Volume of Right Prisms with Fractional Edge Lengths - Seventh Grade Mathematics
The volume of a three-dimensional prism measures its internal cubic capacity, determined by multiplying base area by height and packing fractional unit cubes.
The Concept of Volume and Unit Cubes
Volume is the measure of the three-dimensional space enclosed within a closed solid boundary, quantified in cubic units (such as cubic centimeters, cubic inches, or cubic meters). A unit cube is a cube whose length, width, and height each measure precisely one unit.
A UNIT CUBE (1 x 1 x 1)
+--------+
/ /|
+--------+ | Height = 1
| | |
| | + Width = 1
| |/
+--------+
Length = 1
Volume = 1 cubic unit
Finding the volume of a solid means determining how many unit cubes (or fractional unit cubes) are required to pack the interior space completely without gaps or overlaps.
The Universal Prism Volume Theorem: V = Bh
For any right prism, regardless of whether its base is a rectangle, triangle, hexagon, or irregular polygon, the volume is equal to the area of the base B multiplied by the height h perpendicular to that base.
THE UNIVERSAL PRISM VOLUME FORMULA
V = B * h
B = Area of the 2D Base Polygon (in square units)
h = Perpendicular Height of the Prism (in linear units)
Think of a right prism as a stack of flat polygonal sheets: the base provides the two-dimensional cross-sectional area, and multiplying by the height accumulates that area across three dimensions.
+-----------------------+-----------------------------+------------------------+
| Prism Type | Base Area Formula (B) | Complete Volume Formula|
+-----------------------+-----------------------------+------------------------+
| Rectangular Prism | B = l * w | V = l * w * h |
| Cube | B = s * s = s^2 | V = s^3 |
| Triangular Prism | B = (1/2) * b * h_tri | V = (1/2) * b * h_tri * H|
| Trapezoidal Prism | B = (1/2) * (b1 + b2) * h_tr| V = B * H |
+-----------------------+-----------------------------+------------------------+
Packing Prisms with Fractional Edge Lengths
In seventh grade, edge lengths frequently involve fractions and mixed numbers. We can visualize packing these prisms using fractional unit cubes.
Consider a prism measuring: Length = 1 and 1/2 = 3/2 units
Width = 1 unit = 2/2 units
Height = 1/2 unit = 1/2 units
If we pack this prism with miniature cubes whose edges measure one-half unit: The volume of one mini-cube is one-half times one-half times one-half equals one-eighth cubic unit. How many mini-cubes fit along each dimension? Length: (3/2) divided by (1/2) = 3 mini-cubes. Width: (2/2) divided by (1/2) = 2 mini-cubes. Height: (1/2) divided by (1/2) = 1 mini-cube. Total mini-cubes packed: three times two times one equals six mini-cubes.
Volume = (Number of mini-cubes) * (Volume of one mini-cube)
= 6 * (1/8) = 6/8 = 3/4 cubic unit.
Standard Formula Check:
V = l * w * h = (3/2) * (1) * (1/2) = 3/4 cubic unit.
Both methods arrive at the exact same volume of three-fourths cubic unit, demonstrating that the standard formula V = l * w * h remains valid for all fractional dimensions.
Volume of Triangular Prisms
For a triangular prism, first calculate the area of the triangular base, then multiply by the prism length H.
Example: A triangular prism with:
Triangle base b = 6 cm, Triangle height h = 4 cm
Prism length H = 10 cm
Step 1: Compute Base Area B:
B = (1/2) * b * h = (1/2) * 6 * 4 = 12 square centimeters
Step 2: Multiply by Prism Height H:
V = B * H = 12 * 10 = 120 cubic centimeters
The triangular prism holds one hundred twenty cubic centimeters.
Exercises
Work through each problem carefully in your notebook before reading the provided solutions.
Exercise 1. Calculate the volume of a rectangular prism with fractional edge lengths: length equals two and one-half feet, width equals one and one-third feet, and height equals three-fourths of a foot.
Exercise 2. A cube has edge lengths measuring three-fifths of a meter. Determine the volume of the cube in cubic meters.
Exercise 3. A right triangular prism has a base that is a right triangle with legs measuring five inches and twelve inches. The height of the prism between the triangular bases is fifteen inches. Calculate the volume of the prism.
Exercise 4. A container shaped like a rectangular prism measures four inches by three inches by two inches. How many fractional unit cubes with edge lengths of one-half inch are required to fill the container completely?
Solutions
Solution to Exercise 1. Convert mixed numbers to improper fractions: length is five-halves feet, width is four-thirds feet, and height is three-fourths feet. Multiply the three dimensions to find volume: V equals five-halves times four-thirds times three-fourths. Multiply the numerators: five times four times three equals sixty. Multiply the denominators: two times three times four equals twenty-four. Simplify sixty twenty-fourths by dividing numerator and denominator by twelve: sixty divided by twelve is five, and twenty-four divided by twelve is two. The resulting fraction is five-halves, or two and one-half cubic feet (2.5 cu ft).
Solution to Exercise 2. The volume of a cube is side length cubed: V equals s cubed. Multiply three-fifths times three-fifths times three-fifths. For the numerator: three times three times three equals twenty-seven. For the denominator: five times five times five equals one hundred twenty-five. The volume of the cube is twenty-seven one hundred twenty-fifths of a cubic meter (27/125 m^3).
Solution to Exercise 3. First calculate the area of the triangular base B: B equals one-half times base times height. In a right triangle, the two perpendicular legs serve as base and height: one-half times five times twelve equals one-half of sixty, which is thirty square inches. Now multiply the base area by the prism height of fifteen inches: thirty times fifteen equals four hundred fifty cubic inches. The volume of the prism is four hundred fifty cubic inches.
Solution to Exercise 4. First find the number of one-half inch cubes that fit along each edge: along the four-inch length: four divided by one-half equals eight cubes. Along the three-inch width: three divided by one-half equals six cubes. Along the two-inch height: two divided by one-half equals four cubes. Multiply the counts: eight times six times four. Eight times six is forty-eight; forty-eight times four is one hundred ninety-two. Exactly one hundred ninety-two mini-cubes are required to fill the container. (Check: Volume of container is 4 * 3 * 2 = 24 cu in; volume of one mini-cube is 1/8 cu in; 24 / (1/8) = 192 cubes).