+

Journal Chapter 9 and 10 Majo Díaz-Duran

+areas of a square, rectangle, triangle, parallelogram, trapezoid, kite and rhombus:Square A2, a = length of side

Rectangle w × h w = width h = height

Triangle ½b × h b = base h = vertical height

Parallelogram b × h b = base h = vertical height

Trapezoid ½(b1+b2) × h h = vertical height

Kite ½d1d2

Rhombus ½d1d2

+examples

+find the area of a composite figure. Explain what a composite figure is To find the area you need to break it into individual

pieces.

A composite figure is- any figure made up from 2 or more polygons/circles.

+Examples: By adding:

By Subtracting:

+area of a circle:

You can use the circumference of a circle to find the are

Circumference: 2πr

Area: πr2

+Examples:

The radius of a circle is 3 inches. What is the area?

A=πr2 A= π(3 in)2 A= π(9 in2) A= 28.26 in2

The diameter of a circle is 8

centimeters. What is the area?

D=2r8 cm = 2r 8 cm ÷ 2 =r

 r = 4 cmA=πr2

 A= π(4 cm)2 A= 50.24 cm2

+what a solid is:

A three dimensional figure, can be made up of flat or curved surfaces, each flat surfaces is called a face, an

edge is the segment that is the intersection of two faces, a vertex is the point that is the intersection of

three or more faces.

+Examples:

+find the surface area of a prism. What is a prism? Explain what a “Net” is A prism is formed by two parallel congruent polygonal

faces called bases connected by faces that are parallelograms.

The surface area of a prism = right prism with lateral area(L) and base area(B) L + 2B.

A net is a diagram of the surfaces of a three-dimensional figure that can be folded to form the three-dimensional figure

+Examples:

Net: Pyramid

+find the surface area of a cylinder

A cylinder is formed by two parallel congruent circular vases and a curved surface that connects the bases.

Surface Area of a Cylinder = 2 πr 2 + 2πrh

+Examples: Find the surface area of a cylinder with

a radius of 2 cm, and a height of 1 cm

S=2pir2+2pirh

S=2pi22+2pi(2)(1)

S=6.28(4)+6.28(2)

S=25.12+12.56

Surface area = 37.68 cm2

+find the surface area of a pyramid:

A pyramid is formed by a polygonal base and triangular faces that meet at a common vertex

Lateral area(L) and base area(B) is L+B or P(perimeter)l +B

+Example:

a square pyramid with a base that is 20 m on each side and a slant height of 40 m Find the surface area of the base and the lateral faces.Base: A=s2 or (20)2A=400 SA=400+4(400) SA=2000 m2

+find the surface area of a cone.

A cone is formed by a circular base and a curved surface that connects the base to a vertex.

lateral are L and Base are B L +B or πrl+πr2

+Examples:

a cone with a radius of 4 cm and a slant height of 12 cm:SA=pir2+pirLSA=pi(4)2+pi(4)(12) SA=50.3 +150.8SA =201.1 cm2

+find the volume of a cube

The volume of a cube is (length of side)3.

+Examples:

Example #2

Find the volume if the length of one side is 2 cm

V = 23

V = 2 × 2 × 2

V= 8 cm3

Example #3:

Find the volume if the length of one side is 3 cm

V= 33

V = 3 × 3 × 3

V = 27 cm3

+Cavalieri’s principle

If two objects have the same cross sectional area and the same height they have the same volume.

+examples

+find the volume of a prism

Base area (B) and height (h) is V= Bh

+Examples:

What is the volume of a prism whose ends have an area of 25 in2 and which is 12 in long:Answer: Volume = 25 in2 × 12 in = 300 in3

+find the volume of a cylinder

With base are (B) radius ( r) and height h is V: Bh or V= πr2h

+Examples: What is the volume of the

cylinder with a radius of 2 and a height of 6?Volume= Πr2h Volume = Π2(6) = 24Π

+find the volume of a pyramid

Base area (B) and height(h) V= 1/3Bh

+examples A square pyramid has a

height of 9 meters. If a side of the base measures 4 meters, what is the volume of the pyramid?

Since the base is a square, area of the base = 4 × 4 = 16 m2

Volume of the pyramid = (B × h)/3 = (16 × 9)/3 = 144/3 = 48 m3

A rectangular pyramid has a height of 10 meters. If the sides of the base measure 3 meters and 5 meters, what is the volume of the pyramid?

Since the base is a rectangle, area of the base = 3 × 5 = 15 m2

Volume of the pyramid = (B × h)/3 = (15 × 10)/3 = 150/3 = 50 m3

+find the volume of a cone

Base area (B), radius ( r ) and height (h) v= 1/3Bh or v= 1/3πr2h

+Examples:

Calculate the volume if r = 2 cm and h = 3 cm

V = 1/3 × pi × 22 × 3

V= 1/3 × pi × 4 × 3

V = 1/3 × pi × 12

V = 1/3 × 37.68

V = 1/3 × 37.68/1

V = (1 × 37.68)/(3 × 1)

V = 37.68/3

V= 12.56 cm3

Calculate the volume if r = 4 cm and h = 2 cm

V= 1/3 × pi × 42 × 2

V= 1/3 × pi× 16 × 2

V = 1/3 ×pi× 32

V= 1/3 × 100.48

V = 1/3 × 100.48/1

V= (1 × 100.48)/(3 × 1)

V= 100.48/3

V = 33.49 cm3

+find the surface area of a sphere

surface area = 4πr2

+examples Find the surface area of a sphere

with a radius of 6 cm

SA = 4 × pi × r2

SA = 4 × pi × 62

SA = 12.56 × 36

SA = 452.16

Surface area = 452.16 cm2 Find the surface area of a sphere with a radius of 2 cm

SA = 4 × pi × r2

SA = 4 × pi × 22

SA = 12.56 × 4

SA = 50.24

Surface area = 50.24 cm2

+find the volume of a sphere

V= 4/3πr3

+Examples:

If r = 300 mi (the moon), then the volume would be  V = 4πr3/3 = 4(pi)(300 mi)3/3 = 4(pi)(27,000,000 mi3)/3 = 113,040,000 mi3.

If r = 4 cm (a marble), then the volume would be  V = 4πr3/3 = 4(pi)(4 cm)3/3 = 4(pi)(64 cm3)/3 = 267.9 cm3