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Topic 9: Surfaces and Solids - Coggle Diagram
Topic 9: Surfaces and Solids
9.1 Prisms
A prism composed of bases and lateral faces where these bases and lateral faces are known as collectively as the faces of the prism.
Right prism is a prism in which the lateral edges are perpendicular to the base edges at their points of intersection.
An oblique prism is a prism in which the lateral edges are oblique to the base edges at their point of intersection.
Theorem 9.1.1: The lateral area,
L
of a prism with altitude,
h
, and the perimeter of the base,
P
is given by
L = hP
Theorem 9.1.2: The total area,
A
of a prism with lateral area,
L
and base area,
B
is given by,
A = L+ 2B
Theorem 9.1.3: The volume of a right prism is given by
V = Bh
where
B
is the area of the base and
h
is the length of the altitude.
Postulate 9.1.1:
Volume Postulate
Corresponding to every solid is a unique positive number V known as the volume of that solid.
Postulate 9.1.2:
The volume of a right rectangular prism with the measure of its length, width and altitude is
l, w
and
h,
respectively, is given by
V = lwh.
9.2 Pyramids
A pyramid has any polygon as its base and all the vertices of the base are joined to a point, known as the vertex or the apex of the pyramid.
The slant height,
H
, of regular pyramid is the altitude from the apex of the pyramid to the base of any congruent lateral faces of the regular pyramid.
Theorem 9.2.1: In a regular pyramid with an apothem of length, a, an altitude of length
h
and a slant height of length
H
, they satisfy the Pythagorean Theorem,
Theorem 9.2.2: The lateral area,
L
of a regular pyramid with slant height of length
H
and the perimeter,
P
of the base is given by,
Theorem 9.2.3: The total area,
A
of a pyramid with lateral area,
L
and the base area,
B
is given by
A = L + B
Theorem 9.2.4: The volume,
V
of a pyramid with the base area,
B
and altitude of length
h
is given by,
Theorem 9.2.5: A regular pyramid with altitude of length
h
, radius of the base with length
r
and lateral edge of length
e
, satisfy the Pythagorean Theorem, i.e.,
9.3 Cylinders and Cones
Cylinders have congruent circles as the bases and a quadrilateral as its lateral surface. Axis of a cylinder is the line segment joining the centres of the bases.
Theorem 9.3.1.1: The lateral area,
L
of a right circular cylinder with altitude of length
h
and circumference
C
of the base is given by
L = hC
or
where
r
is the radius of the circular base.
Theorem 9.3.1.2: The total area,
A
of a right circular cylinder with base area,
B
and lateral area,
L
is given by
L = 2B
or
where
r
is the radius of the base and
h
is the length of its altitude.
Theorem 9.3.1.3: The volume
V
of a right circular cylinder with base area,
B
and altitude of length
h
is given by
Cone has a circle as its base where all points on the circle is connected to one point (outside of the plane), and this point is called as the apex or vertex of the cone.
Theorem 9.3.2.1: The lateral area,
L
of a right circular cone with slant height of length
H
and circumference of the base,
C
is given by
where
r
is the radius of the base.
Theorem 9.3.2.2: The total area,
A
of a right circular cone with base area,
B
and lateral area,
L
is given by
A = B + L
or
Theorem 9.3.2.3: In a right circular cone, the radius of the base with length
r
, altitude with length
h
and slant height
H
satisfy the Pythagorean Theorem, that is,
Theorem 9.3.2.4: The volume,
V
of a right circular cone with base area,
B
and altitude of length
h
is given
or
where
r
is the radius of the circular base.
