__Volume Estimation__

**Introduce concepts of volume
relationship between solid shapes with this set of 14 large**
**View-Thru Geometric Solids. Use
the shapes to estimate, measure and compare volumes**
**in a small group or demonstration
setting.**

**Have students list, from least to
greatest, the estimated volume of each solid. Students should**
**check estimates by calculating
the volume or filling each shape with water using a graduated**
**cylinder and recording the
results beside each listed shape.**

__Volume Formulas__

**v – volume r – radius b – base**

**l – length w – width h – height**

**s – side length of base**

**a – apothem (length from the
center of a polygon to one side)**

**Cube: v = l ³**

Sphere: v = (**4 ****⁄****3****) πr ³**

**Cone: v = ****1 ****⁄****3
****(πr²h)**

Cylinder: v = πr²h

**Rectangular prism: v = lwh
**

Hemisphere: v = (**2
****⁄****3****) πr ³**

**Square pyramid: v = ****1 ****⁄****3
****(lw) h **

Triangular pyramid:
v = **1 ****⁄****3 ****(****1 ****⁄****2
****bh) h**

**Pentagonal prism: v = ****5****⁄****2
****ash **

Triangular prism: v = (**1 ****⁄****2
****bh) h**

__Terminology of Solid Geometry__

**base
****face of a geometric shape;
bases of the View-Thru geometric solids are blue**
**cylinder
****two congruent, parallel
circular bases and a single curved, lateral face**
**edge
****intersection of two faces
of a polyhedron where they meet at a line**
**face
****polygon surface of a
polyhedron; shapes in this set are either flat or curved**

**hemisphere
****one half of any sphere**
**polyhedron
****solid figure with a polygon
face**
**prism
****polyhedron with two
congruent, parallel bases and rectangles for the remaining**
**faces; named for the shape of its
bases**

**pyramid
****polyhedron with one base
and triangles for the remaining faces; named for the**
**shape of its bases**

**sphere
****the set of all points in
space equidistant from a given point called the center**
**vertex
****intersection of three or
more faces of a polyhedron where they meet at a point,**
**or corner**
.

__Working with the View-Thru Geometric Solids to Measure Volume__

**The set of 14 View-Thru Geometric
Solids is ideal for measuring and comparing volume**
**relationships between
the various solid shapes. In order to facilitate volume measurement ****relationships, set up the**** following materials**** at a geometry center or centers in your classroom:**

**Materials:
****View-Thru Geometric Solids**

**1000 Milliliters of plastic fill**

**Set of 2 funnels**

**Chart of the 14 solids and their
characteristics**

**Paper and pencil/pen**

**Procedure:
****Have students estimate the
volume of each of the 14 View-Thru Geometric**
**Solids by listing them on a sheet
of paper from largest volume to smallest**
**volume.**
**Volume is expressed in cubic
units of measurement: inches, feet, yards, miles,**
**milliliters, centimeters,
decimeters, meters, kilometers, etc.**
**Using the funnel, fill the
1-liter graduated cylinder with plastic fill.**
**Remove the base of the chosen
solid and fill it with the plastic fill. Note the**
**amount of fill required. Repeat
two or three times to ensure accuracy.**
**Repeat the process with all of
the shapes.**
**Have the students evaluate their
data by listing the solids in descending order**
**from most volume to least volume.
Compare completed list with original**
**estimation.**

**Discuss:
****What other materials could
be used for the measurements?**
**What relationships exist between
the various solids? How does the volume**
**of the cube compare to the volume
of the square pyramid? Explain any other**
**comparisons derived from the
data.**

__Characteristics of Geometric Solids__

**Work with the students to
create a chart similar to the one below (but using vertical and horoaontal lines) to record their own**
**observations:**

**View-Thru****®** ** **
**Shape**
**of Base(s)**
**Number**
**of Faces**
**Number**
**of Vertices**
**Number**
**of Edges**

**Geometric
Solids
**

**1 Large Cube**

**2 Small Cube**

**3 Large Rectangle**

**4 Small Rectangle**

**5 Pentagonal Prism**

**6 Large Triangular Prism**

**7 Small Triangular Prism**

**8 Square Pyramid**

**9 Triangular Pyramid**

**10 Large Cylinder**

**11 Small Cylinder**

**12 Cone**

**13 Sphere**

**14 Hemisphere**

__Euler’s Formula__

**Euler’s Formula is named after
Swiss mathematician Leonard Euler. In the mid-eighteenth**
**century, Euler discovered that
for any polyhedron, F + V = E + 2. In the formula, ****F ****represents**
**the number of faces, ****V ****represents the number of vertex
points, and ****E ****represents
the number of**
**edges. For example, a cube has 6
faces, 8 vertex points, and 12 edges.**

**F
+ V = E + 2**

**6
+ 8 = 12 + 2**

**Have the students use their data
from the preceding chart to discover Euler’s Formula. Euler’s**
**Formula is true for the first
nine solids listed in the table.**

__Intervention Strategies__

**Scaffolded
Instruction: ****Before
providing formulas to students, instead provide the**
**definitions of perimeter and
area, and opportunities to solve problems that allow students**
**to gain data leading to the use
of a formula. Begin with two-dimensional shapes before**
**advancing to three-dimensional
solids.**

**Directed
Orientation: ****Use
different household items that resemble a cube, cone, sphere,**
**cylinder, pyramid, or prism. Have
students sort the items by different attributes you provide.**
**Then, introduce the formal shapes
and have students match the shapes to the corresponding**
**household items.**

**Free
Exploration: ****Have
students fill the solids with rice or water to explore properties of**
**volume. Encourage students to
make estimations and compare which shapes are able to hold**
**more
or less than the others.**

© Learning Resources, Inc.,