the figure is probably a prism. If there are more triangles, the figure is most likely a pyramid. Once the student has
located the lateral faces, then they can make a more detailed inspection of the base or bases.
Height, Slant Height, or Edge –A pyramid contains a number of segments with endpoints at the vertex of the
pyramid. There is the altitude which is located inside right pyramids, the slant height of the pyramid is the height of
the triangular lateral faces, and there are lateral edges, where two lateral faces intersect. Students frequently get these
segments confused. To improve their understanding, give them the opportunity to explore with three-dimensional
pyramids. Have the students build pyramids out of paper or light cardboard. The slant height of the pyramid should
be highlighted along each lateral face in one color, and the edges where the lateral faces come together in another
color. A string can be hung form the vertex to represent the altitude of the pyramid. The lengths of all of these
segments should be carefully measured and compared. They should make detailed observations before and after the
pyramid is assembled. Once the students have gained some familiarity with pyramids and these different segments,
it will make intuitive sense to them to use the height when calculating volume, and the slant height for surface area.
Additional Exercises:
- A square pyramid is placed on top of a cube. The cube has side length 3 cm. The slant height of the triangular
lateral faces of the pyramid is 2 cm. What is the surface area of this composite solid?
Answer:
A= 5 ∗area of the square+ 4 ∗area of the triangle= 5 ∗( 3 ∗ 3 )+ 4 ∗(
1
2
∗ 3 ∗ 2
)
=57 cm^2Note: The top face of the cube is covered by the base of the pyramid so neither square is included in the surface area
of the composite figure.
Cones
Mix’em Up –Students have just learned to calculate the surface area and volume of prisms, cylinders, and cones.
Most students do quite well when focused on one type of solid. They remember the formulas and how to apply them.
It is a bit more difficult when students have to choose between the formulas for all four solids. Take a review day
here. Have the students work in small groups during class on a worksheet or group quiz that has a mixture of volume
and surface area exercises for these four solids. The extra day will greatly help to solidify the material learned in the
last few lessons.
Additional Exercises:
- A cone of height 9 cm sits on top of a cylinder of height 12 cm. Both cone and cylinder have radius with length
4 cm. Find the volume and surface area of the composite solid.
Answer:
V=πr^2 h 1 +1
3
πr^2 h 2 =π∗ 42 ∗ 12 +1
3
∗π∗ 42 ∗ 9 = 240 πcm^3 ≈750 cm^3Chapter 2. Geometry TE - Common Errors