9.4 – perimeter, area, and circumference
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9.4 – Perimeter, Area, and Circumference. Perimeter of a Polygon. The perimeter of any polygon is the sum of the measures of the line segments that form its sides. Perimeter is measured in linear units. Perimeter of a Triangle. a. b. c. - PowerPoint PPT PresentationTRANSCRIPT
9.4 – Perimeter, Area, and Circumference
Perimeter of a PolygonThe perimeter of any polygon is the sum of the measures of the line segments that form its sides. Perimeter is measured in linear units.
Perimeter of a Trianglea b
cThe perimeter P of a triangle with sides of lengths a, b, and c is given by the formula:
P = a + b + c.
9.4 – Perimeter, Area, and Circumference
Perimeter of a Rectangle w
l
The perimeter P of a rectangle with length l and width w is given by the formula:
P = 2l + 2w or P = 2(l + w).
Perimeter of a Square
The perimeter P of a square with all sides of length s is given by the formula:
s
s
P = 4s.
9.4 – Perimeter, Area, and CircumferenceArea of a Polygon
The amount of plane surface covered by a polygon is called its area. Area is measured in square units.
Area of a Rectangle
The area A of a rectangle with length l and width w is given by the formula:
wl
A = l w.
Area of a Square ss
The area A of a square with all sides of length s is given by the formula:
P = s2.
9.4 – Perimeter, Area, and Circumference
Area of a Parallelogram
The area A of a parallelogram with height h and base b is given by the formula:
b
h
A = bh.
Area of a Trapezoid
The area A of a trapezoid with parallel bases b1 and b2 and height h is given by the formula:
b2
h
b1
A = (1/2) h (b1 + b2)
9.4 – Perimeter, Area, and Circumference
Area of a Triangle
The area A of a triangle with base b and height h is given by the formula:
h
b
A = (1/2) b h
9.4 – Perimeter, Area, and CircumferenceFind the perimeter and area of the rectangle.
7 ft15 ft
P = 44
Perimeter
Area A = 105
P = 2l + 2w 2(15) + 2(7)
A = lw
Find the area of the trapezoid.
13 cm.
5 cm.
7 cm.
A = (1/2) h (b1 + b2)
A = (1/2) (5) (7 + 13)
A = (1/2) (5) (20)
A = 50 cm2
6 cm. 6 cm.
ft2
ft
A = (15)(7)
9.4 – Perimeter, Area, and CircumferenceFind the area of the shaded region.
Area of square 4 in.
4 in.
– Area of triangle
s2 – (1/2) b h
42 – (1/2) (4)(4)
16 – 8
8 in2
9.4 – Perimeter, Area, and Circumference
Circumference and Area of a Circle
The circumference C of a circle of diameter d is given by the formula: or where r is a radius.,C d 2 ,C r
d r
The area A of a circle with radius r is given by the formula:2.A r
9.4 – Perimeter, Area, and CircumferenceFind the area and circumference of a circle with a radius that is 6 inches long (use 3.14 as an approximation for ).
Circumference ( 3.14)
Area ( 3.14)
C = 2 r
C = 2 (3.14) 6
C = 37.68 in
A = r2
A = (3.14) 62
A = 113.04 in2
Circumference ()
C = 2 r
C = 2 6
C = 37.699 in
Area ()
A = r2
A = 62
A = 113.097 in2
9.5 – Space Figures, Volume, and Surface Area
9-410
Space figures: Figures requiring three dimensions to represent the figure.Polyhedra: Three dimensional figures whose faces are made only of polygons.
Regular Polyhedra: A polyhedra whose faces are made only of regular polygons (all sides are equal and all angles are equal.
9.5 – Space Figures, Volume, and Surface Area
Other Polyhedra
Pyramids are made of triangular sides and a polygonal base.
Prisms have two faces in parallel planes; these faces are congruent polygons.
9.5 – Space Figures, Volume, and Surface Area
Other Space Figures
Right Circular Cylinders have two circles as bases, parallel to each other and whose centers are directly above each other.
Right Circular Cones have a circle as a base and the surface tapers to a point directly above the center of the base.
9.5 – Space Figures, Volume, and Surface Area
Volume and Surface Area
Volume is a measure of capacity of a space figure. It is always measured in cubic units.
Surface Area is the total region bound by two dimensions. It is always measured in square units.
w
h
l
The volume V and surface area S of a box with length l, width w, and height h is given by the formulas:
V = lwh
Volume and Surface Area of a Rectangular solid (box)
and
S = 2lw + 2lh + 2hw
9.5 – Space Figures, Volume, and Surface Area
9-414
2 in.
3 in.
7 in.
Find the volume and surface area of the box below.
V = 7(2)(3)
V = 42 in.3
S = 2(7)(2) + 2(7)(3) + 2(3)(2)
S = 28 + 42 + 12
S = 82 in.2
9.5 – Space Figures, Volume, and Surface Area
Volume and Surface Area
The volume V and surface area S of a cube with side lengths of s are given by the formulas:
V = s3
Volume and Surface Area of a Cube
s
ss
5 ft.
Find the volume and surface area of the cube below.
V = 53 S = 6(5)2
V = 125 ft.3 S = 625
S = 150 ft.2
and
S = 6s2
9.5 – Space Figures, Volume, and Surface Area
Volume of Surface Area of a Right Circular Cylinder
10 m2 m
The volume V and surface area S of a right circular cylinder with base radius r and height h are given by the formulas:
Volume and Surface Area
Find the volume and surface area of the cylinder below.
V = (2)2(10)
V = 40V = 125.6 m3
S = 2(2)(10) + 2(2)2
S = 40 + 8 = 48
S = 150.72 m2
V = r2hh
r
and
S = 2rh + 2r2
9.5 – Space Figures, Volume, and Surface Area
Volume of Surface Area of a Sphere
r
The volume V and surface area S of a sphere radius r are given by the formulas:
Volume and Surface Area
V = (4/3) r3
Find the volume and surface area of the sphere below.
9 in.S = 4 (9)2V = (4/3)(9)3
V = 972V = 3052.08 in.3
S = 324S = 1017.36 in.2
and
S = 4 r2
9.5 – Space Figures, Volume, and Surface Area
Volume and Surface Area
Volume of Surface Area of a Right Circular Cone
The volume V and surface area S of a right circular cone with base radius r and height h are given by the formulas:
9.5 – Space Figures, Volume, and Surface Area
Find the volume and surface area of the cone below.
V = (1/3)(3)2(4)
V = 12V = 37.68 m3
S = 15 + 9S = 24S = 75.36 m2
h = 4 m
r = 3 m
9.5 – Space Figures, Volume, and Surface Area
Volume and Surface Area Volume of a Pyramid
The volume V of a pyramid with height h and base of area B is given by the formula:
Find the volume of the pyramid (rectangular base) below.
cm3
Note: B represents the area of the base (l w).
7 cm
6 cm3 cm