Kategóriák: Minden - area - volume - theorem - perimeter

a Jackie Lindquist 7 éve

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Basic Measurement for Elementary Students

The document provides fundamental concepts of geometry aimed at elementary students, focusing on perimeter, area, volume, and surface area. It delves into the Pythagorean Theorem, explaining how to determine the length of an unknown side of a right triangle when two sides are known.

Basic Measurement for Elementary Students

Basic Measurement: Perimeter, Area, Volume, and Surface Area

The purpose of this mind map is provide resources for exploring and clarifying the topic. The audience is elementary students.

surface area (SA)

Surface Area (SA or S) is the boundary of a space figure. A space figure is an object that is represented in three dimensions (3D).


Total surface area is the sum of the base area and lateral area (or sides) of a space figure.


Think about a jacket or cover that would enclose the entire shape.

SA of a shpere

Surface Area of a Sphere:

SA=4πr2

Let r be the radius of the sphere.

watch this video to see how the surface area of a sphere is estimated by thinking of a number of small bases covering the sphere

SA of a cone

Surface Area of a Right Circular Cone:

SA=πr2+πrs

Let s be the slant height and r be the radius of the base.


Note: to determine slant height use the Pythagorean Theorem: s= √(x2-y2) given x is the height of the cone and y is the radius of the cone. When thinking about this triangle let s be the hypotenuse, and x and y be the two legs.

SA of right rectangular pyramid

Surface Area of a Right Regular Pyramid:

SA=B+(1/2)ps

Let s be the slant height, and B be the base area, and p be the perimeter of the base.


Note: to determine slant height use the Pythagorean Theorem: s= √(x2-y2) given x is the height of the pyramid and y is the (1/2) the length of one side of the base of the pyramid. When thinking about this triangle let s be the hypotenuse, and x and y be the two legs.

click the link to explore the surface area of pyramids

SA of a cylinder

Surface Area of Right Prism or Cylinder:

SA=2B+ph

Let h be the height, and B be the areas of both bases, and p be the perimeter of each base.


Note: if you unroll the side of a cylinder you have a rectangle

click the link to see the side of a cylinder unrolled

SA of right prism

Surface Area of Right Prism or Cylinder:

SA=2B+ph

Let h be the height, and B be the areas of both bases, and p be the perimeter of each base.


Note: if you unfold a prism you have a collection of 2D shapes called the net of the prism

click the link to see the nets of some rectangular prisms

volume (V)

Volume is the amount of space enclosed within the boundary of a 3D shape (or object).

volume of a sphere

Volume of a sphere: V=(4/3)πr3

where r is the radius of the sphere


Note: the volume of a sphere takes up (2/3) the volume of a cylinder of the same radius and diameter height

click the link to use a sphere volume calculator and a table of radius to volume values

volume of a pyramid or cone

Volume of pyramid or cone: V=(1/3)Bh where B is the area of the base and h is the height.


Notes: a pyramid will fill (1/3) of a rectangular prism with the same base and height


a cone will fill (1/3) of a cylinder with the same base and height

click the link to watch a video explanation of volume of a pyramid

volume of a cylinder

Volume of a cylinder: V=Bh where B is the area of the base and h is the height.


Notes: the base of a cylinder is a circle

this same formula is used to find the volume of right, oblique, or general cylinders!

volume of a box

The volume of a rectangular box (or prism) is found by multiplying length x width x height.


Use the formula: V=Bh where B is the area of the base and h is the height.


Note: to find the volume of a complicated prism, decompose it into simpler shapes (watch the video to see a demonstration of this)

click on the video to hear a song about volume of rectangular prisms

Pythagorean Math

theorem

a2 + b2 = c2


The sum of the areas of squares on the legs of a right triangle is equal to the square on the hypotenuse.

click on the link to explore the Pythagorean Theorem

Uses
Right triangle

If a2 + b2 = c2

then it IS a right triangle.


If a2 + b2  c2

then it IS NOT a right triangle.

unknown side

If two sides of a right triangle are known, then the length of any unknown side can be found using the theorem a2 + b2 = c2

slant height

The slant height of a cone can be determined by thinking about a triangle where one leg is the radius of the base and the other leg is the vertical height of the cone, the slant height would be the hypotenuse.


Thus: Slant Height2=h2+r2

where h is the vertical height and r is the radius of the cone

The Measurement Process

Units
U.S. Customary System of Units

U.S. units of capacity

capacity conversions

U.S. units of volume

Units of volume will be the unit of measurement cubed, for example the volume of a rectangular prism measured in inches will be given as ___ cubic inches (or in.3).


This is because there are three measurements used to calculate volume: length, width, and height. We are considering three dimensions of the object, think 3D therefore cubed.

U.S. units of area

Units of area will be the unit of measurement squared, for example the area of a rectangle measured in inches will be given as ___ square inches (or in.2).


This is because we are considering two dimensions of the object: length and width. Think 2D therefore squared.

U.S. units of length

click on the link to view customary units of length with picture examples

International System of Units

metric units of volume

click on the link, or watch the video, for a look at converting between liters and milliliters

metric units of area

click on the link for picture examples and a great explanation of metric a rea units

SI prefixes

Four steps
4. express as number of units

4. Express the measurement as the number of units used in the comparison.

3. make a comparison

3. Use a measurement device to compare the object to the unit of measurement.


A measurement device is a tool or instrument used to measure: ruler, tape measure, meter stick, gauge, scale, thermometer, etc.

2. select a unit

2. Select an appropriate unit of measurement.


The unit of measure will change based on what you are measuring. Possibilities for measuring length include, but are not limited to: inches, feet, miles, centimeters, meters, kilometers


For weight you could use: ounces, pounds, tons, milligrams, grams, or kilograms.


Remember to chose a unit that makes sense for what you are measuring.

1. choose an attribute
  1. Choose the attribute to be measured: length, area, volume, capacity, temperature, time, or weight.

perimeter (P)

The perimeter of a region is the length of its boundary, just the outer edge of a 2D shape (or region).

perimeter of a circle

The perimeter of a circle is called its circumference.


It is found with the formula C=2πr where C is the circumference and r is the radius of the circle.

Perimeter of polygons

To find the perimeter of a polygon add the lengths of all of its sides.


For example: if I have a square with side lengths of 3 ft., I will add 3+3+3+3= 12 ft.

area (A)

The area of a region is the amount of space inside the boundary (or perimeter) of a 2D object. Think about the number of tiles required to fill a shape with no overlapping.

area of circles

Area of a circle: A=πr2 where r is the radius of the circle.

click the link for area of a circle practice problems

area of trapezoid

Area of trapezoid: A=(1/2)(a+b)h

where a and b are the lengths of the two bases and h is the height of the trapezoid

area of triangles

Area of triangle: A=(1/2)bh

(Area = one half Base x Height)

click on the link to see how two congruent triangles form a parallelogram

area of parallelograms

Area of parallelogram: A=bh (Area=base x height)

click the video link to watch how a the area of a parallelogram can be transformed to make a rectangle of equal area

area of rectangles

For all rectangles: A=wl (Area = Width x Length)

click the link to read about the area of rectangles, and to watch a video tutorial