Solid Geometry CYLINDER CONE AND SPHERE CONE Standards

Solid Geometry CYLINDER, CONE AND SPHERE CONE

Standards 8, 10, 11 SOLIDS PYRAMID PRISM CYLINDER CONE PRESENTATION CREATED BY SIMON PEREZ. All rights reserved SPHERE 2

Cylinder Height radius ▬ Has three faces, they are two bases and a lateral face/curved surface ▬ Has two edges

SURFACE AREA OF CYLINDERS r base r 2 h 2 rh h r r base 2 r Lateral Area: L= 2 r h Total Surface Area = Lateral Area + 2(Base Area) T= 2 r h + 2 r 2 h= height r= radius 5

VOLUME OF CYLINDERS r 2 B= V = Bh V= h r 2 h r RIGHT CYLINDER 6

Find the lateral area, the surface area and volume of a right cylinder with a radius of 20 in and a height of 10 in. Total Surface Area = Lateral Area + 2(Base Area) T= 2 r h + 2 r 2 10 in T = 2 ( 20 in)( 10 in )+2 ( 20 in 2 T= 400 in 2+ 2(400 in )2 T = 400 Lateral Area: 20) in T = 1200 in +2 800 in 2 L= 2 r h L = 2 ( 20 in )( 10 )in Volume: L=400 V= V= in 2 r 2 h ( 20 in ) 2( 10 in) V= (400 in 2)(10 in) V= 4000 in 3 7

cone h s r Has two faces Note: * r = radius * h = height * s = slant height

How to calculate the curved surface area ? l l 2πr r Cut here

How to calculate total surface area of a cone? l l + r Total surface area =πr 2 + πr l r

Surface Area of a Cone • A cone has a circular base and a vertex that is not in the same plane as a base. • In a right cone, the height meets the base at its center. Height The vertex is directly above the center of the circle. Lateral Surface Slant Height r Base r • The height of a cone is the perpendicular distance between the vertex and the base. • The slant height of a cone is the distance between the vertex and a point on the base edge.

SURFACE AREA OF A RIGHT CIRCULAR CONE C= 2 r L= area of sector Area of Circle perimeter of cone’s base l l 2 l h r r C= 2 r r 2 B= perimeter of cone’s base area of sector = area of circle perimeter of circle area of sector 2 l area of sector 2 = l L= area of sector = TOTAL SURFACE AREA: 2 r = 2 l rl l C= 2 l 2 T = area of sector + area of cone’s base h= height T=L+B r = radius T= r l + r 2 l = slant height 12 Lateral Area

VOLUME OF A RIGHT CIRCULAR CONE h r B= r 2 V= 1 Bh 3 V= 1 3 Standards 8, 10, 11 r 2 h h= height r = radius 13

Examples 1 a) If h = 12 cm, r= 5 cm, what is the volume? Answer: Volume = 1 πr 2 h 3 = 1 π (52) ( 12) 3 = 314 cm 3

b) what is the total surface 2 area? = π5 Based Area = 25πcm 2 Slant height = 122 + 5 2 = 13 cm Curved surface area = π(5) ( 13) = 65π cm 2 Total surface area = based area + curved surface area = 25π+65π= 90π = 282. 6 cm 2 (corr. to 1 dec. place)

Find the lateral area, the surface area and volume of a right cone with a height of 12 cm and a radius of 10 cm. Round your answers to the nearest tenth. Lateral Area: l h =12 cm 2 2 2 l 2= 12 + 5 l = 144 + 25 2 l = 169 l = 13 rl L= (12 cm ) (13 cm ) = 489. 8 cm Calculating the base area: 2 B= r 10 cm = r B= we need to find the slant height, using the Pythagorean Theorem: L= r 2 B= 2 ( 5 ) 13 B = (25) (13)(3. 14) B= 1020. 5 cm 2 Calculating surface area: T=L+B T = 489. 8 cm 2 + 1020. 5 cm 2 T = 1510. 3 cm 2 2 Calculating the volume: 1 3 1 V= 3 V= r 2 h 2 ( 5 ) (12) ( 25 ) ( 12 ) V = 314 cm 3

Sphere has no flat surface just has a face

r r

From the experiment that you have done: Determine how many circles could be covered by that rope The area of a hemispherical = the area of 2 circles The area of 2 hemispherical = the area of 4 circles The area of a sphere = the area of 4 circles A = 4 the area of a circle A = 4 πr 2

Volume of Cylinder Volume of the sphere

Solve the following problem: 1. A solid spherical object has a diameter of 4. 2 cm. Find out the surface area of the object 2. What is the radius of the sphere, if the area of the sphere is 78 cm 2 3. A solid spherical object has a radius of 20 cm. Find out the surface area of the object

1. A solid spherical object has a diameter of 4. 2 cm. Find out the surface area of the object. Solution The diameter is 4. 2 cm r = 2. 1 cm The formula of the surface area is A = 4 πr 2 A=4 (2. 1)2 A = 55. 44 cm 2 Therefore, the surface area of that object is 55. 44 cm 2

PROBLEM SOLVING

PROBLEM SOLVING The silo shown below has been built from metal. The top part of the silo is a cylinder of diameter 4 m and height 8 m. The bottom part of the silo is a cone of slant height 3 m. The silo has a circular opening of radius 30 cm on the top. a. What area of metal (to the nearest m 2) was required to build the silo? b. If it costs Rp 25, 000. 00 per m 2 to cover the surface with an anti-rust material, how much will it cost to cover the silo completely?

solution 1. Understanding the problem a. What is the unknown? The area of the metal and cost to cover the surface b. What are the data? A cylinder with a diameter 4 m and height 8 m. The bottom part of the silo is a cone with a slant height 3 m. The silo has a circular opening with a radius 30 cm on the top. It costs Rp 25, 000. 00 per m 2 to cover the surface with an anti-rust material

2. Developing a plan and strategy a. Find the surface area of the top face. b. Find the area of the curved part of cylinder. c. Find the area of the curved section of a cone. d. Find the area of the silo by adding the area of the top face the curved part of cylinder and the curved section of a cone. e. The total cost can be obtained by multiplying total area by the cost per m 2.

3. Carrying out the plan a. The area of the top face = the area of a large circle – the area of a small circle. = 3. 14(22) - 3. 14(0. 32 = 3. 14(4 - 0. 09 = 12. 28 m 2 b. The area of the curved part of cylinder = 2(3. 14)(2)(8) = 100. 48 m 2 c. The area of the curved section of a cone = 3. 14 rs = 3. 14 (2)(3) = 18. 84 m 2 d. The area of the silo = 12. 28 m 2 + 100. 48 m 2 + 18. 84 m 2 = 131. 6 m 2 e. Total cost = 132(Rp 25, 000. 00) = Rp 3, 300, 0