perimeter, area, surface area, and volume examples geometry and andmeasurement

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

GeometryGeometry

and and

MeasurementMeasurement

GeometryGeometry Polyhedron: V + F – E = 2Polyhedron: V + F – E = 2

VerticesVertices EdgesEdges FacesFaces

Should be able to draw Should be able to draw ALLALL of the of the following:following: SphereSphere Prisms – Cube, Rectangular, Prisms – Cube, Rectangular,

TriangularTriangular CylinderCylinder ConeCone Pyramids – Triangular, SquarePyramids – Triangular, Square


RectangleRectangle Perimeter Perimeter P = 2l + 2w, where l = length and P = 2l + 2w, where l = length and

w = widthw = width Example: l = 5 ft and w = 3 ftExample: l = 5 ft and w = 3 ft

PP rectangle rectangle == 2l + 2w2l + 2w

PP == 2(5 ft) + 2(3 ft)2(5 ft) + 2(3 ft) PP == 10 ft + 6 ft10 ft + 6 ft PP == 16 ft16 ft

3 ft

5 ft


RectangleRectangle Area Area A = lw where l = length and w = A = lw where l = length and w =

widthwidth Example: l = 5 ft and w = 3 ftExample: l = 5 ft and w = 3 ft

AA rectangle rectangle = = lwlw

AA == (5 ft)(3 ft)(5 ft)(3 ft) A A == 15 ft15 ft22

3 ft

5 ft


SquareSquare Perimeter Perimeter P = 4s, where s = length of a sideP = 4s, where s = length of a side Example: s = 3 ftExample: s = 3 ft

PP square square == 4s4s

PP == 4(3 ft)4(3 ft) PP == 12 ft12 ft

3 ft


SquareSquare Area Area A = sA = s22 where s = length of a side where s = length of a side Example: s = 3 ftExample: s = 3 ft

AA square square = = ss22

AA == (3 ft)(3 ft)22

A A == 9 ft9 ft22

3 ft


TriangleTriangle PerimeterPerimeter PP = a + b + c, where a, b, and c = a + b + c, where a, b, and c

are the lengths of the sides of the are the lengths of the sides of the triangletriangle

Example: a = 3 m; b = 4 m; c = 5 Example: a = 3 m; b = 4 m; c = 5 mm PP triangle triangle == a + b + ca + b + c PP == 3 m + 4 m + 5 m3 m + 4 m + 5 m PP == 12 m12 m

3 m

5 m4 m


TriangleTriangle AreaArea A = ½ bh, where b is the base and A = ½ bh, where b is the base and

h is the height of the triangleh is the height of the triangle Example: b = 3 m; h = 4 mExample: b = 3 m; h = 4 m

AA triangle triangle == ½ bh½ bh AA == ½ (3 m) (4 m)½ (3 m) (4 m) AA == 6 m6 m22

3 m

5 m

4 m


CircleCircle CircumferenceCircumference

CC circle circle = = d or C = 2d or C = 2r, where d = r, where d =

diameter and r = radiusdiameter and r = radius Example: r = 3 cmExample: r = 3 cm

CC circle circle == 2 2rr

CC == 2 2(3 cm)(3 cm) CC == 6 6 cm cm

3 cm


CircleCircle AreaArea A = A = rr22, where r = radius, where r = radius Example: r = 3 cmExample: r = 3 cm

A A circlecircle == rr22

AA == (3 cm)(3 cm)22

AA == 9 9 cm cm22

3 cm


Rectangular PrismRectangular Prism Surface AreaSurface Area: sum of the areas of all of the : sum of the areas of all of the

facesfaces ExampleExample: There are 4 lateral faces: 2 lateral : There are 4 lateral faces: 2 lateral

faces are 6 cm by 7 cm (Afaces are 6 cm by 7 cm (A11= wh) and 2 = wh) and 2 lateral faces are 5 cm by 7 cm (Alateral faces are 5 cm by 7 cm (A22 = lh). = lh). There are 2 bases 6 cm by 5 cm (AThere are 2 bases 6 cm by 5 cm (A33 = lw) = lw)

AA11 = (6 cm)(7 cm) = 42 cm = (6 cm)(7 cm) = 42 cm22

AA22 = (5 cm)(7 cm) = 35 cm = (5 cm)(7 cm) = 35 cm22

AA33 = (6 cm)(5 cm) = 30 cm = (6 cm)(5 cm) = 30 cm22

SA SA rectangular prismrectangular prism = 2wh + 2lh + 2lw = 2wh + 2lh + 2lw

SA = 2(42 cmSA = 2(42 cm22) + 2(35 cm) + 2(35 cm22) + 2(30 cm) + 2(30 cm22)) SA = 84 cmSA = 84 cm22 + 70 cm + 70 cm22 + 60 cm + 60 cm22

SA = 214 cmSA = 214 cm22

7 cm

6 cm

5 cm


Rectangular Prism Rectangular Prism VolumeVolume: : V = lwh where l is length; w is width; V = lwh where l is length; w is width;

and h is heightand h is height ExampleExample: l = 6 cm; w = 5 cm; h = 7 cm: l = 6 cm; w = 5 cm; h = 7 cm

V V rectangular prismrectangular prism = Bh = lwh = Bh = lwh

VV == (6 cm)(5 cm)(7 cm)(6 cm)(5 cm)(7 cm) VV == 210 cm210 cm33

7 cm

6 cm

5 cm


CubeCube Surface AreaSurface Area: sum of the areas of all : sum of the areas of all

6 congruent faces6 congruent faces ExampleExample: There are 6 faces: 5 cm by : There are 6 faces: 5 cm by

5 cm (A = s5 cm (A = s22))

SA SA cubecube = 6A = 6s = 6A = 6s22

SA = 6(5 cm)SA = 6(5 cm)22

SA = 6(25 cmSA = 6(25 cm22)) SA = 150 cmSA = 150 cm22

5 cm


Cube Cube VolumeVolume: : V = sV = s33 where s is the length of a side where s is the length of a side ExampleExample: s = 5 cm: s = 5 cm

V V cubecube = Bh = s= Bh = s33

VV == (5 cm)(5 cm)33

VV == 125 cm125 cm33

5 cm


Triangular PrismTriangular Prism Surface AreaSurface Area: sum of the areas of all of the : sum of the areas of all of the

facesfaces ExampleExample: There are 3 lateral faces: 6 m by : There are 3 lateral faces: 6 m by

7 m (A7 m (A11= bl). There are 2 bases: 6 m for the = bl). There are 2 bases: 6 m for the

base and 5 m for the height (2Abase and 5 m for the height (2A22 = bh). = bh). AA11 = (6 m)(7 m) = 42 m = (6 m)(7 m) = 42 m22

2A2A22 = (6 m)(5 m) = 30 m = (6 m)(5 m) = 30 m22

SA SA triangular prismtriangular prism = bh + 3bl = bh + 3bl

SA = 30 mSA = 30 m22 + 3(42 m + 3(42 m22)) SA = 30 mSA = 30 m22 + 126 m + 126 m22

SA = 156 mSA = 156 m22

7 m

6 m

5 m


Triangular Prism Triangular Prism VolumeVolume: : V = ½ bhl where b is the base; h is V = ½ bhl where b is the base; h is

height of the triangle; and l is length of height of the triangle; and l is length of the prismthe prism

ExampleExample: b = 6 m; h = 5 m; l = 7 m: b = 6 m; h = 5 m; l = 7 m

V V triangular prismtriangular prism = Bh = ½ bhl = Bh = ½ bhl

VV == ½ (6 m)(5 m)(7 m)½ (6 m)(5 m)(7 m) VV == 105 m105 m33

7 m

6 m

5 m


CylinderCylinder Surface AreaSurface Area: area of the circles plus : area of the circles plus

the area of the lateral facethe area of the lateral face ExampleExample: r = 3 ft; h = 12 ft: r = 3 ft; h = 12 ft

SA SA cylindercylinder= = 22rh +2rh +2rr22

SA = 2SA = 2 (3 ft)(12 ft) + 2 (3 ft)(12 ft) + 2 (3 ft) (3 ft)22 SA SA == 7272 ft ft22 + 2 + 2 (9 ft (9 ft22)) SASA == 7272 ft ft22 + 18 + 18 ft ft22

SASA = = 9090 ft ft22

3 ft

12 ft


CylinderCylinder Volume of a CylinderVolume of a Cylinder: V = : V = rr22h h

where r is the radius of the base where r is the radius of the base (circle) and h is the height.(circle) and h is the height.

ExampleExample: r = 3 ft and h = 12 ft.: r = 3 ft and h = 12 ft. VV cylinder cylinder == Bh = Bh = rr22hh VV == (3 ft)(3 ft)22 (12 ft) (12 ft) VV == (9 ft(9 ft22)(12 ft))(12 ft) VV == 108108 ft ft33

3 ft

12 ft


ConeCone Surface AreaSurface Area: area of the circle plus : area of the circle plus

the area of the lateral facethe area of the lateral face ExampleExample: r = 5 ft; t = 13 ft: r = 5 ft; t = 13 ft

SA SA conecone= = rt +rt +rr22

SA = SA = (5 ft)(13 ft) + (5 ft)(13 ft) + (5 ft) (5 ft)22 SA SA == 6565 ft ft22 + + (25 ft (25 ft22)) SASA == 6565 ft ft22 + 25 + 25 ft ft22

SASA = = 9090 ft ft22

5 ft

13 ft

12 ft


ConeCone VolumeVolume: V = : V = rr22h/3 where r is the h/3 where r is the

radius of the base (circle) and h is the radius of the base (circle) and h is the height.height.

ExampleExample: r = 5 ft; h = 12 ft: r = 5 ft; h = 12 ft V V conecone= = rr22h/3h/3 V V = = [[(5 ft)(5 ft)22 12 ft ]/ 3 12 ft ]/ 3 V V == [(25[(25 ft ft22)(12 ft)]/3)(12 ft)]/3 VV == (25(25 ft ft22)(4 ft))(4 ft) VV = = 100100 ft ft33

5 ft

13 ft

12 ft


SphereSphere Surface AreaSurface Area: 4: 4rr22 where r is the where r is the

radiusradius ExampleExample: r = 8 mm: r = 8 mm SA SA sphere sphere = = 44rr22

SASA = = 44(8 mm)(8 mm)22 SASA = = 44(64 mm(64 mm22)) SA SA = = 256256 mm mm22

8 mm


SphereSphere Volume of a SphereVolume of a Sphere: V = (4/3): V = (4/3) r r33

where r is the radiuswhere r is the radius ExampleExample: r = 6 mm: r = 6 mm V V spheresphere == 44rr33/3/3 VV == [4[4 x (6 mm) x (6 mm)33]/3]/3 VV == [4[4 x 216 mm x 216 mm33]/3]/3 VV == [864[864 mm mm33]/3]/3 VV == 288288 mm mm33

6 mm


Triangular PyramidTriangular Pyramid

Square PyramidSquare Pyramid

Bh3V

perimeter, area, surface area, and volume examples geometry and andmeasurement

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