Category: geometry

Finding Area – Classroom Exercises

Area is All Around Us!  Simple Geometry Classroom Activities


One challenge that students often face is realizing that math DOES actually relate to the real world and that they will actually use the information they are learning at some point in their life. Area happens to be one of those topics that students struggle to understand the reasoning behind.  Before introducing the topic of area, you may want to ask your students some engaging questions such as: “If you wanted to put new tile down on the floor, how would you know how much to buy?” or “If you wanted to put wallpaper on the walls of the classroom, how much wallpaper would you buy?”. These questions will be sure to get them thinking about how to calculate these answers.

Identifying Polygons

Polygons can be defined as two-dimensional, closed figures that are described by the number of sides, length of sides, and the kinds of angles. Many polygons have a respective name depending on their description. Some common polygons you will work with include: triangles, squares, rectangles, trapezoids, quadrilaterals, rhombus, pentagons, hexagons, parallelogram, and octagons.

When learning the names of polygons, students can easily be confused by terms that are used interchangeably at times. For example, a quadrilateral is considered to be a four-sided figure. So one might easily confuse this by calling all quadrilaterals squares or rectangles. However, by definition a rectangle is a special quadrilateral because it has opposite sides that are congruent or the same length and each angle is a right angle that measures 90 degrees. Another special quadrilateral is a square. A square has four sides that are all congruent or the same length as well as four angles that are all right angles measuring 90 degrees. This becomes easily confusing for a student when they are trying to identify polygons by name and descriptors.

Golden Mean Ratio: Egyptian Sculpture

Math in Ancient Sculpture is one of the most interesting and intriguing applications of mathematics, both formula equation and geometric structure.  This article examines the Golden Mean Ratio and Fibonacci FibonacciChamomile wikip Oct 7 2013  Nefertiti Neues Museum Wikip Oct 7 2015Sequence found in the Bust of Nefertiti in Egypt in the early 1900’s. Nefertiti literally, means, “the beautiful one has come”, and is the 14th-century BC Great Royal Wife (chief co-regent) of the Egyptian Pharaoh Akhenaten of the Eighteenth dynasty of Egypt, 1352 BC to 1336 BC.  The iconic Bust of Nefertiti is in Neues Museum of Berlin; though originally in Egypt (En.   Wikipedia.   Org / wiki/Nefertiti_Bust   (230px-Nofretete_Neues_Museum 2 .jpg)  During excavation of Nefertit’s Bust, one of the items found noted Thutmose as being the court sculptor of Egyptian Pharaoh Akhenaten, Nefertiti’s Royal Husband.  In Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science, Stakho speaks of the Berlusian philosopher Edward Soroko’s attempts to determine what mathematical ideas were used in the creation of Her Bust, “…Harmony was the perogative of the Divine order that dominated the universe, and geometry was the main tool of its expression. 

Nefertiti, playing role of Goddess, thus Her image personifying the Wisdom of the world, must have been formed with geometrical perfection and irreproachable harmony, beauty and clarity.  As a matter of fact, the main idea of ancient Egyptian aesthetic philosophy was to glorify the eternal, the measured, and the perfect in a constantly changing universe…In his analysis, he found a harmonious system of regular geometric figures such as triangles, squares, and rhombi.”  (Alekseĭ Petrovich Stakho; Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science; Scott Anthony Olsen. Book. English. 2009; pp 57-59)  Soroko believed that the Bust of Nefertiti was created based upon the Geometries of the Golden Mean Ratio and the Fibonacci Sequence.  A graphical image of Her Bust, as was drawn by Soroko is shown in this link:  Highly recommended to click on this link to have a finer and more detailed idea of the geometries that Thutmose is thought to have in mind when sculpting Her Bust.  One can appreciate, in particular, how the front top vertex of Her headdress is in perfect alignment with Her Heart and Breast Bone.

Golden Mean Squares Wiki Oct 7 2014Depicted here, the Golden Mean Ratio, and in the geometries thought to be applied by Thutmose, the Egyptian Court Sculptor, in the creation of Nefertiti’s Bust:

n is to m, as m is to n − m,

or, algebraically,


n            =         m

_____            ______

m                     n – m

A more clear example of the pattern he believed Her Bust was created upon, the Fibonacci Sequence, can more easily be seen in examining nature’s pattern in pine cones, and seashells.  This ratio is also the most efficient mathematical equation for trees absorbing the most amount of sustenance from the sun’s rays, hence growing in a spiral.  In the same way, today, we can apply solar panels onto rooftops mimicking this pattern to most efficiently absorb the sun’s rays.  For further discussion on Fibonacci Sequence, visit here:

See in the photo at the top, the chamomile spiral, as well as in this link on “fibonacci-sequence” showing the pattern.  Adding two consecutive numbers from the sequence to equal the next one following, the basic mathematical sequence looks like this:

0,         1,            1,            2,            3,            5,            8,            13,            21,            34,            55,            89,            144,            233,            377, etc., etc.

That is:



2+1 (the “number” before)=3

3+2 (the number before)=5








etc., etc., etc.

Whereas the fibonacci sequence, cannot technically be exacted in real time, (similarly, there is no real exact end found as of yet to pi’s “3.14159265359…..”), mathematicians drew swirling squares around the spiral in the attempt to exact the formula – basically giving the left side of the human brain a way of understanding the right-brained infinite spiral, being that the left side of the brain requires finiteness / exactness to feel satisfied, shall we say.  

What kind of intriguing application can your class find to prove that math is interesting and fun?

We wonder if the geometries of Nefertiti’s Bust were also the same geometries of Nefertiti’s real-life human head.  The skulls of Ancient Egyptians, as well as those of Ancient Greece and other cultures, have definitely been shown to have geometrically perfect structures.  How can that be?

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Primary Geometry All Around

Here’s a fun way to integrate math, technology and language in one project for your youngest learners. Mathematically, students learn about solid figures and their properties. Technologically, they use digital cameras and work with word processing programs to insert pictures, word process and format documents. Students practice writing original thoughts and grammatically correct sentences as they describe the objects they have discovered.

Start the project by having students take digital pictures of solid figures in their everyday lives. They can either do this at home or (if they do not have a digital camera) at school. The following chart lists the common solids primary students learn about and some everyday items children would be familiar with. Hopefully, your students will find numerous examples of each.




Rectangular Prisms


-ice cream cone

-construction cone

-cone used for sporting events

-party hat



-scoop of ice cream


-play block

-sugar cube


-tissue box

-cereal box

-stick of butter

-pack of gum

-juice box

-soup can


-dowel rod

-stove pipe

-rolling pin


Once students have taken their pictures, visit the computer lab so students can create their “Book of Geometric Solids.” Students can organize their books in a variety of ways. However, each picture should have one to three sentences to serve as a caption. Captions must identify the type of solid the object is and must tell something about that object. For example, the student inserts a picture of a soccer ball. He then writes, “A soccer ball is an example of a sphere. I play soccer every Saturday morning. It is my favorite sport.”

This activity not only allows students to discover math in their everyday lives, but it also helps develop multiple academic skills.

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Geometry Math Stars for Christmas!

Learn and Practice Geometry by Making Your own Family Christmas Tree Star Ornaments!  Great for decorating your Christmas Tree or for making a Gift to give to a friend or family member.

Materials Needed:

Colored Markers or Crayons

Glue or Scotch Tape


Colored Yarn or String


Colorful Recycled Paper (tissue boxes, flyers, old wrapping, etc.)

Directions for Christmas Star Cutout:

1.Cut out the outline.

2.Cut along all heavy lines.

3.Score plain lines on the front.

4.Score dotted lines on the back.

5.Fold triangles upword along plain lines.

6.Fold triangles downword along dotted lines.

7.Glue or tape tabs to form small tetrahedrons.

8.Continue until you have your Geometry Christmas Star Tetrahedron.

Everyone has their own Star!  Everyone has their own Inner Light!  With Favorite colors, draw your stars, or print this page and cut the pattern of the star out.

Inside each shape on the side of the star, write your name and birthday.  Or – write the names of each member of your family (If you Wish, your family / family Tree Star can be made of relatives (close or extended), friends and/or adopted family – as long as you write each name on each.  You could make many stars – a pretty star for each member of your Christmas Tree, or write everyone’s name on the same Star.

Color and Decorate each one with colored markers, crayons, sparkles, gluing pieces of recycled Christmas paper from last year.  You can also write happy words all over your star like Love, Divine Wisdom, Infinity, Pure Spirit, Fun and Harmony!

Remember to Decorate your Stars!

Glue Yarn / Strong at the top  into a 2 inch loop and tie a knot at the end

Hang your Christmas Geometry Math Stars and Decorate your Tree at Home or in your Classroom!

Have a Merry Christmas and Happy Holiday!

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Reference for Star Tetrahedron Geometry Template: “The Ancient Secret of the Flower of Life,” Vol. 2, by Drunvalo Melchizedek

Drawing with Pythagoras

“Pythagorean Theorem?!!”…I know what you are thinking…..(You can say to your class)…..”OMG  Teacher…..What could we possibly ever use this for, in the future of our lives?“……Well, in this article, we talk about real life examples of using the Pythagorean Theorem.  First let’s learn a basic calculation, corresponding to the diagram here to the <<left, followed by a bit of Pythagorean history.  The basis of the theorem is that the Area of the 2 Squares of the 2 Lines that form the Right Angle of a Right Triangle are the equivalent ( = ) Area of the Square that opposes them in the Triangle. If you look at the diagram here to the << left, the Areas of “a” added to “b” = the Area of “c” (the Square of the opposite line (or “leg”). That opposite line (or  “c” “leg”) is called the hypotenuse. Also, if one of the lines “a” of the right triangle is 4 inches and the other line “b”  is 6 inches, we can calculate how long the hypotenuse is, or the “third leg”. Letting a = 4, b=6, and c= the length of the hypotenuse.  (4)^2 + (6)^2 = c^2. Accordingly, 4 x 4 =16, and 6 x 6 = 36. Thus, 16 + 36 = 52. The square root of 52 is approximately 7.21 , hence the length of the hypotenuse or “third leg“ of the right triangle, is 7.21 inches.

The Pythagorean Theorem is named after the Greek mathematician Pythagoras. Many believe the first discovery and proving of this ancient math theorem came before Pythagoras, but since no tangible account of this has yet been documented, it is named as such. If that was true, however, we wonder what another name of the theorem would have been, and from what country and nation? (a fun question to ponder).  The Pythagorean Theorem can be used with any shape and for any formula that squares a number. And, in fact, the area of any shape can be computed from any line segment squared.  (reference also for Diagram Above).   Apparently, even the teenage Brainiac Lisa Simpson from The Simpsons television series knows all about Pythagoras.  On this site, there are some cool diagrams showing the differences of Lines (Segments / Legs), Radius, and Area:  And, to assist in figuring out these math scenarios, this site offers tabs for entering in numbers to calculate the square root of the numbers in question:

Finally, here are a few Real Life Examples:

Meet Me at The Corner:
Let’s say Stephanie and Maria are meeting at the Hiking Trail Entrance on the corner of Saanich Rd. and Cedar Rd. One phones the other on her mobile phone and asks, “How long will it be before you arrive at the entrance?”… “Well, let’s estimate by first finding out how far away we are from each other.” In present time, Stephanie is on Saanich Rd. to and is 10 miles away. Meanwhile, Maria is on Cedar Rd. and is 4 miles away.  How far away from one another are Stephanie and Maria?  The distance between them = a^2 + b^2 = c^2 or, respectively: 10^2 (10 squared) + 4^2 (4 squared); or respectively, 100 +16 = 116 miles. The square root of 116 is 10.77. Thus, Stephanie and Maria are 10.77, almost 11 miles away from one another. Hence, they figure they will be another hour on their bicycles to meet one another to go hiking.

Firefighters Needing to Know Height of a Building:
3 Firefighters receive a call to help Ann rescue her cat Tia from the Oak Tree outside her window. The tree is about 3 stories tall, and the Tia, after chasing a squirrel, is stuck on a branch at about the height of 2 stories of her house. The height to the branch may be 20 feet, and the firefighters have to put the ladder about 10 feet away from the Oak Tree in order to go around Ann‘s shed. How long of a ladder do the firefighters need in order to rescue Ann‘s cat? a^2 + b^2 = c^2 or, respectively: (20)^2 + (10)^2 = 2^2, the length of ladder required. 400 + 100 = 500. The square root of 500 is approximately 22.37. The firefighters extend their expandable ladder to be approximately 23 feet, whereas they need at least 22.37 feet to safely reach the Oak branch. Ann’s cat Tia is rescued and All are Happy!

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Teaching Math: A Breeze when Incorporating Favorite Interests

Teaching math in the Elementary classroom can be a challenge, as all elementary math teachers know, but when favorite activities of students are incorporated into the curriculum, it can be a breeze. When students have a required learning skill to attain, and they incorporate their favorite interests, then there is inspiration – and the student becomes “self-motivated”.    This is a great First Item to address with the rounds of new students returning for the fall season.  Find out Learners’ Favorites, and keep the list in a special file.

Then, as the year progresses, if a student is having a challenge learning a particular new math lesson, Teachers can refer to the student’s personal file of “Favorites Activities List”.  At this time, then, introducing the association and how that interest relates to the new lesson.

Example No. 1:  Young Matthew enjoys playing or watching the game of baseball. That is included in their list of favorite activities.  November rolls around, and the lesson of drawing shapes in geometry arises, but Matthew is not grasping the concepts.  If looking down onto a Baseball Diamond from an aerial perspective, the shape of Square is easily seen in the formation of the 4 bases on the ball field.  As well, the shape of the bases individually, is a square.

Then show in sequence what happens when the player runs to first base, second, third and fourth, demonstrating the making of a Straight Line 4 times, and in consecutive order. Within each corner, while the player stands on the base, the player looks down at both straight lines that connect, and the player can then see a perfect Right Angle of 90 degrees.  Drawing a line across from the base to the left to the base t the right demonstrates a perfect Right Triangle.

Suddenly a light is switched on in the child’s brain, and Matthew is on the way to understanding the concept of geometry.  Not only do they understand it on paper in 2-D form, but now in 3-D form, in the context of a baseball game, in a real life scenario.

Example No. 2:
  Ashley likes archery.  Archery is included in her favorite activities list.  While imaging and practicing her archery skills, she sees concentric circles – one inside the other.  When a line is drawn from her bow to the target, she demonstrates a perfect straight line.  Hence, she has a different yet equally effective association of a favorite interest to relate to the concepts in geometry –  as Mathew’s love of baseball.  Imagine now that Susie, not only is attaining the required skills the in geometry lesson, but is also having fun while doing it, and has developed self-motivation and interest in learning math.

In these examples, both sides of the Brain are exercised, (Left Right Brain Learning and Thinking) new neural connectors and dendrites grow, and you have encouraged the growth of a healthy developing young brain.

Start the school year off right, and find out what your students’ favorite activities are. Keep the lists on file, and refer to them from time to time during the school year.  You may be surprised at the effectiveness of this subtle teaching tool.

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Flower Geometry on Summer Vacation

Summer can be fun Learning Math on your camping trip or at home in your backyard.  An interesting approach to learning geometry in elementary math is by the study of flowers found in your backyard (or schoolyard in Spring or Fall).  Have your class walk around the local grounds, or give them an assignment to go home and document in their notebooks, different flowers, counting the number of petals in the flowers.

A second item, if time permits is to have the Learners identify the flowers as well.  Upon arrival back in the classroom, Learners identify, with their geometry charts, which geometric pattern or shape the flower has grown into.



Eyes  (peepers for finding flowers)

Camera, if available


Pencil and Good Eraser

Pencil Crayons in various colors


Compass (if you wish to measure angles in the shapes)



Next, Learners draw in their notebooks the geometric shape the flower is equated with, and beside the shape, a simple drawing of the flower, coloring the flower drawing with the corresponding color of the petals.  If possible, 3-D forms can be cut out and interlocked together, with a string glued into the top of the start and made into Christmas ornaments.


Yellow Blue-eyed Grass:  (photo Above) 6-Petaled Yellow Wild Flower (that also grows in Bluish Purple and White); found in tall grasses who / that opens up only with the sun, and closes at the end of the day when the sun sets, or on cloudy days.

Geometric pattern:  6-pointed Star Tetrahedron; Two 3-Dimensional Interlocking Equilateral Triangles with a conjoining dot in the middle.  These 2 photos show the star tetrahedron (6-pointed) both in 2-dimensional form (as it would be if drawn flat on a piece of paper).  The second photo is a rendition of a 3-dimensional form (as if it were hanging as an ornament in a tree).   SourceURL:file:///Users/sheila/Desktop/Summer%20Flower%20Geometry.doc

When looking, aim for the pattern that is found when counting the Number of Petals in the Flowers.  In the Yellow Blue-Eyed Grass, there are 6 petals which if gazed at in a 3-Dimensional way, one can see the pattern of the 6-pointed star tetrahedron.

This can be a fun activity to do while on summer vacation – or during the schoolyear in Spring and Fall, and depending what climate area your school is, it can be done during winter as well.

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Geometry Practice Questions

Geometry questions are very common on high school exit exams, some nursing entrance exams, and college entrance exams.  Topics vary, but most of the following are covered:

– identifying solid figures
– solving problems with solid and plane figures
– solving problems using Pythagorean principles
– solve problems using scale drawings
– calculating area, circumference, volume and perimeter
– solve problems using geometric transformations

Geometry Games, Glossaries, and Facts!

My son is starting a new geometry unit. I want to share with you the best Web sites I have found for kids. On the next geometry blog, I will share photos of handmade geometry books I am in the process of making, such as a circle book and a spinner book to teach other geometry facts. Great geometry Web sites for teaching kids: Diana Dell: Geometry Vocabulary – This is lesson one of my very favorite geometry site! Children can hear it, see it, and play geometry games!! Diana Dell: Geometry Uni

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Geometry Games, Glossaries, and Facts!