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**Perimeter, Area, Surface Area, and Volume Examples**

Geometry and Measurement

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**Geometry Polyhedron: V + F – E = 2**

Vertices Edges Faces Should be able to draw ALL of the following: Sphere Prisms – Cube, Rectangular, Triangular Cylinder Cone Pyramids – Triangular, Square

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**Measurement Rectangle Perimeter**

3 ft 5 ft Rectangle Perimeter P = 2l + 2w, where l = length and w = width Example: l = 5 ft and w = 3 ft P rectangle = 2l + 2w P = 2(5 ft) + 2(3 ft) P = 10 ft + 6 ft P = 16 ft

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**Measurement Rectangle Area A = lw where l = length and w = width**

3 ft 5 ft Rectangle Area A = lw where l = length and w = width Example: l = 5 ft and w = 3 ft A rectangle = lw A = (5 ft)(3 ft) A = 15 ft2

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**Measurement Square Perimeter P = 4s, where s = length of a side**

3 ft Square Perimeter P = 4s, where s = length of a side Example: s = 3 ft P square = 4s P = 4(3 ft) P = 12 ft

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**Measurement Square Area A = s2 where s = length of a side**

3 ft Square Area A = s2 where s = length of a side Example: s = 3 ft A square = s2 A = (3 ft)2 A = 9 ft2

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**Measurement Triangle Perimeter**

P = a + b + c, where a, b, and c are the lengths of the sides of the triangle Example: a = 3 m; b = 4 m; c = 5 m P triangle = a + b + c P = 3 m + 4 m + 5 m P = 12 m

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**Measurement Triangle Area**

A = ½ bh, where b is the base and h is the height of the triangle Example: b = 3 m; h = 4 m A triangle = ½ bh A = ½ (3 m) (4 m) A = 6 m2

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**Measurement Circle Circumference**

3 cm Circle Circumference C circle = d or C = 2r, where d = diameter and r = radius Example: r = 3 cm C circle = 2r C = 2(3 cm) C = 6 cm

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**Measurement Circle Area A = r2, where r = radius Example: r = 3 cm**

A circle = r2 A = (3 cm)2 A = 9 cm2

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**Measurement Rectangular Prism**

7 cm 6 cm 5 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 cm cm cm2 SA = 214 cm2

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**Measurement Rectangular Prism Volume:**

7 cm 6 cm 5 cm 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

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Measurement 5 cm 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

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**Measurement Cube Volume: V = s3 where s is the length of a side**

5 cm 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

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**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 m m2 SA = 156 m2

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**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

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

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Measurement 3 ft 12 ft 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

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5 ft 13 ft 12 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

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5 ft 13 ft 12 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

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**Measurement Sphere Surface Area: 4r2 where r is the radius**

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

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

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Measurement Triangular Pyramid Square Pyramid

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