+ Journal Chapter 9 and 10 Majo Díaz-Duran

+ areas of a square, rectangle, triangle, parallelogram, trapezoid, kite and rhombus: SquareA 2, a = length of side Rectanglew × h w = width h = height Triangle½b × h b = base h = vertical height Parallelogramb × h b = base h = vertical height Trapezoid½(b1+b2) × h h = vertical height Kite½d1d2 Rhombus½d1d2

+ 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: πr 2

+ Examples: The radius of a circle is 3 inches. What is the area? A= πr2 A= π (3 in) 2 A= π (9 in 2 ) A= in 2 The diameter of a circle is 8 centimeters. What is the area? D=2r 8 cm = 2r 8 cm ÷ 2 =r r = 4 cm A=πr2 A= π(4 cm)2 A= 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 π r π rh

+ Examples: Find the surface area of a cylinder with a radius of 2 cm, and a height of 1 cm S=2pir 2 +2pirh S=2pi2 2 +2pi(2)(1) S=6.28(4)+6.28(2) S= Surface area = cm 2

+ 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)2 A=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+πr 2

+ Examples: a cone with a radius of 4 cm and a slant height of 12 cm: SA=pir2+pirL SA=pi(4)2+pi(4)(12) SA= SA =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 = 2 3 V = 2 × 2 × 2 V= 8 cm 3 Example #3: Find the volume if the length of one side is 3 cm V= 3 3 V = 3 × 3 × 3 V = 27 cm 3

+ 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 in 2 and which is 12 in long: Answer: Volume = 25 in 2 × 12 in = 300 in 3

+ find the volume of a cylinder With base are (B) radius ( r) and height h is V: Bh or V= πr 2 h

+ Examples: What is the volume of the cylinder with a radius of 2 and a height of 6? Volume= Πr 2 h 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 m 2 Volume of the pyramid = (B × h)/3 = (16 × 9)/3 = 144/3 = 48 m 3 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 m 2 Volume of the pyramid = (B × h)/3 = (15 × 10)/3 = 150/3 = 50 m 3

+ find the volume of a cone Base area (B), radius ( r ) and height (h) v= 1/3Bh or v= 1/3πr 2 h

+ Examples: Calculate the volume if r = 2 cm and h = 3 cm V = 1/3 × pi × 2 2 × 3 V= 1/3 × pi × 4 × 3 V = 1/3 × pi × 12 V = 1/3 × V = 1/3 × 37.68/1 V = (1 × 37.68)/(3 × 1) V = 37.68/3 V= cm 3 Calculate the volume if r = 4 cm and h = 2 cm V= 1/3 × pi × 4 2 × 2 V= 1/3 × pi× 16 × 2 V = 1/3 ×pi× 32 V= 1/3 × V = 1/3 × /1 V= (1 × )/(3 × 1) V= /3 V = cm 3

+ find the surface area of a sphere surface area = 4πr 2

+ examples Find the surface area of a sphere with a radius of 6 cm SA = 4 × pi × r 2 SA = 4 × pi × 6 2 SA = × 36 SA = Surface area = cm 2 Find the surface area of a sphere with a radius of 2 cm SA = 4 × pi × r 2 SA = 4 × pi × 2 2 SA = × 4 SA = Surface area = cm 2

+ find the volume of a sphere V= 4/3πr 3

+ Examples: If r = 300 mi (the moon), then the volume would be V = 4πr 3 /3 = 4(pi)(300 mi) 3 /3 = 4(pi)(27,000,000 mi 3 )/3 = 113,040,000 mi 3. If r = 4 cm (a marble), then the volume would be V = 4πr 3 /3 = 4(pi)(4 cm) 3 /3 = 4(pi)(64 cm 3 )/3 = cm 3