## Plane and Solid Geometry |

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**tetraedron**; one of six faces , a hexaedron ; one of eight faces , an octaedron ; one of twelve faces , a dodecaedron ; and one of twenty faces , an icosaedron . It is instructive to construct the regular polyedrons as shown in the ... Page 374

... Faces Number of Faces Sum of of Edges of Vertices at Each Vertex Face Angles

... Faces Number of Faces Sum of of Edges of Vertices at Each Vertex Face Angles

**Tetraedron**4 Equi . A 6 4 3 720 ° Hexaedron Octaedron Dodecaedron Icosaedron 756. Euler's Theorem . This theorem is stated because of 374 SOLID GEOMETRY. Page 375

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**tetraedron**3 in . on an edge . Of a regular octaedron 4 in . on an edge . Of a regular icosaedron 6 in . on an edge ...**tetraedron**is e , show that its volume is given by the formula Ve3√2 . = 12 SOLUTION . VON × area of △ ABC ... Page 376

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**tetraedron**O - ABC so that triedral 20 will coincide with triedral ZO ' . Draw CN and C'M perpendicular to O'A'B ' . . CN and C'M determine a plane which intersects O'A'B ' in O'M . Why ? V O'AB CN O'AB CN Now = = Why ? V ' O'A'B ' C'M ... Page 378

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**tetraedron**is its altitude above the base . Find the center of gravity of a regular**tetraedron**8 in . on an edge . 6. The edge of a regular**tetraedron**is e . Find the edge of a regular**tetraedron**that has a volume n times as great as ...### Contents

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### Common terms and phrases

AABC ABCD acute angle altitude angles are equal angles equal base bisects chord circumference circumscribed congruent diagonals diameter diedral angles distance divided Draw drawn equal respectively equilateral triangle equivalent EXERCISES face angles figure Find the area Find the length Find the number Find the radius Find the volume formula frustum given circle given line given point hypotenuse inch interior angles intersecting isosceles trapezoid isosceles triangle lateral area lateral edges locus median middle point number of sides parallel lines parallelepiped parallelogram perimeter plane geometry plane Q polyedral angle polyedron prism prismatoid Proof prove pyramid quadrilateral radii ratio rectangle regular polygon rhombus right angle right circular cone right triangle segment semicircle Show similar slant height sphere spherical angles spherical polygon spherical triangle straight line surface tangent tetraedron Theorem trapezoid triangle ABC triedral vertex angle vertices

### Popular passages

Page 169 - In any triangle, the square of a side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides and the projection of the other side upon it.

Page 75 - The straight line joining the middle points of two sides of a triangle is parallel to the third side, and equal to half of it.

Page 19 - If two triangles have two angles and the included side of one equal respectively to two angles and the included side of the other, the triangles are congruent.

Page 155 - The area of a parallelogram is equal to the product of its base and its height: A = bx h.

Page 89 - Two triangles are congruent if two sides and the included angle of one are equal respectively to two sides and the included angle of the other.

Page 164 - Two triangles which have an angle of one equal to the supplement of an angle of the other are to each other as the products of the sides including the supplementary angles.

Page 155 - ... any two parallelograms are to each other as the products of their bases by their altitudes. PROPOSITION V. THEOREM. 403. The area of a triangle is equal to half the product of its base by its altitude.

Page 248 - ... as the squares of their radii, or as the squares of their...

Page 296 - Axiom. Through a given point only one straight line can be drawn parallel to a given straight line.

Page 39 - In an isosceles triangle the angles opposite the equal sides are equal.