1. MATH 3190 Surface Area and andVolume

2. Measurement Rectangular Prism Rectangular Prism Surface Area: sum of the areas of all of the faces 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 (A 1 = wh) and 2 lateral faces are 5 cm by 7 cm (A 2 = lh). There are 2 bases 6 cm by 5 cm (A 3 = lw) Example: There are 4 lateral faces: 2 lateral faces are 6 cm by 7 cm (A 1 = wh) and 2 lateral faces are 5 cm by 7 cm (A 2 = lh). There are 2 bases 6 cm by 5 cm (A 3 = lw) A 1 = (6 cm)(7 cm) = 42 cm 2 A 1 = (6 cm)(7 cm) = 42 cm 2 A 2 = (5 cm)(7 cm) = 35 cm 2 A 2 = (5 cm)(7 cm) = 35 cm 2 A 3 = (6 cm)(5 cm) = 30 cm 2 A 3 = (6 cm)(5 cm) = 30 cm 2 SA rectangular prism = 2wh + 2lh + 2lw SA rectangular prism = 2wh + 2lh + 2lw SA = 2(42 cm 2 ) + 2(35 cm 2 ) + 2(30 cm 2 ) SA = 2(42 cm 2 ) + 2(35 cm 2 ) + 2(30 cm 2 ) SA = 84 cm 2 + 70 cm 2 + 60 cm 2 SA = 84 cm 2 + 70 cm 2 + 60 cm 2 SA = 214 cm 2 SA = 214 cm 2 7 cm 6 cm 5 cm

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

4. Measurement Triangular Prism Triangular Prism Surface Area: sum of the areas of all of the faces Surface Area: sum of the areas of all of the faces Example: There are 3 lateral faces: 6 m by 7 m (A 1 = bl). There are 2 bases: 6 m for the base and 5 m for the height (2A 2 = bh). Example: There are 3 lateral faces: 6 m by 7 m (A 1 = bl). There are 2 bases: 6 m for the base and 5 m for the height (2A 2 = bh). A 1 = (6 m)(7 m) = 42 m 2 A 1 = (6 m)(7 m) = 42 m 2 2A 2 = (6 m)(5 m) = 30 m 2 2A 2 = (6 m)(5 m) = 30 m 2 SA triangular prism = bh + 3bl SA triangular prism = bh + 3bl SA = 30 m 2 + 3(42 m 2 ) SA = 30 m 2 + 3(42 m 2 ) SA = 30 m 2 + 126 m 2 SA = 30 m 2 + 126 m 2 SA = 156 m 2 SA = 156 m 2 7 m 6 m 5 m

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

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

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

8. Measurement Rectangular Prism Rectangular Prism Volume: Volume: V = lwh where l is length; w is width; and h is height V = lwh where l is length; w is width; and h is height Example: l = 6 cm; w = 5 cm; h = 7 cm Example: l = 6 cm; w = 5 cm; h = 7 cm V rectangular prism = Bh = lwh V rectangular prism = Bh = lwh V=(6 cm)(5 cm)(7 cm) V=(6 cm)(5 cm)(7 cm) V=210 cm 3 V=210 cm 3 7 cm 6 cm 5 cm

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

10. Measurement Triangular Prism Triangular Prism Volume: Volume: V = ½ bhl where b is the base; h is height of the triangle; and l is length of the prism 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 Example: b = 6 m; h = 5 m; l = 7 m V triangular prism = Bh = ½ bhl V triangular prism = Bh = ½ bhl V=½ (6 m)(5 m)(7 m) V=½ (6 m)(5 m)(7 m) V=105 m 3 V=105 m 3 7 m 6 m 5 m

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

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

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

14. Measurement Triangular Pyramid Triangular Pyramid Square Pyramid Square Pyramid

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