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Solution. The figure 54,687 lies between 54,600 and 54,700, which are both three significant figures, but it is closer to 54,700. The number of spectators was 54,700 to 3 sf. The two zeros in the answer are not significant figures, but need to be written to maintain the size of the number. The two zeros are placeholders.
References: Elevation, Isometric.
References: Coplanar, Edge.
The Platonic solids were named after Plato, who lived about 400 BC. They are also called regular polyhedra, or regular solids. A Platonic solid is one in which every face is the same regular polygon, and as a result all of its edges and vertices are exactly the same. There are only five possible platonic solids, and it was Plato who clearly defined them and showed how to construct the solids from their nets.
The five platonic solids are:
1. Cube, made up of six congruent squares.
2. Regular dodecahedron, made up of 12 regular congruent pentagons.
3. Regular icosahedron, made up of 20 congruent equilateral triangles.
4. Regular octahedron, made up of eight congruent equilateral triangles.
5. Regular tetrahedron, made up of four congruent equilateral triangles.
For information on these solids, search for them under their names. Crystals grow in the shape of some of the regular polyhedra. For example, the crystals of common salt grow as cubes, and chrome alum crystals grow as regular octahedrons.
References: Coplanar, Edge.
Reference: Cartesian Coordinates.
POINT OF INFLECTION
A polygon is a closed plane figure with straight sides, and no two sides cross over. The sides intersect in points called vertices. When all the sides are the same length, the polygon is said to be regular, and the angles of each regular polygon are the same size. There is a formula for the sum of the interior angles of a polygon, and it is derived in the following way (see the figure).
The three interior angles of a triangle add up to 180°. This means a + b + c = 180°.
A quadrilateral is made up of two triangles that have a common vertex V: 2 x 180° = 360°, which is the angle sum of a quadrilateral.
A pentagon is made up of three triangles that have a common vertex V: 3 x 180° = 540°, which is the angle sum of a pentagon.
Using the pattern of the triangle, quadrilateral, and the pentagon, we can extend the process to include a polygon with n sides. A polygon with n sides is made up of (ft — 2) triangles that have a common vertex.
The angle sum of a polygon with n sides is (ft — 2) x 180°. The formula is
Sum of interior angles of a polygon with n sides = (ft — 2) x 180°
Example. A decagon is a polygon with 10 sides. Find the interior angle sum of a decagon.
Angle sum of a decagon = (10 — 2) x 180° Substituting ft = 10 into
(ft - 2) x 180°
8 x 180°
The interior angle sum of a decagon is 1440°.
The table gives information about some of the well-known polygons. For more information search under the name of the given polygon. From the table, you can see that each time the number of sides of the polygon increases by one, the angle sum increases by 180°.
Name of Polygon Number of Sides Angle Sum
Triangle 3 180°
Quadrilateral 4 360°
Pentagon 5 540°
Hexagon 6 720°
Heptagon 7 900°
Octagon 8 1080°
Nonagon 9 1260°
Decagon 10 1440°
Hendecagon 11 1620°
Dodecagon 12 1800°
References: Exterior Angle of a Polygon, Regular Polygon.
The plural of polyhedron is polyhedra. A polyhedron is a solid figure with plane faces that are polygons. The faces meet in edges, and the edges meet in vertices. For regular polyhedra see the entry Platonic Solids. Some examples of polyhedra are shown in the figure.
Tetrahedron Pentahedron Hexahedron Octahedron 4 faces 5 faces 6 faces 8 faces
The polyhedra drawn in the figure have the following, more commonly known names:
♦ Tetrahedron, triangle-based pyramid
♦ Pentahedron, square-based pyramid
♦ Hexahedron, cuboid
♦ Octahedron, hexagonal prism
For more information about polyhedra search for them under their respective names.
References: Edge, Plane, Platonic Solids, Prism, Regular, Polyhedron, Vertex.
338 POLYOMINOES POLYNOMIAL
When squares are joined together edge to edge they form polyominoes. The various kinds of polyominoes are as follows:
♦ Monomino. This type of polyomino is made up of just one square (see figure a). There is only one type of monomino.
♦ Domino. Two squares joined together form a domino (see figure b). There is only one type of domino.
♦ Triomino. This is made up of three squares joined together (see figure c). There are two types of triominoes.
♦ Tetromino. This is made of four squares joined together (see figure d). There are five types of tetrominoes.
♦ Pentomino. This is made up of five squares joined together. There are 12 types of pentominoes (see the entry Pentominoes).