3D Measures: – Finding surface area and volume of cubes and cuboids.

3D - Measures. Grade 8 Mathematics: 3D Measures - Surface Area and Volume Subtopics Navigator Introduction to 3D Shapes Cube Properties Cuboid Properties Surface Area Volume Real-World Applications Cumulative Exercises Conclusion Lesson Objectives Understand the properties of cubes and cuboids Calculate surface area of cubes and cuboids Calculate volume of cubes and cuboids Apply formulas to solve real-world problems Distinguish between surface area and volume Introduction to 3D Shapes Three-dimensional (3D) shapes are solid objects that have length, width, and height. Unlike 2D shapes that are flat, 3D shapes occupy space and have volume. In this lesson, we'll focus on two important 3D shapes: cubes and cuboids. CUBE Cube A cube has: 6 equal square faces 12 equal edges 8 vertices (corners) All angles are 90° CUBOID Cuboid A cuboid has: 6 rectangular faces 12 edges (in 3 groups of 4) 8 vertices (corners) All angles are 90° Real-World Examples: Cubes: Dice, sugar cubes, Rubik's cube Cuboids: Books, bricks, cereal boxes, rooms Understanding 3D Shapes How many faces does a cube have? What is the difference between a cube and a cuboid? Name three real-world objects that are cuboids. Cube Properties and Formulas A cube is a special type of cuboid where all edges are equal in length. If we denote the length of one edge as 'a', then: Cube Formulas Surface Area = [latex]6a^2[/latex] Volume = [latex]a^3[/latex] Space Diagonal = [latex]asqrt{3}[/latex] Example 1: Finding Cube Surface Area Find the surface area of a cube with edge length 5 cm. Surface Area = [latex]6 times (5)^2 = 6 times 25 = 150[/latex] cm² Answer: 150 cm² Example 2: Finding Cube Volume Find the volume of a cube with edge length 7 m. Volume = [latex]7^3 = 7 times 7 times 7 = 343[/latex] m³ Answer: 343 m³ Exercises (Cubes) Find the surface area of a cube with edge length 4 cm Calculate the volume of a cube with edge length 10 m A cube has surface area 96 cm². What is its edge length? Find the volume of a cube with surface area 150 cm² How many 2 cm cubes can fit inside a 10 cm cube? Cuboid Properties and Formulas A cuboid has three dimensions: length (l), width (w), and height (h). These are also called the cuboid's edges. Cuboid Formulas Surface Area = [latex]2(lw + lh + wh)[/latex] Volume = [latex]l times w times h[/latex] Space Diagonal = [latex]sqrt{l^2 + w^2 + h^2}[/latex] Understanding Surface Area Formula 1 A cuboid has 6 faces: front & back, left & right, top & bottom 2 Area of front and back: [latex]2 times (l times h)[/latex] 3 Area of left and right: [latex]2 times (w times h)[/latex] 4 Area of top and bottom: [latex]2 times (l times w)[/latex] 5 Total Surface Area = [latex]2(lh + wh + lw)[/latex] Example 3: Finding Cuboid Surface Area Find the surface area of a cuboid with length 8 cm, width 5 cm, and height 3 cm. Surface Area = [latex]2 times [(8 times 5) + (8 times 3) + (5 times 3)][/latex] = [latex]2 times [40 + 24 + 15] = 2 times 79 = 158[/latex] cm² Answer: 158 cm² Example 4: Finding Cuboid Volume Find the volume of a cuboid with length 12 m, width 7 m, and height 4 m. Volume = [latex]12 times 7 times 4 = 336[/latex] m³ Answer: 336 m³ Exercises (Cuboids) Find the surface area of a cuboid with dimensions 6 cm × 4 cm × 3 cm Calculate the volume of a cuboid with dimensions 15 m × 8 m × 5 m A cuboid has volume 240 cm³. If its length is 10 cm and width is 6 cm, find its height Find the surface area of a cuboid with volume 180 cm³, length 9 cm, and width 5 cm How many 1 cm cubes can fit in a cuboid measuring 8 cm × 5 cm × 3 cm? Understanding Surface Area Surface area is the total area of all the faces of a 3D shape. It's measured in square units (cm², m², etc.). Think of it as the amount of wrapping paper needed to cover the entire shape. Shape Surface Area Formula Why This Formula Works Cube [latex]6a^2[/latex] 6 faces, each with area a² Cuboid [latex]2(lw + lh + wh)[/latex] Sum of areas of all 6 rectangular faces Example 5: Surface Area Application A gift box measures 20 cm × 15 cm × 10 cm. How much wrapping paper is needed to cover it completely? Surface Area = [latex]2 times [(20 times 15) + (20 times 10) + (15 times 10)][/latex] = [latex]2 times [300 + 200 + 150] = 2 times 650 = 1300[/latex] cm² Answer: 1300 cm² of wrapping paper needed Surface Area Practice What is the surface area of a cube with edge 12 cm? If a cuboid has dimensions 9 m × 7 m × 4 m, what is its surface area? Explain why surface area is measured in square units. Understanding Volume Volume is the amount of space occupied by a 3D shape. It's measured in cubic units (cm³, m³, etc.). Think of it as how much water the shape could hold if it were hollow. Shape Volume Formula Why This Formula Works Cube [latex]a^3[/latex] Length × width × height, all equal to a Cuboid [latex]l times w times h[/latex] Area of base × height Example 6: Volume Application A rectangular water tank measures 2 m × 1.5 m × 1 m. What is its capacity in liters? (Remember: 1 m³ = 1000 liters) Volume = [latex]2 times 1.5 times 1 = 3[/latex] m³ Capacity = [latex]3 times 1000 = 3000[/latex] liters Answer: 3000 liters Example 7: Finding Dimensions from Volume The volume of a cuboid is 360 cm³. If its length is 12 cm and width is 6 cm, find its height. Volume = length × width × height [latex]360 = 12 times 6 times h[/latex] [latex]360 = 72 times h[/latex] [latex]h = 360 div 72 = 5[/latex] cm Answer: Height = 5 cm Volume Practice What is the volume of a cube with edge 8 cm? A cuboid has volume 450 cm³, length 15 cm, and width 6 cm. Find its height. Explain why volume is measured in cubic units. Real-World Applications Understanding surface area and volume is essential in many real-world situations: Packaging - Calculating material needed for boxes Construction - Determining concrete needed for foundations Shipping - Calculating cargo space in containers Heating/Cooling - Determining room volumes for HVAC systems Painting - Calculating paint needed for walls Example 8: Painting Application A room measures 5 m × 4 m × 3 m. If one liter of paint covers 10 m², how much paint is needed for the walls? (Ignore doors and windows for this example) Wall area = [latex]2 times [(5 times 3) + (4 times 3)] = 2 times [15 + 12] = 54[/latex] m² Paint needed = [latex]54 div 10 = 5.4[/latex] liters Answer: 5.4 liters of paint Example 9: Storage Application A storage container measures 2.5 m × 2 m × 2 m. How many boxes measuring 50 cm × 40 cm × 30 cm can fit inside? First, convert all measurements to meters: 0.5 m × 0.4 m × 0.3 m Container volume = [latex]2.5 times 2 times 2 = 10[/latex] m³ Box volume = [latex]0.5 times 0.4 times 0.3 = 0.06[/latex] m³ Number of boxes = [latex]10 div 0.06 approx 166[/latex] boxes Answer: 166 boxes Real-World Problems A cube-shaped box has edge length 30 cm. What is its volume in cubic centimeters? A bookshelf measures 120 cm × 40 cm × 180 cm. What is its surface area? A swimming pool is 25 m long, 10 m wide, and 2 m deep. How much water does it hold in liters? How many 5 cm cubes can fit in a cuboid measuring 20 cm × 15 cm × 10 cm? A gift box needs to be wrapped. It measures 18 cm × 12 cm × 8 cm. Calculate the wrapping paper needed. Cumulative Exercises Find the surface area of a cube with edge length 9 cm Calculate the volume of a cuboid with dimensions 14 m × 8 m × 5 m A cube has volume 512 cm³. Find its surface area A cuboid has surface area 376 cm². If its length is 10 cm and width is 8 cm, find its height How many 3 cm cubes can fit in a cuboid measuring 15 cm × 12 cm × 9 cm? A room is 6 m long, 5 m wide, and 3 m high. Find the area of its four walls A water tank measures 2.4 m × 1.8 m × 1.5 m. What is its capacity in liters? A cube and a cuboid have the same volume. The cube has edge length 8 cm. The cuboid has length 16 cm and width 4 cm. Find the height of the cuboid Find the surface area of a cuboid with volume 480 cm³, length 12 cm, and height 5 cm Which has greater surface area: a cube of edge 10 cm or a cuboid measuring 12 cm × 10 cm × 8 cm? A box measures 60 cm × 40 cm × 30 cm. How many books measuring 20 cm × 15 cm × 5 cm can it hold? If the edge of a cube is doubled, what happens to its surface area and volume? A swimming pool is 20 m long, 15 m wide, and 2 m deep. Calculate the cost of tiling its floor and four walls at $15 per m² Find the volume of a cube whose surface area is 294 cm² A cuboid has dimensions in ratio 2:3:4. If its volume is 192 cm³, find its dimensions Show/Hide Answers Problem: Find the surface area of a cube with edge length 9 cm Step 1: Surface Area = [latex]6a^2[/latex] Step 2: = [latex]6 times 9^2 = 6 times 81 = 486[/latex] cm² Answer: 486 cm² Problem: Calculate the volume of a cuboid with dimensions 14 m × 8 m × 5 m Step 1: Volume = length × width × height Step 2: = [latex]14 times 8 times 5 = 560[/latex] m³ Answer: 560 m³ Problem: A cube has volume 512 cm³. Find its surface area Step 1: Volume = [latex]a^3 = 512[/latex], so [latex]a = sqrt[3]{512} = 8[/latex] cm Step 2: Surface Area = [latex]6 times 8^2 = 6 times 64 = 384[/latex] cm² Answer: 384 cm² Problem: A cuboid has surface area 376 cm². If its length is 10 cm and width is 8 cm, find its height Step 1: Surface Area = [latex]2(lw + lh + wh) = 376[/latex] Step 2: [latex]2[(10 times 8) + (10 times h) + (8 times h)] = 376[/latex] Step 3: [latex]2[80 + 10h + 8h] = 376[/latex] Step 4: [latex]2[80 + 18h] = 376[/latex] Step 5: [latex]160 + 36h = 376[/latex] Step 6: [latex]36h = 216[/latex], so [latex]h = 6[/latex] cm Answer: Height = 6 cm Problem: How many 3 cm cubes can fit in a cuboid measuring 15 cm × 12 cm × 9 cm? Step 1: Volume of cuboid = [latex]15 times 12 times 9 = 1620[/latex] cm³ Step 2: Volume of small cube = [latex]3^3 = 27[/latex] cm³ Step 3: Number of cubes = [latex]1620 div 27 = 60[/latex] Answer: 60 cubes Problem: A room is 6 m long, 5 m wide, and 3 m high. Find the area of its four walls Step 1: Area of four walls = [latex]2 times [(6 times 3) + (5 times 3)][/latex] Step 2: = [latex]2 times [18 + 15] = 2 times 33 = 66[/latex] m² Answer: 66 m² Problem: A water tank measures 2.4 m × 1.8 m × 1.5 m. What is its capacity in liters? Step 1: Volume = [latex]2.4 times 1.8 times 1.5 = 6.48[/latex] m³ Step 2: Capacity = [latex]6.48 times 1000 = 6480[/latex] liters Answer: 6480 liters Problem: A cube and a cuboid have the same volume. The cube has edge length 8 cm. The cuboid has length 16 cm and width 4 cm. Find the height of the cuboid Step 1: Volume of cube = [latex]8^3 = 512[/latex] cm³ Step 2: Volume of cuboid = [latex]16 times 4 times h = 512[/latex] Step 3: [latex]64h = 512[/latex], so [latex]h = 8[/latex] cm Answer: Height = 8 cm Problem: Find the surface area of a cuboid with volume 480 cm³, length 12 cm, and height 5 cm Step 1: Volume = length × width × height Step 2: [latex]480 = 12 times w times 5[/latex], so [latex]60w = 480[/latex], [latex]w = 8[/latex] cm Step 3: Surface Area = [latex]2[(12 times 8) + (12 times 5) + (8 times 5)][/latex] Step 4: = [latex]2[96 + 60 + 40] = 2 times 196 = 392[/latex] cm² Answer: 392 cm² Problem: Which has greater surface area: a cube of edge 10 cm or a cuboid measuring 12 cm × 10 cm × 8 cm? Step 1: Cube surface area = [latex]6 times 10^2 = 600[/latex] cm² Step 2: Cuboid surface area = [latex]2[(12 times 10) + (12 times 8) + (10 times 8)][/latex] Step 3: = [latex]2[120 + 96 + 80] = 2 times 296 = 592[/latex] cm² Step 4: 600 > 592, so the cube has greater surface area Answer: The cube has greater surface area Problem: A box measures 60 cm × 40 cm × 30 cm. How many books measuring 20 cm × 15 cm × 5 cm can it hold? Step 1: Volume of box = [latex]60 times 40 times 30 = 72000[/latex] cm³ Step 2: Volume of one book = [latex]20 times 15 times 5 = 1500[/latex] cm³ Step 3: Number of books = [latex]72000 div 1500 = 48[/latex] Answer: 48 books Problem: If the edge of a cube is doubled, what happens to its surface area and volume? Step 1: Original surface area = [latex]6a^2[/latex], new surface area = [latex]6(2a)^2 = 24a^2[/latex] (4 times) Step 2: Original volume = [latex]a^3[/latex], new volume = [latex](2a)^3 = 8a^3[/latex] (8 times) Answer: Surface area becomes 4 times, volume becomes 8 times Problem: A swimming pool is 20 m long, 15 m wide, and 2 m deep. Calculate the cost of tiling its floor and four walls at $15 per m² Step 1: Area of floor = [latex]20 times 15 = 300[/latex] m² Step 2: Area of four walls = [latex]2 times [(20 times 2) + (15 times 2)] = 2 times [40 + 30] = 140[/latex] m² Step 3: Total area = [latex]300 + 140 = 440[/latex] m² Step 4: Cost = [latex]440 times 15 = 6600[/latex] dollars Answer: $6600 Problem: Find the volume of a cube whose surface area is 294 cm² Step 1: Surface Area = [latex]6a^2 = 294[/latex] Step 2: [latex]a^2 = 294 div 6 = 49[/latex], so [latex]a = 7[/latex] cm Step 3: Volume = [latex]7^3 = 343[/latex] cm³ Answer: 343 cm³ Problem: A cuboid has dimensions in ratio 2:3:4. If its volume is 192 cm³, find its dimensions Step 1: Let dimensions be 2x, 3x, and 4x Step 2: Volume = [latex]2x times 3x times 4x = 24x^3 = 192[/latex] Step 3: [latex]x^3 = 192 div 24 = 8[/latex], so [latex]x = 2[/latex] Step 4: Dimensions = 4 cm, 6 cm, and 8 cm Answer: 4 cm × 6 cm × 8 cm Conclusion/Recap In this lesson, we've explored surface area and volume of cubes and cuboids. Remember these key points: Surface Area is the total area of all faces, measured in square units Volume is the space occupied, measured in cubic units Cube Formulas: Surface Area = [latex]6a^2[/latex], Volume = [latex]a^3[/latex] Cuboid Formulas: Surface Area = [latex]2(lw + lh + wh)[/latex], Volume = [latex]l times w times h[/latex] These concepts have practical applications in packaging, construction, and many other fields With practice, calculating surface area and volume will become second nature. Always remember to include the correct units in your answers! Clip It! Share your ANSWER in the Chat. 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