Solved Exercises on Volume of Prisms in Grade 7

Master volume of prisms: rectangular, triangular, and complex prisms through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rectangular Prism
Exercise 1
Find the volume of a rectangular prism with length 8 cm, width 5 cm, and height 3 cm.
Definition:

Rectangular Prism: A 3D shape with six rectangular faces. It has length, width, and height.

Method for rectangular prism volume:
  1. Identify the length, width, and height
  2. Apply the volume formula: V = length × width × height
  3. Substitute the values
  4. Calculate the result
  5. Include the units (cm³)
Given
l=8cm, w=5cm, h=3cm
Formula
V = lwh
Result
120 cm³
Step 1: Identify the dimensions

Length = 8 cm, Width = 5 cm, Height = 3 cm

Step 2: Apply the volume formula

Volume = length × width × height

Step 3: Substitute values

Volume = 8 × 5 × 3

Step 4: Calculate

Volume = 40 × 3 = 120 cm³

Step 5: Verify the calculation

Check: 8 × 5 = 40, then 40 × 3 = 120 ✓

Volume = 120 cm³
Final answer:

The volume of the rectangular prism is 120 cm³.

Applied rules:

• Prism Volume Formula: V = Base Area × Height

• Rectangular Prism: V = lwh

• Cubic Units: Volume units are cubed (cm³)

2 Triangular Prism
Exercise 2
A triangular prism has a base area of 24 cm² and a height of 10 cm. Find its volume.
Definition:

Triangular Prism: A 3D shape with two parallel triangular bases and three rectangular lateral faces.

Given
Base Area = 24 cm², Height = 10 cm
Formula
V = Bh
Result
240 cm³
Step 1: Identify the known values

Base Area (B) = 24 cm²

Height (h) = 10 cm

Step 2: Apply the general prism volume formula

Volume of prism = Base Area × Height

V = B × h

Step 3: Substitute values

V = 24 × 10

Step 4: Calculate

V = 240 cm³

Step 5: Verify with alternative approach

If the triangular base had sides a, b, c and we knew them, we could calculate area using Heron's formula first, then multiply by height. But since area is given, we skip this step.

Volume = 240 cm³
Final answer:

The volume of the triangular prism is 240 cm³.

Applied rules:

• General Prism Formula: V = Base Area × Height

• Triangular Base: Area = ½bh or other area formula

• Volume Units: Cubic units (cm³)

3 Finding Missing Dimension
Exercise 3
A rectangular prism has a volume of 180 cm³. If its length is 9 cm and its width is 5 cm, find its height.
Definition:

Missing Dimension: When volume and some dimensions are known, rearrange the volume formula to find the unknown dimension.

Given
V=180, l=9, w=5
Formula
V = lwh → h = V/(lw)
Result
4 cm
Step 1: Write the volume formula

V = l × w × h

Step 2: Substitute known values

180 = 9 × 5 × h

180 = 45 × h

Step 3: Solve for the unknown dimension

h = 180 ÷ 45

h = 4 cm

Step 4: Verify the solution

Check: 9 × 5 × 4 = 180 ✓

Step 5: Express the answer with units

Height = 4 cm

Height = 4 cm
Final answer:

The height of the rectangular prism is 4 cm.

Applied rules:

• Formula Rearrangement: Isolate the unknown variable

• Algebraic Division: Divide both sides by known factors

• Verification: Substitute back to check the answer

Prism Volume Properties and Methods
\(V = \text{Base Area} \times \text{Height}\)
Prism Volume Formula
Rectangular Prism
V = lwh
Three perpendicular dimensions
Triangular Prism
V = ½bhl
Triangular base × length
Cylinder
V = πr²h
Circular base × height
Key definitions:

Prism: A 3D shape with two parallel, congruent polygonal bases connected by rectangular faces.

Base: The polygonal face of the prism that is repeated at the top and bottom.

Height: The perpendicular distance between the two bases.

Volume: The amount of space inside a 3D shape, measured in cubic units.

Rectangular Prism: A prism with rectangular bases (also called a rectangular box or cuboid).

Triangular Prism: A prism with triangular bases.

Base Area: The area of the polygonal base of the prism.

Complete methodology:
  1. Identify the Prism Type: Determine the shape of the base
  2. Find Base Area: Calculate the area of the base using appropriate formula
  3. Identify the Height: Find the perpendicular distance between bases
  4. Apply Volume Formula: V = Base Area × Height
  5. Substitute Values: Replace variables with known measurements
  6. Calculate Result: Perform the multiplication
  7. Include Units: Express answer in cubic units
Tip 1: Always ensure the height is perpendicular to the base area.
Tip 2: The base can be any of the parallel faces; height is always perpendicular to it.
Tip 3: For rectangular prisms, you can multiply length, width, and height in any order.
Tip 4: Remember that volume units are always cubed (cm³, m³, etc.).
Formulas to know by heart:

• General Prism: V = Base Area × Height

• Rectangular Prism: V = length × width × height

• Cube: V = side³

• Triangular Prism: V = ½ × base × height × length

• Cylinder: V = πr²h

• Finding Missing Dimension: Dimension = Volume ÷ (other dimensions)

Solution: Exercises 4 to 5
4 Complex Prism Problem
Exercise 4
A prism has a trapezoidal base with parallel sides of 8 cm and 12 cm, and a height of 5 cm. If the prism's height is 10 cm, find its volume.
Definition:

Trapezoidal Prism: A prism with trapezoidal bases. Volume = Base Area × Height.

Trapezoid Base
½(8+12)×5 = 50 cm²
Prism Height
10 cm
Volume
500 cm³
Step 1: Calculate the area of the trapezoidal base

Area of trapezoid = ½(sum of parallel sides) × height

Base Area = ½(8 + 12) × 5

Base Area = ½(20) × 5 = 10 × 5 = 50 cm²

Step 2: Apply the prism volume formula

Volume = Base Area × Prism Height

Step 3: Substitute values

Volume = 50 × 10

Step 4: Calculate

Volume = 500 cm³

Step 5: Verify the solution

Check: ½(8+12)×5×10 = ½×20×5×10 = 500 ✓

Volume = 500 cm³
Final answer:

The volume of the trapezoidal prism is 500 cm³.

Applied rules:

• General Prism Volume: V = Base Area × Height

• Trapezoid Area: A = ½(b₁ + b₂)h

• Volume Calculation: Multiply base area by height

5 Multi-Step Problem
Exercise 5
A water tank is shaped like a rectangular prism measuring 2 m by 1.5 m by 1 m. It is filled to 75% capacity. How many liters of water does it contain? (1 m³ = 1000 liters)
Definition:

Capacity: The maximum volume a container can hold. Often expressed in liters when referring to liquids.

Dimensions
2m × 1.5m × 1m
Volume
3 m³
Water
2250 liters
Step 1: Calculate the total volume of the tank

Volume = length × width × height

Volume = 2 × 1.5 × 1 = 3 m³

Step 2: Find 75% of the total volume

Water volume = 75% × Total volume

Water volume = 0.75 × 3 = 2.25 m³

Step 3: Convert cubic meters to liters

1 m³ = 1000 liters

2.25 m³ = 2.25 × 1000 = 2250 liters

Step 4: Verify the calculation

Check: 2 × 1.5 × 1 = 3 ✓

Check: 0.75 × 3 = 2.25 ✓

Check: 2.25 × 1000 = 2250 ✓

Step 5: State the final answer

The tank contains 2250 liters of water.

Water volume = 2250 liters
Final answer:

The tank contains 2250 liters of water.

Applied rules:

• Rectangular Prism Volume: V = lwh

• Percentage Calculation: Part = Percent × Whole

• Unit Conversion: 1 m³ = 1000 liters

Complete Guide: Volume of Prisms
\(V = A_{base} \times h\)
General Prism Volume
Key definitions:

Prism: A polyhedron with two parallel, congruent polygonal bases connected by rectangular lateral faces.

Base Area: The area of the polygonal base of the prism.

Height: The perpendicular distance between the two parallel bases.

Volume: The measure of the space occupied by a 3D shape, expressed in cubic units.

Right Prism: A prism where the lateral faces are rectangles and the height is perpendicular to the base.

Oblique Prism: A prism where the lateral faces are parallelograms and the height is not perpendicular to the base (Grade 8+).

Complete methodology:
  1. Identify the Base Shape: Determine the polygonal shape of the base
  2. Calculate Base Area: Use the appropriate area formula for the base shape
  3. Measure the Height: Find the perpendicular distance between the bases
  4. Apply the Formula: V = Base Area × Height
  5. Perform Calculations: Multiply base area by height
  6. Express Answer: Include proper cubic units
  7. Verify Solution: Check calculations and reasonableness
Tip 1: The height of a prism is always perpendicular to the base area.
Tip 2: You can think of volume as stacking layers of the base shape.
Tip 3: For complex bases, break them into simpler shapes to find the area.
Tip 4: Always check that your answer has cubic units (m³, cm³, etc.).
Formulas to know by heart:

• General Prism: V = Base Area × Height

• Rectangular Prism: V = length × width × height

• Cube: V = side³

• Triangular Prism: V = ½ × base × height of triangle × length of prism

• Volume from Area: V = A_base × h

• Finding Missing Dimension: Dimension = V ÷ (other areas/dimensions)

Visualizing Prism Volumes: Different Base Shapes
Exercise 6: Volume Relationships
Consider how volume changes with different base areas:
Same height (5 cm) for rectangular, triangular, and trapezoidal bases
Showing the direct relationship between base area and volume

Analysis: The chart shows how volume increases linearly with base area when height is constant.

  • Volume is directly proportional to base area (V = Bh)
  • Greater base area means greater volume
  • Height acts as a constant multiplier
  • Same height allows direct comparison of base effects

Questions & Answers

Question: What's the difference between the height of the base and the height of the prism?

Answer: These are two completely different measurements:

Height of the Base:

  • Belongs to the 2D polygon that forms the base
  • Used to calculate the base area
  • For a triangle base: it's the perpendicular height of the triangle
  • For a trapezoid base: it's the perpendicular distance between the parallel sides

Height of the Prism:

  • Is the 3D measurement between the two parallel bases
  • Used in the volume formula: V = Base Area × Height
  • Perpendicular distance connecting the two bases

Example: A triangular prism might have a base triangle with height 6 cm (for area calculation) and a prism height of 10 cm (for volume calculation).

Question: Can the base of a prism be any shape? How do I find the volume if the base is irregular?

Answer: Yes, the base of a prism can be any polygon:

Common base shapes:

  • Triangle (triangular prism)
  • Rectangle (rectangular prism)
  • Square (cube)
  • Parallelogram
  • Trapezoid
  • Pentagon, hexagon, etc.

For irregular bases:

  • Break the shape into simpler polygons (triangles, rectangles)
  • Calculate the area of each part
  • Sum the areas to get the total base area
  • Multiply by the prism height

The volume formula V = Base Area × Height works for any prism regardless of base shape!

Question: How do I convert between different cubic units like cm³, m³, and liters?

Answer: Here are the key conversions:

Cubic unit relationships:

  • 1 m = 100 cm, so 1 m³ = 1,000,000 cm³ (100³)
  • 1 dm = 10 cm, so 1 dm³ = 1,000 cm³ (10³)
  • 1 liter = 1 dm³ = 1,000 cm³
  • 1 mL = 1 cm³

Conversion examples:

  • To convert cm³ to m³: divide by 1,000,000
  • To convert m³ to cm³: multiply by 1,000,000
  • To convert cm³ to liters: divide by 1,000
  • To convert liters to cm³: multiply by 1,000

Always double-check that your units match the context of the problem!