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GEOMETRY

Counting  Arithmetic  Fractions  Graphing  Algebra  Geometry

Polygons Area Perimeter

Circles Area Arc Radius Pi

Distance Between Two Points

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Geometry starts with identifying shapes of 8 regular polygons and calulating perimeters. Examples showing how to find area and perimeter are helpful. A composite polygons tool generates endless for practice with these calculations. The Unitland Activity requires printing an Activity sheet for each student having them graph (X,Y) Coordinates which make up the shape, then breaking up the polygon into squares and rectangles to calculate the Area and Perimeter.

Circles Arcs Areas Radius and Pi, Examples defining terms and formulas may be reviewed first. The Circles tool in Easy mode and the Spheres tool operate similarly, generating a radius or diameter and asking for: arcs, areas and volume, rounded to 2 decimals. The circles tool in hard mode generates a circle with 6 elements, one is given, students find the other 5 elements, answering in terms of Pi → no calculators! The Steps around half circle and Pi Mosaic Activities may be used anytime.

Distance Between Two Points features a tool generating two points on a grid asking for the distance between them rounded to 2 decimals, detailed solutions are shown for each set of points. An algebraic example shows step-by-step operations to solve for distance and a geometric example shows a proof of the Pythagorean Theorem → A²+B²=C².

Play Section Overview


Area of a Square

Area of a Triangle

Area of a Rectangle

Area of a Composite

← BASE →← HEIGHT → 6u6uArea = Base • Height Area = 6u • 6uArea = 36u² ← BASE →← HEIGHT → 7u4uArea = Base • Height Area = 7u • 4uArea = 28u² ← BASE →← HEIGHT → 7u4uArea = ½ (Base • Height) Area = ½ • 7u • 4uArea = 14u² Squares and Rectangles: Area= b•hTriangles: Area= ½ • b•h 7u6u3 • 6 = 182•6=122•2=4Total Area = 18 + 12 + 4 = 34u²

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How to Calculate Distance Between Two Points
The Algebra The Geometry

Distance =

(Answer to 2 Decimal Places)

-10-9-8-7-6-5-4-3-2-11234567891010987654321-1-2-3-4-5-6-7-8-9-10

Find Length C with endpoints A and B

ABC Lines Parallel to Grid Lines can be Measured Diagonal Lines Are Calculated A² + B² = C² ← 7 →↑1.5↓ (7)² + (1.5)² = C²49 + 2.25 = C²C = √51.25C = 7.16 units

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Distance Worksheet / Key

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Find Length C with endpoints a b

1: Draw a Square with Area = C²

2: Draw outer square touching C at corners

Cab Area=C² NOTICE 4 Triangles Legs a and b Length=(a + b) Area=(a + b)² Area = C²+ 4 Triangles Pythagorean Theorem abba abba Area = (A + B)² Area = C² + (4) (½) ab

(A + B)² = C² + (4) ½ AB

A² + 2AB + B² = C² + 2AB

A² + B² = C²

Try Some Calculations

Quadralaterals have 4 Sides

7 Types shown below

Square 4 equal sides and all 90° angles Rectangle Opposite sides equal and all 90° angles Rhombus All sides equal and opposite sides parallel Kite Adjacent pairs of sides equal Parallelogram Opposite sides are parallel Trapezoid Two sides are parallel Complex Sides intersect each other

Steps to determine Perimeter and Area

Perimeter:

• Determine missing side lengths

• Add all side lengths together

Area:

• Section polygon into squares and rectangles

• Determine the area of each section

• Add all areas together



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How To Calculate Areas

     Area = u²

Perimeter = u

All Spheres have a Center, the distance from the Center to the edge is a Radius. Surface Area and Volume, are related to Radius in the following formulas:

Surface Area = 4 π r²

Volume   = 4/3 π r³

3 Examples

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  Goto Circles
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Spheres Tutorial


Print Activity Sheet

Surface Area and Volume

Calculate Surface Area and Volume for each Sphere
or Ball, using the given Radius in inches.

Pool = 1.13Bowling = 4.13Beach = 14.00

Surface Area = 4 π R²

Volume = 4/3 π R³

Pool Ball

Surface Area = (4) (3.14) (1.13)² = 16.04 in²

Volume = (4/3) (3.14) (1.13)³ = 6.05 in³

Bowling Ball

Surface Area = (4) (3.14) (4.13)² = 214.23 in²

Volume = (4/3) (3.14) (4.13)³ = 295.16 in³

Beach Ball

Surface Area = (4) (3.14) (14.00)² = 2,461.76in²

Volume = (4/3) (3.14) (14.00)³ = 11,497.36 in³

CONGRATULATIONS

You are a new developer in Unitland


Your first assignment is to calculate the area and perimeter, of the town center. Engineering has supplied you coordinates of the shape, you must graph them and calculate, Area and Perimeter.

Continue

Print Activity Sheet

TOWN'S LOGO

051015202530051015202530 422222521225246 (X,Y) Points (3,26)(3,28)(15,28)(15,26)(10,26)(10,2)(4,2)(4,6)(6,6)(6,4)(8,4)(8,26)(3,26) Plot the Points Connect the Dots Calculate Perimeter Calculate Area Print Answer Key Reset 4 +  2 + 2 +2 + 22 + 5 +2 + 12 + 2 +5 + 24 + 688 unitsPERIMETER AREA • 12 x 2 = 24 • 22 x 2 = 44 • 6 x 2 = 12 • 2 x 2 = 4 24+44+12+484 units² (3,26)•(3,28)• • (15,28)• (15,26)•(10,26)•(10,2)•(4,2)•(4,6)•(6,6)•(6,4)•(8,4)•(8,26)

standard hyperbola

(x-h)² - (y-k)² = 1

 a²        b²    

• • • • • Line of Symmetry y=k y = (a/b) x y = -(a/b) x asymptotes branch vertices (h-a,k) (h+a,k) focus focus (h,k+c) (h,k-c) origin (h,k) Transverse Axis x=h

The Central Hyperbola
h=k=0 & a=b

central hyperbola

h=k=0 & a=b


(x)² - (y)² = 1

 a²    b²    

• • • • • Line of Symmetry y-axis y = x y = -x asymptotes branch vertices (-a,0) (a,0) focus focus (0,c) (0,-c) origin (0,0) Transverse Axis x-axis

The Geometric Definition

The geometric definition of a hyperbola

Two identical curves such that the difference between any point on the curve and the foci is a constant

• • • • • • focus focus

The Geometric Definition

There are seven shapes for students to identify by clicking on the shape name.

In harder mode students calculate a perimeter for the shapes

POLYGONS

 Square Rectangle   TriangleHexagon

 RhombusParallagram PentagonOctagon

←90° Angles ←90° Angles

4 sides equal length, all 90° angles
Opposite sides equal length, all 90° angles
Parallelagram with 4 sides equal length
Adjcent pairs of sides equal
Opposite sides are parallel
Triangles have 3 sides : More Info
Five equal length sides
Six equal length sides
Eight equal length sides


    Shape: Sqr   Rect   Tri   Pent   Hex   Oct
  Identify: 0/0   0/0   0/0   0/0   0/0   0/0
Perimeter:0/0   0/0   0/0   0/0   0/0   0/0
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Identify Shapes
Calculate Perimeter
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Areas and Perimeters

8 Regular Polygons

Composite Polygons

Unit Land Activity

Below are 3.14 Squares with side length R. The Circle has raduis R, so the area of squares equals area of circle.

1.Decorate squares    2.Cut out    3.Paste into circle



Radius = R Area = (3.14) * R²← R → R²R²R².14 R² →

Print Activity Sheet

If it takes 10 steps to walk from center of a circle to edge,
How many steps should it take to walk half way around the circle ?

Steps Around the Half Circle

Answer -> 10 π = 31.4

--- 10 Steps →- Radius Steps →← How Many Steps to walk ½ Circle? →

Using a large circle like the ones at the center of a court or field, have students walk from the center to an edge, counting their steps. This is the number of Radius Steps.

How many same-sized steps would it take to walk half the circle? The answer should be Pi (3.14) times the Radius Steps.

Try It !


Radius and Pi (π) Area Circumference
π is ratio of Arc to Radius Pi = 3.14159... ← Radius (r) → 1 ← ½Arc = 3.14 → Area is Inside the Circle Area = π R² 2 π R Circumference ← is around the outside →

Pi Mosaic

Steps Around ½ Circle

Easy Mode : Generates a Radius or Diameter and asks students to calculate Circumference and Area using 3.14 for Pi and The following equations and entering answers rounded to 2 decimals.

Circumference = 2 π R

Area = π R²

Hard Mode : Generates 1 of 6 components either, Radius, Diameter, ½ Circumference, ¼ Circumference, ½ Area or ¼ Area, students find other values, answer in terms of Pi, no calculators!

Run Score Page
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Easy   Hard
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Circles Tutorial