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

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.


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.


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)


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


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.


Other Polyhedra

Pyramids are made of triangular sides and a polygonal base.

Prisms have two faces in parallel planes; these faces are congruent polygons.


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.


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


9-414

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



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


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:


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


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:


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



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:


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


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

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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