Surface Area and Surface Area and VolumeVolume

At exactly 11:00 (12:30) I will put up At exactly 11:00 (12:30) I will put up the warm up. At your tables, do as the warm up. At your tables, do as

many as you can in 3 minutes!many as you can in 3 minutes!

Fill in the blanks with conversions--no peeking!

• 16: C ___ in a G ___• 32: D ___ F___ at which W ___ F ___• 60: M ___ in a(n) H ___• 16: O ___ in a P ___• 1000: M ___ in a M ___• 8: L ___ O ___ in a C ___• 2.2: K ___ in a P ___• 0.6: M ___ in a K ___• 1760: Y ___ in a M ___• 1: Q ___ in a L ___• 2.54: C in a(n) I ___

Agenda• Go over warm up• Check HW• Volume of prisms and cylinders• Exploration 10.14• Assign Homework

Hints• 2a. P: half a circle + 2 legs of the triangle.

Use Pythagorean Theorem.A: half a circle + area of the triangle.P = 3π + 6.7 + 6.7 m; A = (9/2)π + 18 m2

• 2b. P: Extend the top horizontal line to form a right triangle. Use Pythagorean Theorem.A: area of the rectangle + area of the right triangle.P = 11.66 + 60 m; A = 240 + 30 m2

• 5. Find 128/360 of the circumference. 11.16 ft• 7a. (1) Enclose the figure in a rectangle. Find are of

entire rectangle and subtract the white region.(2) Draw lines to form 3 rectangles. 175 cm2

• 7b. (1) Enclose the figure in a rectangle. Find are of entire rectangle and subtract the white region.(2) Draw in a vertical line to form two trapezoids. (3) Draw in a horizontal line to form 2 triangles and a rectangle.

• 9. Draw a square--subdivide the length and width into 12 1-inch segments. Then, count the squares that are formed. 144 square inches

• 10. Find the area of the backyard. How many full bags are needed to cover the backyard? 9600 sq. ft to be covered; 10 bags needed; $39.90

• 11. The shed has area 6 • 10. The new rectangle with the border is (6 + 2x) • (10 + x). Subtract the area of the shed, and set the remaining area to 18 square feet. 0.65 ft. (7.8 in.)

• 14. The area of the square is 36 sq. m. So, the length of a side is ___. Then, the perimeter of the square is ___. Now, use this amount of fence in a circle--that is, find the distance around the circle. Determine the radius, and then find the area of that circle.

Fence = 24 m; radius of circle = 3.82; area = 45.82 m2

• 15. Since the 12-inch diagonal cuts create an isosceles right triangle, (12, 12, and x), use the Pythagorean Theorem to find x, the length of the original square. 16.97 in.

• 20. The radius of the flower bed is 3 m. The radius of the flower bed plus the sidewalk is (3 + 1) m. Find the difference in the areas.16π - 9π = 21.98 m2.

• 26. Think of the work we did in Exploration 10.12. a. 9 x 1 and 5 x 5 b. 9 x 1 and 4 x 5c. Same as a. d. Same as b.

• 29b. Count the “square units” in each region.Each is 2 sq. un.

• 45. Think of the work we did in Exploration 10.12. Remember, the figures may be non-convex.P = 20 un., A = 9 sq. un. P = 20 un., A = 25 sq. un

• 48a. What will it look like if you put all the white areas together?25π cm

• 48b and c. How many missing lengths can you find?b. Need distance between horizontal lines.c. 34 + 3π in.

• 49a and b. How many missing lengths can you find?

a. 84 sq. ft. b. Need height of the rectangle.

• 50a. Draw a picture. 24.99 sq. in.• 50b. Think of Exploration 10.12. • P of big rectangle is 28 in. So, P of little rectangle is 14 in.

1 x 6, 2 x 5, 3 x 4, etc.• 50c. Think of Exploration 10.12.

If area is 20 sq. cm, then length of 1 side must be less than 4.47 cm. P is between 17.88 cm (4.7 x 4.7) and very large 160.5 cm (40 x .5) and bigger.

• 50d. If the length is double the width, draw a picture of where the posts must go.Posts: 19 on horizontal, 8 vertical, not counting the corners. Area = 180 x 90 = 16,200 sq. ft.

Volume of a Cube• Take a block. Assume that each edge

measures 1 unit.• So, the volume of that block is

1 unit3. We also call this a cubic unit.• Use the blocks to make 2 other cubes.

How many cubic units are needed?

Volume of a cube• Answer:• 1 cubic unit, 8 cubic units, 27 cubic units• Any “cube” will be formed with x3 blocks. • Ex: a cube with an edge that measures 13

units will have volume of 133, or 2179 cubic units.

Make rectangular prisms• Make 3 different rectangular prisms, each with a

base of 6 cubes.• The base must be a rectangle. Why?• The area of the base remains constant. Why?• The only thing that changes is the height. Why?• What is the volume (number of cubes) of each

prism? Is this related to the L, W, and H? If so, how?

Dimensions of Rectangular Prisms

• Do your prisms look like this?• 3 x 2 x 1 3 x 2 x 2

• 3 x 2 x 3 3 x 2 x 4

Rectangular prisms• Volume: Volume is defined as area of

the base multiplied by the height. • Why do we say L • W • H for a

rectangular prism?

lengthwidth

height

Exploration 10.15• Do 3 and 4. • Show your work and find each answer.• In 1 - 2 sentences, describe how to

imagine the solution to someone who is sight-impaired or blind.

Other 4-sided prisms• Suppose we had a trapezoidal prism.• Does the area of the base • height still

make sense? (Hint: what is the base?)

3-sided prism• What is the base? What is the height?

Other prisms• Can you find the base and height of

each prism?

What is a prism with a circular base?

• A cylinder.• Does area of the

base • height of the cylinder (prism)still make sense?

• What is area of the base • height?

In prisms and cylinders…• The bases are congruent.• In a prism, the faces are all rectangles.• Why aren’t the faces of a cylinder also

rectangles?

Surface Area of a Cube• In a cube, all six faces are congruent.• So, to find the surface area of a cube, we

simply need to find the area of one face, and then multiply it by 6.

• If we don’t have a cube, but we have a rectangular prism, there are still 6 faces: but they are not all congruent.

• Front and back, top and bottom, right and left.

Volume and Surface Area• Assume that each block has volume

1 unit3. Make 4 different polyhedra, each containing 12 cubes.

• Do all four have the same volume?• Do all four have the same surface

area?