One-, Two-, Three-Dimensional Shapes Duane B. Karlin CEP 811 June 12, 2011
What is DIMENSION ? Dimension is a measure in one direction. What is GEOMETRY ? Geometry is the study of shapes. Geometric figures can have one, two, or three dimensions.
MEASUREMENTS c an be in U.S. STANDARD o r METRIC. U.S. STANDARD : inches, feet, yards, miles METRIC : meter, decimeter, centimeter, millimeter 12 inches = 1 foot 3 feet = 1 yard 1,760 yards = 1 mile 1 meter = 10 decimeters = 100 centimeters = 1,000 millimeters U.S. STANDARD conversions are trickier to memorize because they do not have a common converting number. METRIC conversions are easier to understand because they are multiples of 10.
READY TO LEARN ABOUT… One-dimensional shapes? Two-dimensional shapes? Three-dimensional shapes? Or are you ready to TEST YOUR KNOWLEDGE?TEST YOUR KNOWLEDGE?
One-dimensional shapes are measured in only one direction. This is defined as the LENGTH. LINES are a one-dimensional shape. One-Dimensional Shapes
Two-Dimensional Shapes Two-dimensional shapes can be measured in two directions. Their measurements are LENGTH (or BASE) and WIDTH (or HEIGHT). Click on a shape or capital word to learn more. The distance around is PERIMETER. PERIMETER The enclosed space is AREA. AREA Want a hint about INTERIOR ANGLES? INTERIOR ANGLES
CENTER Center CENTER: the middle of a circle. It is the same distance from the center to any point on the circle.
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 CIRCUMFERENCE: the distance around a circle.
CIRCUMFERENCE = 2πr π = 3.14 r = radius CIRCUMFERENCE, instead of PERIMETER, is used to measure the distance around a CIRCLE. 3 inches C = 2 x 3.14 x 3 C = 6.28 x 3 C = 18.84 CIRCUMFERENCE = 18.84 inches
AREA of a CIRCLE is the INTERIOR space. AREA = πr 2 3 inches A = 3.14 x 3 2 A = 3.14 x 3 x 3 A = 3.14 x 9 A = 28.26 AREA = 28.26 square inches
TRIANGLE 3 sides 3 interior angles The sum of the 3 interior angles always equal 180°. The prefix “TRI-” means 3. INTERIOR means inside.
BASE HEIGHT AREA of a TRIANGLE = ½ BASE (b) x HEIGHT (h) A = ½b x h (6 inches) A = ½ x 6 x 6 A = 3 x 6 A = 18 square inches This formula works for ALL TRIANGLES.
EquilateralIsoscelesScalene RightAcuteObtuse 6 types of TRIANGLES. Click on a shape to learn more, or learn about AREA.AREA
EQUILATERAL TRIANGLE All interior angles equal 60°. All three sides are the same length. (60° + 60° + 60° = 180°) 60°
ISOSCELES TRIANGLE Two sides are equal. The angles opposite of the equal sides are also equal. REMEMBER: the sum of the interior angles will always equal 180° in a triangle.
SCALENE TRIANGLE All three sides are different lengths. All interior angles are different, but they still equal 180°.
RIGHT TRIANGLE One angle, opposite the longest side, measures 90°. It is signified by the ☐ symbol.
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.
OBTUSE TRIANGLE One interior angle in an obtuse triangle is greater than 90°.
QUADRILATERALS The prefix “QUAD-” means 4, as in a 4-sided figure or shape. Click on a shape to learn more.
PERIMETER of any shape is calculated by adding the sides together. PERIMETER = distance around a shape 3 inches PERIMETER = 3 + 3 + 3 + 3 P = 12 inches
AREA of a QUADRILATERAL is calculated by multiplying the Length (or Base) by the Width (or Height). AREA = square units it takes to fill a shape 3 inches AREA = 3 x 3 A = 9 square inches
SQUARE All 4 sides are equal and parallel. Parallel means the lines always maintain the same distance apart. Parallel lines will never touch. All interior angles equal 90°. REMEMBER: A square is a rectangle, but a rectangle is not a square!
RECTANGLE Opposite sides are equal and parallel. All interior angles equal 90°.
RHOMBUS, or DIAMOND A special type of PARALLOGRAM.All 4 sides are equal and parallel. Interior angles equal 90°.
PARALLELOGRAM Opposite sides are equal and parallel. Opposite angles are equal.
Area = ½ x (b1 + b2) x h AREA OF A TRAPEZOID = ½ x (BASE 1 + BASE 2) x HEIGHT 15 inches 10 inches 5 inches A = ½ x (15 + 10) 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°. An OCTAGON, 8-sided shape, has 180 x (8 – 2) = 180 x 6 = 1080°. A HEXAGON, 6-sided shape, has 180 x (6 – 2) = 180 x 4 = 720°.
SHAPES WITH MORE THAN 4 SIDES Click on a shape to learn more.
PENTAGON No parallel sides. All 5 sides can be equal, but they don’t have to be. Interior angles all equal 540°. The prefix “PENTA-” means 5. If each side is equal, then each interior angle equals 108°.
AREA of a PENTAGON Divide the pentagon into 5 equal triangles. Divide those triangles in half. You now have 10 right angle triangles. The formula for finding the area of a triangle is A = ½ b x h A = ½ x 3 x 5 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. A = 7.5 x 10 AREA = 75 square inches BASE = 3 inches HEIGHT = 5 inches
HEXAGON Parallel sides are opposite each other. The prefix “HEXA-” means 6. Interior angles all equal 720°. 3 pairs of parallel sides. If each side is equal, which they do not have to be, then each interior angle equals 120°.
OCTAGON The prefix “OCTA-” means 8. Interior angles all equal 1080°. 4 pairs of parallel sides. Parallel sides are opposite each other. If each side is equal, which they may or may not be, then each interior angle equals 135°.
Three-Dimensional Shapes Three-dimensional shapes are measured in three directions: length, width, and height. Three-dimensional shapes also have FACES, VERTICES, and EDGES. FACESVERTICES EDGES Click on a shape or capital word to learn more.
FACES FACES refers to the sides of a shape. In this example, the CUBE has 6 faces, but we can only see 3. REMEMBER: In a three-dimensional shape, you may not always be able to see all of the faces (sides) of the shape.
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! 3 inches SURFACE AREA = (L x W) x 6 = (3 x 3) x 6 = 9 x 6 SURFACE AREA = 54 square inches SURFACE AREA is the measurement we would use to cover the outside of the shape, like a wrapped package.
CUBE To find the VOLUME of a shape, use this formula: Length x Width x Height. VOLUME is the amount of space a three-dimensional shape occupies. VOLUME = L x W x H 4 inches VOLUME = 4 x 4 x 4 VOLUME = 64 cubic inches HINT: “CUBIC” measurement is used with volume because 64 equal-sized cubes would fit into the shape.
SPHERE To find the SURFACE AREA of a sphere, use this formula: SURFACE AREA = 4πr 2 8 inches DIAMETER = 8 inches, so the RADIUS equals 4 inches. = 4π4 2 = 4π(4 x 4) = 4π(16) =12.56 x 16 SURFACE AREA = 200.96 square inches Ready to learn about the VOLUME of a SPHERE?
SPHERE 8 inches To calculate the VOLUME of a SPHERE, things get a little tricky. VOLUME = 4/3 πr 3 = 4/3 π (4 x 4 x 4) = 4/3 x π x 64 = 4.187 x 64 VOLUME = 267.95 cubic inches The RADIUS is half of the DIAMETER, so half of 8 is 4.
CYLINDER 2 inches 6 inches A CYLINDER is actually two circles (one on the top and one on the bottom) and a rectangle in the middle. If we cut the middle and lay it flat, it would form a rectangle. Click on the dotted line to see what the cylinder would look like if it was “dissected.”
To see the CYLINDER in this shape makes calculating the SURFACE AREA easier to understand. SURFACE AREA = 2πr 2 + 2πrh CYLINDER The formula looks confusing, but it is simply finding the surface area of two circles and one rectangle. 2 inches 6 inches The circumference of the circle actually forms the base of the rectangle. = 2π2 2 + 2π2 x 6 = 2π4 + 2π12 = 6.28 x 4 + 6.28 x 12 = 25.12 + 75.36 SURFACE AREA = 100.48 square inches
CYLINDER To calculate the VOLUME of a CYLINDER, use this formula: V = πr 2 h 2 inches 6 inches V = π x 2 2 x 6 V = π x 4 x 6 V = π x 24 V = 75.36 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.
TEST YOUR KNOWLEDGE OF SHAPES QUESTION 1 How many dimensions does a line have? O NE T WO T HREE A S 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 = πr 2
QUESTION 3 How many faces does a cylinder have? ThreeTwoFiveEight
QUESTION 4 On a three-dimensional shape, what is it called where two or more points meet? FaceVertexMysteryParty
QUESTION 5 How many parallel sides are on a pentagon? 5320
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. TRUEFALSE
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? 19232515
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