### M Warm Up

1. Find the area of a square with 5” to a side 2. Find the area of a triangle with a base of 7 feet and height of 5 feet 3. Find the area of a rectangle with base of 3cm and height of 9 cm 4. Find the area of a trapezoid with the large base of 9 m, small base of 5 m, and height of 3m 5. Find the area of a circle with a radius of 4 yards

• • • Triangle – A = ½ bh Square – A = s 2 Rectangle – A = bh

### Areas

• • Trapezoid – A = 1/2(b 1 + b 2 )h Circle – A =  r 2

### Volumes – Two Bases

• • • • • Base: different from “base” in area Base in Volume = “what the shape sits on” The bases in two base solids are parallel The height is the perpendicular distance between the two bases The “sides” are quadrilaterals, in our case, rectangles, unless we are talking about a cylinder

• • •

### Volumes – Two Bases

Two base solids are called Prisms (or cylinders) Names are based on shape of base: – Right means the sides are perpendicular to the bases, the sides are rectangles – Oblique means the sides are not perpendicular to the bases, the shape “leans” ( volume stays the same: Cavalieri’s principle) – Rectangular, Triangular, etc., is the shape of the base – Cylinder – the base is a circle V = B times h

### Volumes – One Base

• • • The end opposite the base in one base solids is a point The height is the perpendicular distance between the base and the point The “sides” are normally triangles, unless we are talking about a cone.

### Volumes – One Base

• • • One base solids are called Pyramids or Cones Names are based on shape of base, as for two bases: – Right means the sides are congruent, or the point is directly over the center of the circle base (cone) – Oblique means the sides are not congruent, the shape “leans” – Rectangular, Triangular, etc., is the shape of the base – Cone – the base is a circle V = 1/3 B times h

1. 15 yd cubed 11. 1200 m cubed 2. 10 mi cubed 3. 15 yd cubed 12. 1,325.3 m cubed 13. 13,823 mi cubed 4. 3.1 km cubed 5. 37.7 in cubed 6. 8 m cubed 7. 22.5 yd cubed 8. 0.7 in cubed 9. 1026 cm cubed 10. 680 cm cubed 14. 1693.3 yd cubed 15. 836 in cubed 16. 1734 mi cubed 17. 197 cm cubed 18. 2094 cm cubed 19. 718.4 yd cubed 20. 840 m cubed

### T Warm Up

1. Find the volume and surface area of a sphere with a circumference of 12  ”.

2. Find the volume and surface area of the above sphere if the radius is tripled.

3. What is the scale factor between the two spheres?

4. Write a proportion to show the ratio of the radii to the ratio of the surface areas.

5. Write a proportion to show the ratio of the radii to the ratio of the volumes 6. What conclusions can we draw to relate scale factor to area and volume of spheres?

### Today’s Standards

MCC9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

### Wed Warm Up

• • If a sphere has a volume of 1,000 cubic inches, what would the volume be if the radius were made 5 times larger?

If a sphere has an area of 1,000 square inches, what would the area be if the radius were made 5 times smaller?

1. Find the area and volume of a sphere with a radius of 3 inches 2. Find the area and volume of a sphere with a radius 5 times larger that the above 3. Write a proportion to show the ratio of the radii to the ratio of the surface areas.

4. Write a proportion to show the ratio of the radii to the ratio of the volumes 5. What conclusions can we draw to relate scale factor to area and volume of spheres?

• • If a sphere has a volume of 8,000 cubic inches, what would the volume be if the radius were made 4 times larger?

If a sphere has an area of 8,000 square inches, what would the area be if the radius were made 4 times smaller?

### Volume of Composite Shapes

from Student CCGPS Frameworks page 38 • Approximate the Volume of the Backpack that is 17in x 12in x 4in. Vol = Prism + ½ cylinder Vol = Bh + ½ Bh Vol = bh*h + ½  r 2 h Vol = 12*4*(17-12/2) + ½  *6 2 *4 Vol = 528 + 72  Vol  754.2 in 3

### Volume of Composite Shapes

from Student CCGPS Frameworks page 38 • Find the Volume of the Grain Silo shown below that has a diameter of 20ft and a height of 50ft Vol = Bh + ½ Vol sphere Vol = (  r 2 )h + ½ (4/3  r 3 ) Vol =  *10 2 *40 + ½ *4/3*  *10 3 Vol = 4666 2/3   14,661 ft 3

The diameter of a baseball is about 1.4 in. How much leather is needed to cover the baseball? How much rubber is needed to fill it?

### Find the Volume

• 165  • 230  • 7pi/9 radians • 4pi/3 radians

### Convert:

• • • • What is volume of a sphere that has a circumference of 32  What is the volume of a cylinder that can just hold the above sphere?

What is the ratio of the volume of the sphere to the volume of the cylinder?

Prove it with variables, not numbers

• • • • I have posted the graphic organizers for angles and segments in circles on my wiki, along with some helpful video links I have a link to Ms. Brassard’s wiki, which has more instructional videos for your use You may log onto and use carnegielearning.com for review You may use USATestPrep

• • • • What is surface area of a sphere that has a circumference of 32  ?

What is the surface area of a cylinder (including the “ends”) that can just hold the above sphere?

What is the ratio of the surface area of the sphere to the surface area of the cylinder?

Prove it with variables, not numbers

• • In his work On the Sphere and Cylinder, Archimedes proved that the ratio of the volume of a sphere to the volume of the cylinder that contains it is 2:3. In that same work he also proved that the ratio of the surface area of a sphere to the surface area of the cylinder that contains it, together with its circular ends, is also 2:3. Because expressions for the volume and surface area of a cylinder were known before his time, Archimedes’’ results established the first exact expressions for the volume and surface area of a sphere.

From http://www.math.nyu.edu/~%20crorres/Archime des/Tomb/Cicero.html

• • • • Assume a cylindrical watering can hold 100cm 3 of water.

How much water can it hold if the height is tripled and the radius stays the same?

How much water can it hold if the radius is made 1/5 the original and the height stays the same?

How much water can it if both the radius and height were made 4 times larger?

• • • Go to Geo Sketch 5 Various Problems Go to Geo Sketch 5 Angle Problems Go to Geo Sketch 5 arc length and sector area

find area between triangle and circle r = 7' angle =23 deg

• Find the volume of a pyramid with a regular octagonal base that would be inscribed in a 12 meter diameter circle and 14 meters high.

110 4x 2x 20y C 10y 24x 12x