Perimeter, Area, Surface Area, and Volume Examples Geometry and andMeasurement.
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Transcript of 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
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