles may 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
(a) area of the points: we note first that the side AB of the inscribed pentagon is in a golden ratio φ with the length L of the lines VA and VB from the vertex, that is, AB = L/φ. The area Ap of a single ‘point’ is therefore
Ap = (1/2).(L/φ).L.cos(π/10) = (L²/4.φ).√(2 + φ) = 2.6450 m²
and since there are 5 points, their total area A1 = 5 x Ap = 13.225 m².
(b) area of the inscribed pentagon: the area of the isosceles triangle AOB making up one-fifth that of the inscribed pentagon is
Aq = (1/2).(L/φ).(L/2.φ).[1/tan(π/5)] = [L²/(4.φ²)].cos(π/5)/sin(π/5)
. . .= [L²/(4.φ)]/[√(3 - φ)]
giving the area A2 of the inscribed pentagon as 5.Aq = 5.914 m². As a partial check I calculated the area of a regular pentagon with side (L/φ) from the standard formula
A2 = (L/φ)².√(25 + 10√5)/4
which gave an identical answer.
Hence the total area of the pentagram is 13.225 + 5.914 = 19.139 m². As the mural was of uniform thickness 0.50 m, the volume of marble contained was 9.57 m³.
Marble Mural II
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PICASSO DREAM MARBLE MOSAIC STONE ART TILE MURAL DECOR $450.00 |
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a nice marble mosaic mural floor or table top inlay $185.00 |
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48″ x 48 Peacocks Marble Mosaic Tiles Stone Art Wall Mural Home Decore $775.00 |
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48″ x 40″ Peacocks Marble Mosaic Tiles Stone Art Wall Mural Home Decore $675.00 |
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48″ x 24″ Peacocks Marble Mosaic Tiles Stone Art Wall Mural Home Decore $525.00 |
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52″ x32″ Peacocks Marble Mosaic Tiles Stone Art Wall Mural Home Decore $575.00 |
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52″ x34″ Peacocks Marble Mosaic Tiles Stone Art Wall Mural Home Decore $625.00 |
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40″ x 20″ Peacock Marble Mosaic Tiles Stone Art Wall Mural Home Decore $360.00 |
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40″ x 40 Peacocks Marble Mosaic Tiles Stone Art Wall Mural Home Decore $575.00 |
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LOVELY MOSAIC MARBLE MURAL HANGING ART TILE $165.00 |
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SUPER MARBLE MOSAIC MEDALLION ART DECOR MURAL FLOORING $350.00 |
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LOVELY MOSAIC MARBLE Mural Hanging ART TILE $317.00 |
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27.3″x13.7′FLOWER MOSAIC MARBLE MURAL HANGING ART TILE $100.00 |
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Art Rose Mural Tumbled Marble Backsplash Bath Tile #160 $108.00 |
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48″ x 24″ Tigers Marble MosaicTiles Stone Art Wall Mural Home Decore $525.00 |
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40″ x 22″ Dolphins Marble Mosaic Tiles Stone Art Mural Pool Bathroom $350.00 |
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25″ Flower Handmade Marble MosaicTile stone wall Mural $170.00 |
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56″ x 36″ Dolphins Sea Landscape Marble Mosaic Tiles Stone Art Mural Pool $785.00 |
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40″ x 30″ Dolphins Sea Landscape Marble Mosaic Tiles Stone Art Mural Pool $650.00 |
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44″ x 38″ Dolphins Sea Landscape Marble Mosaic Tiles Stone Art Mural Pool $620.00 |
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52″ x 44″ Dolphins Sea Landscape Marble Mosaic Tiles Stone Art Mural Pool $825.00 |
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36″ x 36″ Dolphins Marble Mosaic Tiles Stone Art Mural Pool Wall $485.00 |
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48″ x 48″ Dolphins Marble Mosaic Tiles Stone Art Mural Pool Wall Floor $785.00 |
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84″ x 48″ Dolphins Marble Mosaic Tiles Stone Art Mural Pool Wall Floor $1,650.00 |
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24″ x 24″ Rooster Marble Mosaic Stone Art Tile Wall Mural Home Decore $270.00 |
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26″ x 20″ Rooster Marble Mosaic Stone Art Tile Wall Mural Home Decore $280.00 |
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40″ x 24″ Swans Marble Mosaic Tiles Stone Mural Wall Pool Bathroom $410.00 |
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20″ x 20″ Swan Marble Mosaic Tiles Stone Mural Wall Pool Bathroom $225.00 |
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40″ x 28″ Swans Marble Mosaic Tiles Stone Mural Wall Pool Bathroom $465.00 |
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40″ x 40″ Swans Marble Mosaic Tiles Stone Mural Wall Pool Bathroom $565.00 |
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40″ x 38″ Swans Marble Mosaic Tiles Stone Mural Wall Pool Bathroom $555.00 |
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44″ x 26″ Swans Marble Mosaic Tiles Stone Mural Wall Pool Bathroom $475.00 |
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50″ x40″ Swans Marble Mosaic Tiles Stone Mural Wall Pool Bathroom $650.00 |
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36″ x 24″ Swans Marble Mosaic Tiles Stone Mural Wall Pool Bathroom $425.00 |
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Ching Vineyard Wine Art Decor Tumbled Marble Tile Mural $95.00 |
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Green Wine Art Tumbled Marble Tile Mural Backsplash $125.00 |
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32″ x 24″ Kitchen Backsplash Marble Mosaic Stone Art Tile Mural $370.00 |
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48″FANTASTIC MARBLE MOSAIC FISH MURAL DECORATION ART $530.00 |
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34″ x 20″ Horse Marble Mosaic Tiles Stone Art Wall Mural Home Decore $385.00 |
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40″ x 40″ Horse Marble Mosaic Tiles Stone Art Wall Mural Home Decore $585.00 |
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40″ x 32″ Horse Marble Mosaic Tiles Stone Art Wall Mural Home Decore $485.00 |
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48″ x 24″ Horses Marble Mosaic Tiles Stone Art Wall Mural Home Decore $485.00 |
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52″ x 38″ Horses Marble Mosaic Tiles Stone Art Wall Mural Home Decore $650.00 |
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104″ x 48″ Horses Marble Mosaic Tiles Stone Art Wall Mural Home Decore $2,200.00 |
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Amalfi Coast Italy Tumbled Marble Mural Tile #170 $108.00 |
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86″ x 40″ Horses Marble Mosaic Tiles Stone Art Wall Mural Home Decore $1,450.00 |
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84″ x 48″ Horse Marble Mosaic Tiles Stone Art Wall Mural Home Decore $1,375.00 |
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24 x 18 Mural Tumbled Marble Sea Backsplash Tile #65 $158.00 |
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44″ x 32″ Horses Marble Mosaic Tiles Stone Art Wall Mural Home Decore $525.00 |
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26″ x 20″ Butterfly Marble Mosaic Tiles Stone Art Wall Mural Home Decore $325.00 |
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28″ x 28″ Butterfly Marble Mosaic Tiles Stone Art Wall Mural Home Decore $390.00 |
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52″ x 34″ Butterflies Marble Mosaic Tiles Stone Art Wall Mural Home Decore $675.00 |
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28″ x 28″ Butterfly Marble Mosaic Tiles Stone Art Wall Mural Home Decore $350.00 |
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LOVELY MOSAIC MARBLE MURAL HANGING ART TILE FIGURE $750.00 |
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Taylor Horses Equine Tumbled Marble Tile Mural Art $210.00 |
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SUPER MARBLE MOSAIC DOLPHIN ART DECOR MURAL FLOOR $150.00 |
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31″ Peacocks Marble Mosaic Stone Mural Hanging Art Decor $240.00 |
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82″ Flower Handmade Marble MosaicTile stone wall Mural $1,299.00 |
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25″ Lovely Animal Marble Mosaic Mural Art Tile Decor $170.00 |
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64″ x 32″ Flowers Handmade Marble Mosaic Tiles Stones Art Wall Mural $785.00 |
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40″ x 28″ Peacocks Marble Mosaic Tiles Stone Art Wall Mural Home Decore $475.00 |
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24 x 16 Mural Tumbled Marble Glass of Wine Tile #465 $138.00 |
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Tumbled Marble Mural Seafront Bath Backsplash Tile #88 $108.00 |
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Art Mural Tumbled Marble Bath Backsplash Tile #66 $108.00 |
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LOVELY CHERUB MOSAIC MURAL HANGING ART TILE $335.00 |
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48″x 32″ Chicken Rooster Marble Mosaic Stones Art Wall Mural Home Decore $555.00 |
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Van Gogh’s Sunflower Tumbled Marble Mural/Coaster Set – Set of 6 Perfect gifts to match style and interests. Handcrafted tumbled marble Mural/Coaster set featuring paintings by Van Gogh’s Sunflower Perfect decoration – One of kind Mural – for your table. And it is functional coasters and trivet as well. Mural can be used for your wall decoration. Mural is only for indoor wall decoration only. Use non-sanded grouts for tile installation. Clean tile with non-… |
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Melk abbey marble room fresco, Melk, Basse-Autriche, Austria, Europe Photo Mugs Melk abbey marble room fresco, Melk, Basse-Autriche, Austria, Europe…. |
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Van Gogh’s Cafe Terrace at Night Tumbled Marble Mural/Coaster set – Set of 6 Perfect gifts to match style and interests. Handcrafted tumbled marble Mural/Coaster set featuring paintings by Van Gogh’s Cafe Terrace at Night Perfect decoration – One of kind Mural – for your table. And it is functional coasters and trivet as well. Mural can be used for your wall decoration. Mural is only for indoor wall decoration only. Use non-sanded grouts for tile installation. Clean ti… |
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12×12 Fish Mosaic Art Tile Wall Floor Pool Decor $115.00 Mosaics have endless uses and infinite possibilities! They can be used indoors or outdoors, be part of your kitchen, decorate your bathroom and the bottom of your pools, cover walls and ceilings, or serve as frames for mirrors and paintings…. |
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12×14 Marble Mosaic Stone Fish Art Tile Pool Bathroom $133.00 Mosaics have endless uses and infinite possibilities! They can be used indoors or outdoors, be part of your kitchen, decorate your bathroom and the bottom of your pools, cover walls and ceilings, or serve as frames for mirrors and paintings…. |
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12×20 Fish Mosaic Art Tile Wall Floor Pool Decor $149.00 Mosaics have endless uses and infinite possibilities! They can be used indoors or outdoors, be part of your kitchen, decorate your bathroom and the bottom of your pools, cover walls and ceilings, or serve as frames for mirrors and paintings…. |
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Photo Jigsaw Puzzle of Melk abbey marble room fresco, Melk, Basse-Autriche, Austria, Europe from Robert Harding $24.99 Photo Puzzle, Melk abbey marble room fresco, Melk, Basse-Autriche, Austria, Europe. Melk abbey marble room fresco, Melk, Basse-Autriche, Austria, Europe. Chosen by Robert Harding. 10×14 Photo Puzzle with 252 pieces. Packed in black cardboard box of dimensions 5 5/8 x 7 5/8 x 1 1/5. Puzzle image 5×7 affixed to box top. Puzzle pieces printed on RA4 paper at 300 dpi. This item is shipped from our Ame… |
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Michelangelo V: The Final Period: Last Judgement, Frescoes of the Pauline Chapel, Last Pietas $135.00 … |
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Handbook of the mural and monumental brasses, stained glass windows and other ecclesiastical relics in the Greyfriars Kirk, Edinburgh … |
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Skill: Marble Mural.: An article from: The Architectural Review $9.95 This digital document is an article from The Architectural Review, published by EMAP Architecture on March 1, 2011. The length of the article is 566 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available immediately after purchase. You can view it with any web browser.Citation DetailsTitle: Skill: Marble Mural.Author: James … |
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