# Three-Dimensional Shapes

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Three-Dimensional Shapes
One-, Two-, Three-Dimensional Shapes State of Montana Math Standards: Number Sense and Operation Content Standard 1: 1.4 Common Fractions and Decimals: Identify and model common fractions such as, tenths, fourths, thirds, and halves; and decimals such as money and place value to 0.001; and recognize and compare equivalent representations; 1.5 Length, Time, and Temperature: Select and apply appropriate standard units and tools to measure length, time, and temperature within relevant scientific and cultural situations, including those of Montana American Indians. Geometric Reasoning Content Standard 3: 3.1 Two-Dimensional Attributes: Describe, compare, and analyze attributes of two-dimensional shapes; 3.2 Three-Dimensional Attributes: Describe attributes of three-dimensional shapes such as cubes and other rectangular prisms, cylinders, cones, and spheres; 3.4 Linear Measurement: Estimate and measure linear attributes of objects in metric units such as centimeters and meters and customary units such as inch, foot, and yard; 3.5 Area and Perimeter: Define and determine area and perimeter of common polygons using concrete tools such as grid paper, objects, or technology and justify the strategy used. Algebraic and Functional Reasoning Content Standard 4: 4.1 Patterns and Relations: Describe, extend, and make generalizations about geometric or numeric patterns Duane B. Karlin CEP 811 June 12, 2011

What is GEOMETRY? Geometry is the study of shapes. Geometric figures can have one, two, or three dimensions. What is DIMENSION? Dimension is a measure in one direction.

MEASUREMENTS can be in U.S. STANDARD or METRIC.
U.S. STANDARD: inches, feet, yards, miles 12 inches = 1 foot 3 feet = 1 yard 1,760 yards = 1 mile U.S. STANDARD conversions are trickier to memorize because they do not have a common converting number. METRIC: meter, decimeter, centimeter, millimeter 1 meter = 10 decimeters = 100 centimeters = 1,000 millimeters METRIC conversions are easier to understand because they are multiples of 10.

One-Dimensional Shapes
One-dimensional shapes are measured in only one direction. This is defined as the LENGTH. LINES are a one-dimensional shape.

Two-Dimensional Shapes
Two-dimensional shapes can be measured in two directions. Their measurements are LENGTH (or BASE) and WIDTH (or HEIGHT). The distance around is PERIMETER. The enclosed space is AREA. Want a hint about INTERIOR ANGLES? Click on a shape or capital word to learn more.

CENTER CENTER: the middle of a circle. It is the same distance
from the center to any point on the circle. Center

DIAMETER Diameter DIAMETER: a line segment that passes through the center of a circle and has its endpoints on opposite sides of the circle.

Radius RADIUS RADIUS: a line segment with one endpoint at the center of a circle and the other endpoint on the circle.

CIRCUMFERENCE CIRCUMFERENCE: the distance around a circle.

CIRCUMFERENCE, instead of PERIMETER, is used to
measure the distance around a CIRCLE. CIRCUMFERENCE = 2πr π = 3.14 r = radius 3 inches C = 2 x 3.14 x 3 C = 6.28 x 3 C = 18.84 CIRCUMFERENCE = inches

AREA of a CIRCLE is the INTERIOR space.
AREA = πr2 A = 3.14 x 32 3 inches A = 3.14 x 3 x 3 3 inches A = 3.14 x 9 A = 28.26 AREA = square inches

TRIANGLE The prefix “TRI-” means 3. 3 interior angles 3 sides
INTERIOR means inside. The sum of the 3 interior angles always equal 180°.

AREA of a TRIANGLE = ½ BASE (b) x HEIGHT (h)
A = ½b x h A = ½ x 6 x 6 A = 3 x 6 A = 18 square inches HEIGHT (6 inches) BASE This formula works for ALL TRIANGLES. (6 inches)

6 types of TRIANGLES. Equilateral Isosceles Scalene Right Acute Obtuse Click on a shape to learn more, or learn about AREA.

EQUILATERAL TRIANGLE 60° All three sides are the same length.
All interior angles equal 60°. (60° + 60° + 60° = 180°) 60° 60° EQUILATERAL TRIANGLE

ISOSCELES TRIANGLE REMEMBER: the sum of the
interior angles will always equal 180° in a triangle. Two sides are equal. The angles opposite of the equal sides are also equal. ISOSCELES TRIANGLE

SCALENE TRIANGLE All three sides are different lengths.
All interior angles are different, but they still equal 180°. SCALENE TRIANGLE

RIGHT TRIANGLE One angle, opposite the longest side,
measures 90°. It is signified by the ☐ symbol. RIGHT TRIANGLE

ACUTE TRIANGLE All 3 interior angles are less than 90°.
Equilateral triangles are an example of an acute triangle, but not all acute triangles are equilateral triangles. ACUTE TRIANGLE

One interior angle in an obtuse triangle is greater than 90°.

PERIMETER = distance around a shape
P = 12 inches 3 inches 3 inches 3 inches 3 inches PERIMETER of any shape is calculated by adding the sides together.

AREA = square units it takes to fill a shape
AREA = 3 x 3 A = 9 square inches 3 inches 3 inches AREA of a QUADRILATERAL is calculated by multiplying the Length (or Base) by the Width (or Height).

SQUARE All 4 sides are equal and parallel.
All interior angles equal 90°. REMEMBER: A square is a rectangle, but a rectangle is not a square! SQUARE Parallel means the lines always maintain the same distance apart. Parallel lines will never touch.

RECTANGLE All interior angles equal 90°.
Opposite sides are equal and parallel.

RHOMBUS, or DIAMOND Interior angles equal 90°.
A special type of PARALLOGRAM. All 4 sides are equal and parallel.

PARALLELOGRAM Opposite sides are equal and parallel.
Opposite angles are equal.

Has one pair of parallel sides.
TRAPEZOID

AREA OF A TRAPEZOID = ½ x (BASE 1 + BASE 2) x HEIGHT
10 inches 5 inches 15 inches Area = ½ x (b1 + b2) x h A = ½ x ( ) x 5 A = ½ x (25) x 5 A = 12.5 x 5 AREA = 62.5 square inches

HINT! Remember, the number of degrees in any geometric shape is
180 x (N – 2), where “N” is equal to the number of sides. So, with a PENTAGON, 5-sided shape, we would write: 180 x (5 – 2) = 180 x 3 = 540, so the number of degrees in a PENTAGON is 540°. A HEXAGON, 6-sided shape, has 180 x (6 – 2) = 180 x 4 = 720°. An OCTAGON, 8-sided shape, has 180 x (8 – 2) = 180 x 6 = 1080°.

SHAPES WITH MORE THAN 4 SIDES

PENTAGON The prefix “PENTA-” means 5. No parallel sides.
If each side is equal, then each interior angle equals 108°. Interior angles all equal 540°. All 5 sides can be equal, but they don’t have to be.

AREA of a PENTAGON Divide the pentagon into 5 equal triangles. A = ½ x 3 x 5 Divide those triangles in half. A = 1.5 x 5 A = 7.5 But this is only the area for one triangle, so we need to multiply this number by the total number of triangles within the pentagon. BASE = 3 inches HEIGHT = 5 inches A = 7.5 x 10 You now have 10 right angle triangles. AREA = 75 square inches The formula for finding the area of a triangle is A = ½ b x h

HEXAGON The prefix “HEXA-” means 6. Interior angles all equal 720°.
If each side is equal, which they do not have to be, then each interior angle equals 120°. 3 pairs of parallel sides. Parallel sides are opposite each other.

OCTAGON The prefix “OCTA-” means 8. Interior angles all equal 1080°.
If each side is equal, which they may or may not be, then each interior angle equals 135°. 4 pairs of parallel sides. Parallel sides are opposite each other.

Three-Dimensional Shapes
Three-dimensional shapes are measured in three directions: length, width, and height. Three-dimensional shapes also have FACES, VERTICES, and EDGES. Click on a shape or capital word to learn more.

FACES REMEMBER: In a three-dimensional shape, you may not
always be able to see all of the faces (sides) of the shape. FACES refers to the sides of a shape. In this example, the CUBE has 6 faces, but we can only see 3.

VERTEX (singular), or VERTICES (plural)
A VERTEX is where two or more points meet; a corner. This example of a RECTANGULAR PRISM has 8 VERTICES. Once again, not every VERTEX may be visible in a three-dimensional shape.

EDGES The EDGE of a shape is the line where two surfaces meet.
This CYLINDER has 2 EDGES.

CUBE The CUBE has 6 sides, 8 vertices, and 12 edges.
To find the SURFACE AREA of a CUBE, find the area of one side (L x W), and then multiply by the total number of sides (6). Remember to count all the hidden sides! SURFACE AREA = (L x W) x 6 3 inches = (3 x 3) x 6 = 9 x 6 SURFACE AREA = 54 square inches 3 inches 3 inches SURFACE AREA is the measurement we would use to cover the outside of the shape, like a wrapped package.

CUBE VOLUME is the amount of space a three-dimensional shape occupies.
VOLUME = L x W x H VOLUME = 4 x 4 x 4 VOLUME = 64 cubic inches 4 inches HINT: “CUBIC” measurement is used with volume because 64 equal-sized cubes would fit into the shape. 4 inches 4 inches To find the VOLUME of a shape, use this formula: Length x Width x Height.

SPHERE DIAMETER = 8 inches, so the RADIUS equals 4 inches.
To find the SURFACE AREA of a sphere, use this formula: SURFACE AREA = 4πr2 = 4π42 8 inches = 4π(4 x 4) = 4π(16) =12.56 x 16 SURFACE AREA = square inches Ready to learn about the VOLUME of a SPHERE?

SPHERE To calculate the VOLUME of a SPHERE,
things get a little tricky. VOLUME = 4/3 πr3 = 4/3 π (4 x 4 x 4) = 4/3 x π x 64 8 inches = x 64 VOLUME = cubic inches The RADIUS is half of the DIAMETER, so half of 8 is 4.

CYLINDER 2 inches If we cut the middle and lay it flat, it would
form a rectangle. 6 inches Click on the dotted line to see what the cylinder would look like if it was “dissected.” A CYLINDER is actually two circles (one on the top and one on the bottom) and a rectangle in the middle.

CYLINDER To see the CYLINDER in this shape
makes calculating the SURFACE AREA easier to understand. The formula looks confusing, but it is simply finding the surface area of two circles and one rectangle. SURFACE AREA = square inches 6 inches The circumference of the circle actually forms the base of the rectangle. SURFACE AREA = 2πr2 + 2πrh = 2π22 + 2π2 x 6 2 inches = 2π4 + 2π12 = 6.28 x x 12 =

CYLINDER 2 inches To calculate the VOLUME of a CYLINDER, use this
formula: V = πr2h 6 inches V = π x 22 x 6 V = π x 4 x 6 V = π x 24 V = cubic inches

RECTANGULAR PRISM The RECTANGULAR PRISM has
6 sides, 8 vertices, and 12 faces. To calculate the SURFACE AREA or VOLUME or the RECTANGULAR PRISM, use the same formula as you would for the CUBE.

QUESTION 1 How many dimensions does a line have? One Two Three As many as it needs

QUESTION 2 Which of the following formulas would be used to calculate the area of a trapezoid? A = ½ B x H A = L x W A = ½ (Base 1 + Base 2) x Height A = πr2

How many faces does a cylinder have?
QUESTION 3 How many faces does a cylinder have? Three Two Five Eight

On a three-dimensional shape, what is it called
QUESTION 4 On a three-dimensional shape, what is it called where two or more points meet? Face Vertex Mystery Party

QUESTION 5 How many parallel sides are on a pentagon? 5 3 2

QUESTION 6 Which of these figures is a scalene triangle?

QUESTION 7 True or false? A square is a rectangle and a rectangle is a square. TRUE FALSE

QUESTION 8 What is geometry? The study of numbers.
The study of shapes. An example of counting. What the acorn said when it grew up.

QUESTION 9 If I had a quadrilateral, two octagons, and a triangle, how many sides would I have? 19 23 25 15

QUESTION 10 WHICH FORMULA WILL HELP ME FIGURE OUT HOW MANY DEGREES
ARE IN ANY GIVEN GEOMETRIC SHAPE? 180 x (number of sides - 2) ½ Base x Height x the number of sides 2πr add the number of sides together

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Your knowledge of shapes is out of this world!
CONGRATULATIONS! Your knowledge of shapes is out of this world!