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In the previous sections, we covered two dimensional geometry; now, let's expand these same concepts into three dimensions, building on what we already know about describing space in two dimensions.
One of the most intuitive transitions from two- to three-dimensional geometry is in moving from circles to spheres.
Spheres
A sphere is a perfectly round, three-dimensional geometric shape where every point on its surface is exactly the same distance (the radius) from a single central point. It's the three-dimensional analog of a circle. For example, a ball or globe is spherical in shape. The surface area A and volume V of a sphere are given by:
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Checkpoint
Find the volume of a sphere with diameter 2cm.
Select the correct option
Cylinders are another kind of object which are related to circles and are thus described by formulas similar to those of circles.
Cylinders
A cylinder is a three-dimensional geometric shape formed by two identical circular bases connected by a curved lateral surface. The segment connecting the centers of the circular bases is called the axis, which is perpendicular to each base in a right cylinder (the type usually studied).
The volume V of a cylinder with radius r is given by:
The curved surface of a cylinder (excluding the circular ends) is given by:
If we include the circular ends, each with area πr2, we get
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Exercise
A cylinder has volume 36π and curved surface area 24π.
Find the radius r.
Select the correct option
Discussion
Imagine a sphere fit within a cylinder of height 2r and base-radius r.
What is the connection between the surface area of the sphere and the curved surface of the cylinder?
Why does this make sense?
The curved (lateral) surface area of the cylinder of radius r and height 2r is
and the surface area of the sphere of radius r is also
hence they are equal.
If you “cut” the cylinder along a vertical line and flatten its side you get a rectangle of width 2πr and height 2r, so area 2πr⋅2r=4πr2. Imagine slicing the sphere into many very thin horizontal “belts.” Each belt on the sphere has almost the same circumference as the cylinder (2πr) and a small “height” matching that of the cylinder belts. As you add up all those little belt‐areas around the sphere, you get exactly the same total as for the cylinder’s side.
Thus the sphere’s surface “unfolds” in effect to the same area as the side of its "circumscribing" cylinder (the cylinder it fits within).
The process of "attaching" two 2D objects and connecting them in 3D spaces can be done with more shapes than just circles. A prism is the general term for this family of shapes.
Volume of a prism
A prism is a three-dimensional solid shape consisting of two parallel, congruent faces called bases, connected by rectangular lateral faces. Prisms are named according to the shape of their bases—for example, triangular prism, rectangular prism, or hexagonal prism.
The volume V of a prism is calculated by multiplying the area A of its base by its height h:
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Exercise
A prism of height 2cm is formed from a right angled triangle with two sides of length 3cm.
Find the volume of the prism.
Select the correct option
There are also some shapes, like right cones, which are more closely related to other three dimensional objects (like spheres) than two dimensional ones.
Right cones
A right circular cone is a three-dimensional geometric shape whose apex (vertex) lies directly above the center of its circular base.
Key Parts:
Circular Base: Flat circle with radius r.
Apex (Vertex): The point directly above the center of the base.
Height (h): Perpendicular distance from apex to base center.
Slant Height (l): Distance along the cone's surface from apex to edge of base.
Formulas:
Volume:
Surface Area of curved surface:
Slant Height Relationship:
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Discussion
Notice that a cone has volume VCone=31πr2h and surface area ACone=πrl, while a sphere has volume VSphere=34πr3 and surface area ASphere=4πr2.
What do the formulas for a cone look like if it has h=l=r?
What does this tell you about the relationship between a cone and a sphere?
Substitute h=l=r into the cone formulas:
For a sphere of radius r:
So both the volume and surface area of the sphere are exactly four times those of one cone with h=l=r.
Intuitively, you can imagine taking four identical cones (each of height r and base‐radius r) and fitting their curved sides together around a point so that their bases fan out.
In this arrangement:
• Each cone’s curved surface covers exactly one quarter of the sphere’s surface.
• Each cone contributes exactly one quarter of the sphere’s volume.
Thus the sphere behaves as though it were “built” from four of these congruent cones.
Exercise
A cone has volume 6π and height 4.
Find the area of the curved portion of the cone.
Select the correct option
The final kind of three dimensional object of interest is a right pyramid, which mixes triangle and square geometry with some of the terminology we used to describe cones. Right pyramids are kind of like right cones with sharp edges.
Right Pyramid
A right pyramid is a three-dimensional shape with a polygonal base and triangular lateral faces, in which the apex (vertex) is located directly above the center (centroid) of the base.
Key Parts:
Polygonal base: a flat polygon (triangle, square, pentagon, etc.)
Apex (vertex): the point positioned vertically above the base's centroid
Height (h): perpendicular distance from apex to base centroid
Slant height (l): distance along a lateral face from the apex perpendicular to an edge of the base
The volume of a right pyramid is given by
where A is the area of the base.
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Discussion
Two pyramids with the same base area A and height h can have quite different shapes. For example, consider two pyramids, A and B. Both pyramids have A=4 and h=3, but where Pyramid A has a square base with square side lengths x=2, Pyramid B has a right triangular base with triangle side lengths y=4 and z=2.
Pyramids A and B have the same volume V=31(4)(3)=6.
(insert 3d image)
Will any two right pyramids with A1=A2 and h1=h2 have the same volume V? Does the shape of the pyramid's base at all affect its volume? How about the slant height l?
The volume of any right pyramid depends only on its base area and its vertical height. In fact, if two right pyramids have base areas A1,A2 and heights h1,h2, then
If A1=A2 and h1=h2, then
so the two volumes are equal. Hence:
• Any two right pyramids with the same base area and the same height have the same volume.
• The shape of the base does not matter beyond its area.
• The slant height l does not enter the volume formula (it only affects the lateral surface area).
Exercise
A right pyramid has for its base an equilateral triangle of side length 5cm, and a height of h. The volume of the pyramid is 10cm3.
Find h.
Select the correct option
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