Download Geometry Notes 11-4 Pyramids

Geometry
Notes 11-4
Pyramids
vertex
Pyramid: a solid figure with a base that is a polygon
and the lateral faces are triangles.
altitude
Vertex: the point where the lateral edges meet.
Altitude: is the perpendicular line segment from the vertex to the base.
PC in diagram above.
Regular Pyramid: A pyramid whose base is a regular polygon and whose
altitude is perpendicular to the base at the center.
Slant Height: The length of the altitude of a triangular lateral face of a
regular pyramid. PB is the slant height of the pyramid.
Volume of a pyramid: V = 1/3 Bh
Properties of Regular Pyramids:
- the lateral faces of a regular pyramid are isosceles triangles
- the lateral faces of a regular pyramid are congruent
Ex. 1: A regular pyramid has a base that is a hexagon. The length of an edge
of the bases is 22 cm and the slanted height is 32 cm. Find the lateral
area of each pyramid.
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Ex. 2 A regular pyramid has a base that is a square. The length of an edge
of the base is 22 cm and the height is 14 cm. Find the volume of the
figure.
Ex 3: A regular pyramid has a square base and four lateral sides that are
isosceles triangles. The length of an edge of the base is 10 cm and
the height of the pyramid is 12 cm. The length of the altitude to the
base of each lateral side is 13 cm.
a. What is the total surface area of the pyramid?
b. What is the volume of the pyramid?
Ex. 4 A regular pyramid has a base that is the hexagon ABCDEF and
vertex at V. If the length AB is 2.5 cm, and the slant height of the
pyramid is 6 cm, find the lateral area of the pyramid.
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Detailed Answers to the true/false question on P 432.
3. At a given point on a given line, only one line can be drawn perpendicular to the
line.
(False) There are an infinite number of lines. Think about the diameters of a
circle, or the spokes of a bicycle wheel.
4. If A is a point on plane p, and B is a point not in p, then no other point
on AB is in plane p.
(True)
5. A line perpendicular to a plane is perpendicular to every line in the plane.
(False) A line in the ceiling is perpendicular to the front wall, but it is not
perpendicular to the line in the corner of the room.
7. Two intersecting planes that are each perpendicular to a third plane are
perpendicular to each other.
(False) The side walls are each perpendicular to the floor, but they are not
perpendicular to each other.
8. If AB is perpendicular to plane p at A and AB is in plane q, then p is
perpendicular to q.
(True) Let AB be the line in the corner of the room and let plane p be the floor.
Then B could be a point on either of the side walls, and these could be plane q.
The walls and the floor are perpendicular.
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9. At a given point on a given line, only one plane is perpendicular to the
given plane.
(False) Use the point on the floor in the corner of the room. Both side walls
are perpendicular to the floor.
10. If a plane is perpendicular to one of two intersecting lines, it is perpendicular
to the other.
(False) Look at 2 intersecting lines in the ceiling. The side wall is perpendicular
to one of them, but not the other.
11. If a line is perpendicular to one of two intersecting planes, it is perpendicular to
the other.
(False) The front and side walls are intersecting planes. The line containing the
top of the white board is perpendicular to the side wall, but is not perpendicular to
the front wall.
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