Three Dimensional Figures (Popeye)

PopeyeAll examples come from RegentsPrep.org

Practice Problems:

3-D Figures

Applied Problems

 

 

 

 


Right Triangular Prism

Prisms are three-dimensional closed surfaces. 

A prism has two parallel faces, called bases, that are congruent polygons.  The lateral faces are rectangles in a right prism, or parallelograms in an oblique prism.  In a right prism, the joining edges and faces are perpendicular to the base faces. 

Prisms are also called polyhedra since their faces are  polygons.

A regular prism is a cube.


Right Rectangular Prism


Oblique Triangular Prism

 

The volume of a prism is the product of the base area times the height of the prism.

V = Bh
(Volume of a prism:  B = base area,  h = height)

= height(altitude) between bases
= area of the base

The surface area of a prism is the sum of the areas of the bases plus the areas of the lateral faces.  This simply means the sum of the areas of all faces.


The surface area, S, of a right prism can be found using the formula = 2ph.
= area of base, p = perimeter of base, h = height.

  
net is a two-dimensional figure
that can be cut out and folded up to
make a three-dimensional solid.

Pyramid

We will be working with regular pyramids unless otherwise stated.

Pyramids are three-dimensional closed surfaces.

   The one base of the pyramid is a polygon and the lateral faces are always triangles with a common vertex.  The vertex of a pyramid (the point, or apex) is not in the same plane as the base. 

Pyramids are also called polyhedra since their faces are  polygons.  

The most common pyramids are regular pyramids.  A regular pyramid has a regular polygon for a base and its height meets the base at its center.  The slant height is the height (altitude) of each lateral face.

In a regular pyramid, the lateral edges are congruent.
Since the base is a regular polygon, whose sides are all congruent, we know that the lateral faces of a regular pyramid are congruent isosceles triangles.

Pyramids are named for the shape of their base.

        Triangular pyramid                                    Square pyramid              

The volume of a pyramid is one-third the product of the base area times the height of the pyramid.



(Volume of a pyramid:  B = base area,  h = height)


= height (altitude) from vertex to base
B = area of base

The surface area of a pyramid is the sum of the area of the base plus the areas of the lateral faces.
This simply means the sum of the areas of all faces.

The surface area, S, of a regular pyramid can be found using the formula .
B = area of base, p = perimeter of base, s = slant height.

 

net is a two-dimensional figure
that can be cut out and folded up to
make a three-dimensional solid.

 

Cylinder

= height (altitude)
r = radius

We will be working with right circular cylinders unless
otherwise stated.

Cylinders are three-dimensional closed surfaces.

In general use, the term cylinder refers to a right circular cylinder with its ends closed to form two circular surfaces, that lie in parallel planes.

Cylinders are not called polyhedra since their faces are not polygons.   In many ways, however, a cylinder is similar to a prism.  A cylinder has parallel congruent bases, as does a prism, but the cylinder’s bases are circles rather than polygons.

The volume of a cylinder can be calculated in the same manner as the volume of a prism:   the volume is the product of the base area times the height of the cylinder,
V = Bh.
Since the base in a cylinder is a circle, the formula for the area of a circle can be substituted into the volume formula for B:

 
(Volume of a cylinder:  r = radius of base, h = height)

 

net is a two-dimensional figure that can be cut out and folded up to make a three-dimensional solid. 

Lateral = any face or surface that is not a base.

The surface area (of a closed cylinder) is a combination of the lateral area and the area of each of the bases.  When disassembled, the surface of a cylinder becomes two circular bases and a rectangular surface (lateral surface), as seen in the net at the left. 

Note that the length of the rectangular surface is the same as the circumference of the base.  Remember that the area of a rectangle is length times width.

The lateral area (rectangle) = height × circumference of the base.
The base area = area of a circle (remember there are two bases)

(Total Surface Area of a Closed Cylinder)
…which can also be factored and written as

When working with surface areas of cylinders, read the questions carefully.

Will the surface area include both of the bases?

Will the surface area include only one of the bases?

Will the surface area include neither of the bases?


The lateral area only.

 

Cone

= height (altitude)
r = radius
= slant height


We will be working with right circular cones unless
otherwise stated.

Cones are three-dimensional closed surfaces.

In general use, the term cone refers to a right circular cone with its end closed to form a circular base surface.  The vertex of the cone (the point) is not in the same plane as the base.

Cones are not called polyhedra since their faces are not polygons.   In many ways, however, a cone is similar to a pyramid.  A cone’s base is simply a circle rather than a polygon as seen in the pyramid.

 

The volume of a cone can be calculated in the same manner as the volume of a pyramid:  the volume is one-third the product of the base area times the height of the cone,
Since the base of a cone is a circle, the formula for the area of a circle can be substituted into the volume formula for :


(Volume of a cone:  r = radius, h = height)

net is a two-dimensional figure that can be cut out and folded up to make a three-dimensional solid.

Lateral = any face or surface that is not a base.

In a right circular cone, the slant height, s, can be found using the Pythagorean Theorem:

The surface area (of a closed cone) is a combination of the lateral area and the area of the base.  When cut along the slant side and laid flat, the surface of a cone becomes one circular base and the sector of a circle (lateral surface), as shown in the net at the left.

Note that the length of the arc in the sector is the same as the circumference of the small circular base.
By using a proportion, the area of the sector (lateral area) will be:  

(measurements pertain to the larger net figure, the circle containing the sector)


(arc length of the sector equals the circumference of the smaller base circle)

(the radius of the smaller base is r, while the radius of the larger sector is s)

Note:  The formula for the area of the sector (lateral area), , is equal to one half the product of the slant height and the circumference of the base.

The lateral area (sector) = 
The base area = area of a circle


(Total Surface Area of a Closed Cone)

When working with surface areas of cones, read the questions carefully.

Will the surface area
include the base?

Will the surface area
not include the base?


 

Sphere

= radius

Spheres are three-dimensional closed surfaces.

A sphere is a set of points in three-dimensional space equidistant from a point called the center.  The radius of the sphere is the distance from the center to the points on the sphere.

Spheres are not polyhedra.

 Of all shapes, a sphere has the smallest surface area for its volume.

The volume of a sphere is four-thirds times pi times the radius cubed.

(Volume of a sphere:  r = radius)

Note:  A cross section of a geometric solid is the intersection of a plane and the solid.

The surface area of a sphere is four times the area of the largest cross-sectional circle (called the great circle).

A great circle is the largest circle that can be drawn on a sphere.  Such a circle will be found when the cross-sectional plane passes through the center of the sphere. 

The equator is an examples of a great circle.  Meridians (passing through the North and South poles) are also great circles.     The shortest distance between two points on a sphere is along the arc of the great circle joining the points.

The shortest distance between points on any surface is called ageodesic.  In a plane, a straight line is a geodesic.  On a sphere, a great circle is a geodesic.

 

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