3D Solids Notes Definitions Solid: a three-­‐dimensional object Polyhedron: any solid whose surfaces area all flat Lateral: word used to describe a solid’s surfaces which are not the bases (aka, the side faces of prisms and pyramids and the rounded surfaces of cylinders and cones) Prism: a polyhedron with two identical and parallel bases, and all lateral sides are parallelograms/rectangles Cylinder: a solid with two identical circular bases Height of a cylinder or prism: a distance perpendicular to the bases that connects the two bases (is not part of the base itself, and may be any of the lateral edges) Pyramid: a polyhedron with one base and all triangular lateral faces that meet at an apex Cone: a solid with one circular base extending to an apex Height of a cone or pyramid: a distance perpendicular to the base that goes up to the apex Slant height of a cone or pyramid: the distance from the apex to the outside of the base (forms the hypotenuse of a right triangle whose height is the height of the cone/pyramid and base is the radius/apothem of the base-­‐face) Base: the end faces of a prism or cylinder; the bottom face of a pyramid or cone (gives a prism or pyramid the first part of its name, while “prism” or “pyramid” is the second part) *Things to know about the base of a solid: 1. It is one of the 2-­‐D faces of the solid. 2. It has a twin which is parallel to it in prisms and cylinders. 3. It has an apex across from it (but not touching it) in pyramids and cones 4. It is the face of which you find the area and perimeter to use in volume and surface area formulas. 5. It gives pyramids and prisms the first part of their names. 6. It has its own 1-­‐dimensional base (which is one of its edges) and its own 1-­‐dimensional height (which is perpendicular to its base) 7. The height of the solid is perpendicular to it but is NOT part of it or inside it. 8. In prisms, it is always the face (and its parallel twin) which are NOT rectangles. The only exception is if the prism is a rectangular prism, in which case you may choose any pair of parallel rectangles as the bases. 9. In pyramids, it is always the face across from (and not touching) the apex. Unless the pyramid is a triangular pyramid, the base is NOT a triangle. 10. In cylinders and cones, the base is always the circles. Sphere: a ball Right vs. Oblique solids: A right prism/cylinder/pyramid/ cone has a height (perpendicular to bases) that can be drawn center-­‐to-­‐center of the bases or apex-­‐to-­‐center of the base *All prisms and cylinders we work with will be right Regular vs. Irregular solids: the bases of regular solids are regular polygons; irregular solids have bases with sides of different lengths Cross-­section: shape obtained when cutting across an object *We will be dealing only with cross-­‐sections parallel to the base, which are always the same shape as the base. Surface area: the total area of the (outside) surfaces of a solid Volume: the amount of 3-­‐dimensional space taken up by a solid Formulas €
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NOTE: Ab stands for the area of the solid’s base, Pb stands for the perimeter of the solid’s base, h stands for the height of the solid, and r stands for the radius SA = 2(Ab ) + Pb • h *Surface area of right prisms and cylinders: €
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*Volume of right prisms and cylinders: V = Ab • h € Ab • h
*Volume of cones and pyramids: V =
3 €
*Slant height of cones: sl =
h 2 + r 2 € 4
3
V
=
•
π
•
r
*Volume of a sphere: 3
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€