Volume and Capacity. Grade 7 Mathematics: Volume and Capacity Subtopic Navigator Understanding Volume and Capacity Units of Volume and Capacity Volume and Capacity Conversions Volume of Rectangular Prisms Volume of Cylinders Volume of Composite Solids Capacity Calculations Comparing Volumes and Capacities Real-World Applications Cumulative Exercises Conclusion Lesson Objectives Understand the difference between volume and capacity Convert between different units of volume and capacity Calculate volumes of rectangular prisms, cylinders, and composite solids Solve capacity problems involving containers and liquids Compare and order volumes and capacities Apply volume and capacity concepts to real-world situations Volume and Capacity Volume is the amount of three-dimensional space occupied by an object, while capacity refers to the amount of fluid a container can hold. Although related (1 mL = 1 cm³), these concepts have different applications: volume for solids, capacity for containers. Understanding both is essential for fields ranging from engineering to cooking. Units of Volume and Capacity Volume is typically measured in cubic units (cm³, m³, etc.), while capacity is measured in liters (L) and milliliters (mL). The fundamental relationship is: 1 cm³ = 1 mL, and 1000 cm³ = 1 L = 1000 mL. Example 1: Understanding Different Units Complete the following relationships: a) 1 m³ = ______ cm³ b) 1 L = ______ mL c) 1 cm³ = ______ mL d) 1000 L = ______ m³ Solution: a) 1 m³ = 1,000,000 cm³ (since 1 m = 100 cm, so 1 m³ = 100³ cm³ = 1,000,000 cm³) b) 1 L = 1,000 mL c) 1 cm³ = 1 mL (fundamental relationship) d) 1000 L = 1 m³ (since 1000 L = 1000 × 1000 mL = 1,000,000 mL = 1,000,000 cm³ = 1 m³) Example 2: Choosing Appropriate Units What are appropriate units for measuring: a) The volume of a swimming pool b) The capacity of a medicine dropper c) The volume of a textbook d) The capacity of a car's fuel tank Solution: a) Cubic meters (m³) or liters (L) - large volume b) Milliliters (mL) - very small capacity c) Cubic centimeters (cm³) - moderate solid volume d) Liters (L) - moderate liquid capacity Units Problems How many cubic centimeters are in 2.5 cubic meters? Convert 4500 mL to liters What is the relationship between cubic decimeters and liters? Which is larger: 1 cubic meter or 1000 liters? Express 0.75 cubic meters in liters Volume and Capacity Conversions Converting between different units of volume and capacity requires understanding the relationships between units. For metric conversions, we use powers of 10. For conversions between cubic units and capacity units, we use the relationship 1 cm³ = 1 mL. Example 1: Complex Conversions Convert the following: a) 2.5 m³ to liters b) 4500 cm³ to liters c) 3.75 L to cubic centimeters d) 0.008 m³ to milliliters Solution: a) 2.5 m³ = 2.5 × 1000 L = 2,500 L (since 1 m³ = 1000 L) b) 4500 cm³ = 4500 mL = 4.5 L (since 1000 mL = 1 L) c) 3.75 L = 3.75 × 1000 mL = 3750 mL = 3750 cm³ d) 0.008 m³ = 0.008 × 1,000,000 cm³ = 8,000 cm³ = 8,000 mL Example 2: Multi-Step Conversions A container has a capacity of 2.5 cubic meters. How many 250 mL bottles can it fill? Solution: 2.5 m³ = 2.5 × 1000 L = 2,500 L (since 1 m³ = 1000 L) 2,500 L = 2,500 × 1000 mL = 2,500,000 mL Number of bottles = 2,500,000 mL ÷ 250 mL = 10,000 bottles Alternative method: Each bottle: 250 mL = 0.25 L Number of bottles = 2,500 L ÷ 0.25 L = 10,000 bottles Conversion Problems Convert 3.5 cubic meters to liters How many cubic centimeters in 4.25 liters? Convert 15,000 mL to cubic meters A tank holds 0.75 m³. How many 500 mL containers can it fill? Express 2,500,000 cm³ in cubic meters and liters Volume of Rectangular Prisms The volume of a rectangular prism (box) is calculated as V = length × width × height. All measurements must be in the same units before multiplying. The result will be in cubic units. Example 1: Complex Rectangular Prism Calculate the volume of a rectangular prism with dimensions: Length = 2.5 m, Width = 1.2 m, Height = 0.8 m Express your answer in cubic meters, liters, and cubic centimeters. Solution: Volume = length × width × height = 2.5 m × 1.2 m × 0.8 m = 2.4 m³ In liters: 2.4 m³ = 2.4 × 1000 L = 2,400 L In cubic centimeters: 2.4 m³ = 2.4 × 1,000,000 cm³ = 2,400,000 cm³ Example 2: Finding Missing Dimension A rectangular prism has volume 1,440 cm³. If its length is 12 cm and width is 10 cm, what is its height? If this prism represents a container, what is its capacity in liters? Solution: Volume = length × width × height 1,440 cm³ = 12 cm × 10 cm × height 1,440 = 120 × height height = 1,440 ÷ 120 = 12 cm Capacity = 1,440 cm³ = 1,440 mL = 1.44 L Rectangular Prism Problems Find volume of prism: length=15 cm, width=8 cm, height=12 cm. Express in cm³ and liters. A box has volume 2,160 cm³. If length=18 cm and width=10 cm, find height. Calculate volume: 2.5 m × 1.8 m × 0.9 m. Express in m³ and L. Which has greater volume: 30 cm × 20 cm × 15 cm or 25 cm × 22 cm × 18 cm? A container 40 cm × 30 cm × 25 cm is filled with water. What is water volume in liters? Volume of Cylinders The volume of a cylinder is calculated as V = πr²h, where r is the radius and h is the height. π is approximately 3.1416, but we often use $frac{22}{7}$ or 3.14 for calculations. All measurements must be in the same units. Example 1: Complex Cylinder Volume Calculate the volume of a cylinder with radius 7 cm and height 20 cm. Use π = $frac{22}{7}$ and express answer in cm³ and liters. Solution: Volume = πr²h = $frac{22}{7} × (7)^2 × 20$ = $frac{22}{7} × 49 × 20$ = 22 × 7 × 20 (since 49/7 = 7) = 22 × 140 = 3,080 cm³ Capacity = 3,080 cm³ = 3,080 mL = 3.08 L Example 2: Finding Cylinder Height A cylindrical tank has volume 15,400 cm³ and radius 10 cm. Find its height. (Use π = $frac{22}{7}$) What is its capacity in liters? Solution: Volume = πr²h 15,400 = $frac{22}{7} × (10)^2 × h$ 15,400 = $frac{22}{7} × 100 × h$ 15,400 = $frac{2200}{7} × h$ h = 15,400 × $frac{7}{2200}$ h = $frac{107,800}{2200}$ = 49 cm Capacity = 15,400 cm³ = 15,400 mL = 15.4 L Cylinder Problems Find volume of cylinder: radius=14 cm, height=25 cm (use π = $frac{22}{7}$) A cylindrical can has volume 1,540 cm³ and height 10 cm. Find radius (π = $frac{22}{7}$). Calculate volume: diameter=28 cm, height=30 cm. Express in liters. Which holds more: cylinder r=10 cm, h=20 cm or rectangular prism 15 cm × 15 cm × 15 cm? A water tank is cylindrical with r=1.4 m, h=2 m. What is capacity in liters? (π = $frac{22}{7}$) Volume of Composite Solids Composite solids are made by combining basic shapes. To find their total volume, calculate the volume of each part separately, then add them together. Be careful with units and ensure all measurements are consistent. Example 1: L-Shaped Prism An L-shaped solid consists of two rectangular prisms: Prism A: 20 cm × 15 cm × 10 cm Prism B: 30 cm × 10 cm × 10 cm They are joined along the 10 cm height. Calculate the total volume in cm³ and liters. Solution: Volume of Prism A = 20 × 15 × 10 = 3,000 cm³ Volume of Prism B = 30 × 10 × 10 = 3,000 cm³ Total volume = 3,000 + 3,000 = 6,000 cm³ Capacity = 6,000 cm³ = 6,000 mL = 6 L Example 2: Cylinder on Rectangular Base A solid consists of a rectangular base (30 cm × 20 cm × 5 cm) with a cylinder (radius 7 cm, height 15 cm) placed on top. Calculate the total volume. (Use π = $frac{22}{7}$) Solution: Volume of rectangular base = 30 × 20 × 5 = 3,000 cm³ Volume of cylinder = πr²h = $frac{22}{7} × 7^2 × 15$ = $frac{22}{7} × 49 × 15 = 22 × 7 × 15 = 2,310$ cm³ Total volume = 3,000 + 2,310 = 5,310 cm³ Composite Solid Problems A solid has cube (edge=12 cm) on top of rectangular prism (15×12×8 cm). Find total volume. Calculate volume: Cylinder (r=14 cm, h=20 cm) + Rectangular prism (28×14×10 cm). Use π = $frac{22}{7}$. Two cylinders: one r=7 cm, h=15 cm; another r=10 cm, h=12 cm. Find total volume in liters. A T-shaped solid: Vertical part 10×8×25 cm, horizontal part 20×8×10 cm. Find volume. Rectangular tank 40×30×25 cm has cylindrical hole (r=7 cm, h=25 cm) drilled through. Find remaining volume. Capacity Calculations Capacity refers to the amount of fluid a container can hold. For regular containers, capacity equals internal volume. When dealing with liquids, we often work in liters or milliliters. Example 1: Partial Filling A rectangular tank measures 50 cm × 40 cm × 30 cm. a) What is its full capacity in liters? b) If filled to ¾ of its height, how many liters does it contain? c) How many 500 mL bottles can be filled from the tank when ¾ full? Solution: a) Full volume = 50 × 40 × 30 = 60,000 cm³ Full capacity = 60,000 mL = 60 L b) When ¾ full, height of water = ¾ × 30 cm = 22.5 cm Volume of water = 50 × 40 × 22.5 = 45,000 cm³ Water in liters = 45,000 mL = 45 L c) Water available = 45,000 mL Number of 500 mL bottles = 45,000 ÷ 500 = 90 bottles Example 2: Capacity with Thickness A cylindrical water tank has external diameter 1.4 m and height 2 m. The walls are 5 cm thick. What is the internal capacity in liters? (Use π = $frac{22}{7}$) Solution: External radius = 1.4 m ÷ 2 = 0.7 m = 70 cm Wall thickness = 5 cm Internal radius = 70 cm - 5 cm = 65 cm = 0.65 m Height is same: 2 m = 200 cm Internal volume = πr²h = $frac{22}{7} × (65)^2 × 200$ = $frac{22}{7} × 4,225 × 200$ = $frac{22}{7} × 845,000$ = $frac{18,590,000}{7}$ = 2,655,714.29 cm³ (approximately) Capacity = 2,655,714.29 mL ≈ 2,655.71 L Capacity Problems A tank 60×40×50 cm is 60% full. How many liters of water does it contain? Cylindrical can: diameter=14 cm, height=20 cm. What is capacity in liters? (π = $frac{22}{7}$) How many 250 mL cups can be filled from a 5-liter container? A swimming pool 8 m × 4 m × 1.5 m is filled to 80% capacity. How many liters of water? Container holds 15 L when full. When filled to $frac{2}{3}$ capacity, how many 500 mL bottles can be filled? Comparing Volumes and Capacities Comparing volumes and capacities requires converting all measurements to the same units. Once converted, we can determine which is larger, by how much, or arrange them in order. Example 1: Complex Comparison Arrange these volumes from smallest to largest: A: 2.5 m³ B: 3,200 L C: 2,800,000 cm³ D: 0.0035 m³ Solution: Convert all to liters for comparison: A: 2.5 m³ = 2.5 × 1000 L = 2,500 L B: 3,200 L (already in liters) C: 2,800,000 cm³ = 2,800,000 mL = 2,800 L D: 0.0035 m³ = 0.0035 × 1000 L = 3.5 L In order: D (3.5 L) < A (2,500 L) < C (2,800 L) < B (3,200 L) Example 2: Ratio Comparison Container A has volume 1,440 cm³. Container B has dimensions 12 cm × 10 cm × 15 cm. a) Which container has greater volume? b) What is the ratio of their volumes? c) If both are filled with water, how much more water does the larger hold in milliliters? Solution: Volume of A = 1,440 cm³ Volume of B = 12 × 10 × 15 = 1,800 cm³ a) B has greater volume (1,800 cm³ > 1,440 cm³) b) Ratio A:B = 1,440:1,800 = 4:5 (dividing by 360) c) Difference = 1,800 - 1,440 = 360 cm³ = 360 mL Comparison Problems Which is larger: 3.5 m³ or 3,800 L? Arrange: 2,500 cm³, 3.2 L, 0.0028 m³, 2,800 mL from smallest to largest. Cylinder A: r=7 cm, h=10 cm; Cylinder B: r=10 cm, h=5 cm. Which has greater volume? By how much? Find ratio of volumes: Cube edge=12 cm vs Sphere diameter=12 cm (Volume sphere = $frac{4}{3}πr^3$) Container X holds 15 L, Y holds 12,500 cm³, Z holds 0.015 m³. Arrange by capacity. Real-World Applications Volume and capacity calculations are essential in cooking, construction, manufacturing, and environmental science. These applications demonstrate the practical importance of understanding three-dimensional measurement. Example 1: Swimming Pool Application A rectangular swimming pool measures 8 m long, 4 m wide, and has a constant depth of 1.5 m. a) What is the pool's volume in cubic meters? b) What is its capacity in liters? c) If water costs ₦150 per 1000 liters, how much does it cost to fill the pool? d) The pool is filled to 90% capacity. How much water is this in liters? Solution: a) Volume = 8 × 4 × 1.5 = 48 m³ b) Capacity = 48 × 1000 L = 48,000 L c) Cost = (48,000 ÷ 1000) × ₦150 = 48 × ₦150 = ₦7,200 d) 90% of capacity = 0.9 × 48,000 L = 43,200 L Example 2: Manufacturing Application A company produces cylindrical cans with radius 3.5 cm and height 12 cm. a) What is the volume of one can in cm³? (Use π = $frac{22}{7}$) b) What is its capacity in mL? c) How many cans can be filled from a 500 L tank? d) If the cans are packed in boxes measuring 42 cm × 28 cm × 24 cm, how many cans fit in one box? Solution: a) Volume = πr²h = $frac{22}{7} × (3.5)^2 × 12$ = $frac{22}{7} × 12.25 × 12$ = $frac{22}{7} × 147 = 22 × 21 = 462$ cm³ b) Capacity = 462 mL c) 500 L = 500,000 mL Number of cans = 500,000 ÷ 462 ≈ 1,082 cans d) Box volume = 42 × 28 × 24 = 28,224 cm³ Number of cans = 28,224 ÷ 462 = 61.09 ≈ 61 cans (can't have partial can) Real-World Application Problems A fish tank 60×30×40 cm is filled to 80% capacity. How many liters of water? Water flows into a cylindrical tank (r=70 cm, h=2 m) at 10 L/min. How long to fill? (π = $frac{22}{7}$) A recipe needs 2.5 L of broth. How many 350 mL cans are needed? A swimming pool 10×5×1.8 m loses 2% of water daily. How many liters lost per day? Box dimensions 50×40×30 cm. How many 1L bottles can it hold if packed efficiently? Cumulative Exercises Convert 3.75 cubic meters to liters Calculate volume of rectangular prism: 25 cm × 18 cm × 12 cm. Express in liters. Find volume of cylinder: radius=14 cm, height=25 cm (π = $frac{22}{7}$). Express in liters. A tank 80×50×40 cm is 60% full. How many liters of water? Convert 15,000 cm³ to cubic meters and liters Which is larger: 4.2 m³ or 4,500 L? How many 500 mL bottles can be filled from a 25 L container? A composite solid: Rectangular base 30×20×10 cm + Cylinder r=7 cm, h=15 cm on top. Find total volume in cm³. (π = $frac{22}{7}$) Water tank: cylindrical, r=1.4 m, h=2 m. What is capacity in liters? (π = $frac{22}{7}$) Arrange: 3,500 cm³, 4.2 L, 0.0038 m³, 3,800 mL from smallest to largest. Show/Hide Answers Problem: Convert 3.75 cubic meters to liters Answer: 3.75 m³ = 3.75 × 1000 L = 3,750 L Problem: Calculate volume of rectangular prism: 25 cm × 18 cm × 12 cm. Express in liters. Answer: Volume = 25 × 18 × 12 = 5,400 cm³ Capacity = 5,400 mL = 5.4 L Problem: Find volume of cylinder: radius=14 cm, height=25 cm (π = $frac{22}{7}$). Express in liters. Answer: Volume = πr²h = $frac{22}{7} × 14^2 × 25$ = $frac{22}{7} × 196 × 25 = 22 × 28 × 25 = 15,400$ cm³ Capacity = 15,400 mL = 15.4 L Problem: A tank 80×50×40 cm is 60% full. How many liters of water? Answer: Full volume = 80 × 50 × 40 = 160,000 cm³ Full capacity = 160,000 mL = 160 L 60% full = 0.6 × 160 L = 96 L Problem: Convert 15,000 cm³ to cubic meters and liters Answer: 15,000 cm³ = 15,000 ÷ 1,000,000 = 0.015 m³ 15,000 cm³ = 15,000 mL = 15 L Problem: Which is larger: 4.2 m³ or 4,500 L? Answer: 4.2 m³ = 4,200 L 4,500 L > 4,200 L, so 4,500 L is larger Problem: How many 500 mL bottles can be filled from a 25 L container? Answer: 25 L = 25,000 mL Number of bottles = 25,000 ÷ 500 = 50 bottles Problem: A composite solid: Rectangular base 30×20×10 cm + Cylinder r=7 cm, h=15 cm on top. Find total volume in cm³. (π = $frac{22}{7}$) Answer: Rectangular volume = 30 × 20 × 10 = 6,000 cm³ Cylinder volume = πr²h = $frac{22}{7} × 7^2 × 15 = frac{22}{7} × 49 × 15 = 22 × 7 × 15 = 2,310$ cm³ Total volume = 6,000 + 2,310 = 8,310 cm³ Problem: Water tank: cylindrical, r=1.4 m, h=2 m. What is capacity in liters? (π = $frac{22}{7}$) Answer: Volume = πr²h = $frac{22}{7} × (1.4)^2 × 2$ = $frac{22}{7} × 1.96 × 2 = frac{22}{7} × 3.92 = frac{86.24}{7} = 12.32$ m³ Capacity = 12.32 × 1000 L = 12,320 L Problem: Arrange: 3,500 cm³, 4.2 L, 0.0038 m³, 3,800 mL from smallest to largest. Answer: Convert all to liters: 3,500 cm³ = 3,500 mL = 3.5 L 4.2 L = 4.2 L 0.0038 m³ = 0.0038 × 1000 L = 3.8 L 3,800 mL = 3.8 L Order: 3,500 cm³ (3.5 L) < 0.0038 m³ (3.8 L) = 3,800 mL (3.8 L) < 4.2 L Conclusion/Recap Volume and capacity are fundamental concepts in mathematics with wide-ranging applications in everyday life, science, and industry. Volume measures the three-dimensional space occupied by an object, while capacity refers to the amount of fluid a container can hold. The key relationship between these concepts is 1 cm³ = 1 mL, establishing a direct connection between cubic units and liquid measures. Mastery of volume calculations for various shapes (rectangular prisms, cylinders, and composite solids), along with the ability to convert between different units and solve practical problems, provides essential skills for fields ranging from architecture and engineering to cooking and environmental science. These skills enable us to design containers, estimate material quantities, manage resources efficiently, and understand spatial relationships in our three-dimensional world. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c
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