Post on 28-Dec-2015
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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
MeasurementMeasurement
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