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Understanding Surface Area and Volume Calculations for Common Geometric Shapes

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This guide provides clear methods for calculating the surface area and volume of various geometric shapes, including rectangular prisms, cubes, triangular prisms, cylinders, cones, spheres, and pyramids. Detailed examples illustrate each calculation, demonstrating how to apply formulas effectively. Learn about key measurements such as length, width, height, and radius to compute these values accurately. Essential for students and anyone looking to grasp geometry fundamentals, this resource aids in mastering mathematical concepts related to surface area and volume.

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Understanding Surface Area and Volume Calculations for Common Geometric Shapes

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  1. MATH 3190 Surface Area and Volume

  2. 7 cm 5 cm 6 cm Measurement • Rectangular Prism • Surface Area: sum of the areas of all of the faces • Example: There are 4 lateral faces: 2 lateral faces are 6 cm by 7 cm (A1= wh) and 2 lateral faces are 5 cm by 7 cm (A2 = lh). There are 2 bases 6 cm by 5 cm (A3 = lw) • A1 = (6 cm)(7 cm) = 42 cm2 • A2 = (5 cm)(7 cm) = 35 cm2 • A3 = (6 cm)(5 cm) = 30 cm2 • SA rectangular prism = 2wh + 2lh + 2lw • SA = 2(42 cm2) + 2(35 cm2) + 2(30 cm2) • SA = 84 cm2 + 70 cm2 + 60 cm2 • SA = 214 cm2

  3. 5 cm Measurement • Cube • Surface Area: sum of the areas of all 6 congruent faces • Example: There are 6 faces: 5 cm by 5 cm (A = s2) • SA cube = 6A = 6s2 • SA = 6(5 cm)2 • SA = 6(25 cm2) • SA = 150 cm2

  4. 7 m 5 m 6 m Measurement • Triangular Prism • Surface Area: sum of the areas of all of the faces • Example: There are 3 lateral faces: 6 m by 7 m (A1= bl). There are 2 bases: 6 m for the base and 5 m for the height (2A2 = bh). • A1 = (6 m)(7 m) = 42 m2 • 2A2 = (6 m)(5 m) = 30 m2 • SA triangular prism = bh + 3bl • SA = 30 m2 + 3(42 m2) • SA = 30 m2 + 126 m2 • SA = 156 m2

  5. 3 ft 12 ft Measurement • Cylinder • Surface Area: area of the circles plus the area of the lateral face • Example: r = 3 ft; h = 12 ft • SA cylinder= 2rh +2r2 • SA = 2 (3 ft)(12 ft) + 2 (3 ft)2 • SA = 72 ft2 + 2 (9 ft2) • SA = 72 ft2 + 18 ft2 • SA = 90 ft2

  6. 13 ft 12 ft 5 ft Measurement • Cone • Surface Area: area of the circle plus the area of the lateral face • Example: r = 5 ft; t = 13 ft • SA cone= rt +r2 • SA = (5 ft)(13 ft) +  (5 ft)2 • SA = 65 ft2 +  (25 ft2) • SA = 65 ft2 + 25 ft2 • SA = 90 ft2

  7. 8 mm Measurement • Sphere • Surface Area: 4r2 where r is the radius • Example: r = 8 mm • SA sphere = 4r2 • SA = 4(8 mm)2 • SA = 4(64 mm2) • SA = 256 mm2

  8. 7 cm 5 cm 6 cm Measurement • Rectangular Prism • Volume: • V = lwh where l is length; w is width; and h is height • Example: l = 6 cm; w = 5 cm; h = 7 cm • V rectangular prism = Bh = lwh • V = (6 cm)(5 cm)(7 cm) • V = 210 cm3

  9. 5 cm Measurement • Cube • Volume: • V = s3 where s is the length of a side • Example: s = 5 cm • V cube = Bh = s3 • V = (5 cm)3 • V = 125 cm3

  10. 7 m 5 m 6 m Measurement • Triangular Prism • Volume: • V = ½ bhl where b is the base; h is height of the triangle; and l is length of the prism • Example: b = 6 m; h = 5 m; l = 7 m • V triangular prism = Bh = ½ bhl • V = ½ (6 m)(5 m)(7 m) • V = 105 m3

  11. 3 ft 12 ft Measurement • Cylinder • Volume of a Cylinder: V = r2h where r is the radius of the base (circle) and h is the height. • Example: r = 3 ft and h = 12 ft. • V cylinder = Bh = r2h • V = (3 ft)2 (12 ft) • V = (9 ft2)(12 ft) • V = 108 ft3

  12. 13 ft 12 ft 5 ft Measurement • Cone • Volume: V = r2h/3 where r is the radius of the base (circle) and h is the height. • Example: r = 5 ft; h = 12 ft • V cone= r2h/3 • V = [(5 ft)2 12 ft ]/ 3 • V = [(25 ft2)(12 ft)]/3 • V = (25 ft2)(4 ft) • V = 100 ft3

  13. 6 mm Measurement • Sphere • Volume of a Sphere: V = (4/3) r3 where r is the radius • Example: r = 6 mm • V sphere = 4r3/3 • V = [4 x (6 mm)3]/3 • V = [4 x 216 mm3]/3 • V = [864 mm3]/3 • V = 288 mm3

  14. Measurement • Triangular Pyramid • Square Pyramid

  15. Test Taking Tips • Get a good nights rest before the exam • Prepare materials for exam in advance (scratch paper, pencil, and calculator) • Read questions carefully and ask if you have a question DURING the exam • Remember: If you are prepared, you need not fear

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