Chapter 10

Loukia A.

Section 10.1 : Areas of Parallelograms and Trapezoids

Vocab Words...


  • Base of a Parallelogram- The length of any of the parallelogram's sides.
  • Height of a Parallelogram- The perpendicular distance between the base and the opposite side on a parallelogram.
  • Base of a Trapezoid- One of the two parallel sides on a trapezoid.
  • Height of a Trapezoid- The perpendicular distance between the bases on a trapezoid.


Key Concepts...


Area of a Parallelogram: A= b*h

Area of a Trapezoid: A= (1/2) * (b1*b2) *h


****A=area, b= base, h= height, b1 and b2= either one of the bases.

Practice Questions

Parallelograms...

  1. A= 23*42
  2. 20= (1/2) *h

Trapezoids...


  1. A= (1/2) * (4 + 8) *3
  2. A= (1/2) * (6 + 16) *5


Answers

1. A=966 units²

2. H= 40 units²


1. A= 18 units²

2. A= 55 units²

Real Life Examples

In real life, these formulas are used in a lot of architecture and the building of homes. For example, if you are building a house, and you have to put in a roof on your house. You want to figure out what the front side of the roof, you can easily use the formula and figure out the area and how many tiles you need to fill in the area of the front side of the roof. You could also use these for anything shaped in a shape of a parallelogram or trapezoid.
Parallelogram and Trapezoid Area

10.2: Area of Circles

Vocab Words...

  • Area- The amount of surface the figure covers.
  • Circle- The set of all points in a plane that are the same distance from a fixed point called the center.
  • Radius- The distance from the center to any point on the circle though the center, or twice the radius.
  • Diameter- The distance across the circle through the center, or twice the radius.
  • Circumference- The distance around a circle.
  • Pi (π)- An non-terminating number that for every circle, the quotient of its circumference and its diameter are the same.

Key Concepts...

Area of a Circle: A=πr²


****A=Area, r= radius

Practice Questions

Find the area of the circle given its radius (r) or diameter (d). Use 3.14 for π.

  1. r= 18 mi
  2. d= 80 in.
  3. d=11mm
  4. r= 2.9 ft.


Answers


  1. 1,017.36 mi²
  2. 5,024 in.²
  3. 94.985 mm²
  4. 26.4074 ft.²

Real Life Examples

Finding the area of circles can be used for anything from architecture to simple field designs finding the area of circles is a necessary skill to have in modern day life. It is commonly used for sports fields such as basketball courts and hockey rinks.
Area of a Circle - MathHelp.com - Math Help

10.3: Three-Dimensional Figures

Vocab Words...

  1. Solid- A three dimensional figure that encloses a part of a space.
  2. Polyhedron- A solid that is enclosed by polygons.
  3. Face-The polygons that form a polyhedron.
  4. Prism- A polyhedron that has two congruent bases that lie in parallel planes. The other faces are rectangles.
  5. Pyramid- A polyhedron with one base and the other faces are triangles.
  6. Cylinder- A solid with two congruent circular bases that lie in parallel planes.
  7. Cone- A solid with one circular base.
  8. Sphere- A solid is a solid formed by all points in space that are the same distance from a fixed point called the center.
  9. Edge- The segments where faces of a polyhedron meet.
  10. Vertex- A points where three or more edges meet.

Practice Questions

  1. How many faces,edges, and vertices does a hexagonal pyramid have?
  2. Show two ways to represent a cylinder. Tell whether it is a polyhedron.


Answers

  1. 7, 12, 7.
  2. Answers may vary

Real Life Examples

Everything, that is real, in our world is three dimensional. Everything we touch is three dimensional. Whether you are designing a car, or making a cereal box, all of these jobs need to work with three dimensional objects.
3D Shapes I Know (solid shapes song- including sphere, cylinder, cube, cone, and pyramid)

10.4: Surface Areas of Prisms and Cylinders

Vocab Words...

  • Net- A two-dimensional pattern that forms a solid when it is folded.
  • Surface Area- The sum of all the faces areas on a polyhedron.

Key Concepts...

Surface Area of a Prism: S= 2B+Ph

Surface Area of a Cylinder: S= 2B+Ch (or) 2πr²+ 2πrh


****S=Surface Area, B=Base's Area, P= Base's Perimeter, h= height

Practice Questions

Find the surface area of the solids. Use 3.14 for π.


Answers

  1. 2,352 cm²
  2. 72 ft²
  3. 533.8 cm²

Real Life Examples

One example in real life would be making toddler's toys. This relates to this by the game where they put different blocks, the circle, triangle, and rectangle, into the proper slot. Most of the time children end up getting the triangle stuck in the circle slot, but there also involves math in this. The constructors of the blocks must make sure that the blocks are a size safe enough so the children don't accidentally choke on it. They use these formulas to figure out the size.

10.5: Surface Areas of Pyramids and Cones

Vocab Words...

  • Slant Height- (represented as l in equations) The height of a lateral face, any face that is not the base.

Key Concepts...

Surface Area of a Pyramid: S= B+1/2Pl

Surface Area of a Cone: S= πr²+πrl


**** S= Surface Area, B= Base's Area,

Practice Questions

Find the surface area of the solid. Round to the nearest tenth.

  1. A square pyramid with a base side length 12 m and slant height 9 m.
  2. A cone with a radius 8 cm and slant height of 9 cm.
  3. A cone with the diameter 15 m and slant height 8.2 m.


Answers

  1. 282 m²
  2. 427.3 cm²
  3. 239.7 m²

Real Life Examples

Have you ever been to see the Great Pyramids of Giza? How about some of the Aztec's temples, which are pyramids as well? What if you are doing a report on either one of the ancient pyramids, and you need to know the size of one. That's is where archeologist come in. They find the sizes of these pyramids for you, and give the information to people in need of it. They use these type of formulas if they are trying to make a to scale model of the pyramid, or sometimes, just measuring the pyramid itself!

10.6: Volumes of Prisms and Cylinders

Vocab Words...

  • Volume- The measurement in the amount of space a solid occupies.

Key Concepts...

Volume of a Prism: V=Bh

Volume of a Cylinder: V=Bh or V=πr² h


**** V=volume, B= base's area, h=height, r=radius

Practice Questions

Find the volume of the solids using the exercise from 10.4. Use 3.14 for π.

  1. Problem 1, the rectangular prism.
  2. Problem 2, the triangular prism.
  3. Problem 3, the cylinder.


Answers

  1. 960 cm³
  2. 42 ft³
  3. 942 cm³

Real Life Examples

When you think of finding the volume of a prism or cylinder, you don't usually think of recycling bins. Even if it is an unfamiliar notion, math is involved in finding which recycling bin you want for your house. If your neighborhood lets you choose between to recycling bin choices, you would have to compare the choices and decide which one holds the most trash. When doing this you can use these formulas to help figure it out.
Volume of Prism and Cylinders

10.7: Volumes of Pyramids and Cones

Vocab Words...

  • Pyramid- A polyhedron that has one base and all other faces are triangles.
  • Cone- A solid with one circular base.
  • Volume- The measurement in an amount of space a solid occupies.

Key Concepts

Volume of a Pyramid: V= 1/3 Bh

Volume of a Cone: V= 1/3 Bh or V= 1/3 πr² h


**** V= volume, B= Base's area, h=height, r= radius

Practice Questions

Find the volume of the solid. Round to the nearest tenth.

  1. A square pyramid with base side length of 10 ft. and height of 8 ft.
  2. A cone with a radius 18m and height of 6m.
  3. A triangular pyramid with the base side length of 9 cm and 12 cm, and a height of 14 cm.


Answers

  1. 266.7 ft.³
  2. 2035.8 m³
  3. 252 cm³

Real Life Examples

An example I real life would be different structures that are in a pyramid and cone shape. One famous structure would be the tepee. A tepee is a hut like structure in the shape of a cone or pyramid. They have a wooden skeleton and, usually, a birch wood and different animal hide outside. hey could use these formulas for building these and finding out how many people can fit in it.

Spheres...

Volume of a Sphere: V= (4/3)*π*r³

Surface Area of a Sphere: 4*π*r²

Area of a Sphere: 4*π*r²