Measurement Project Concept Map

Jasmine- Geometric FiguresAdam- Surface area, volume, pythagorean theorem Kylie- Area and Perimeter

Pythagorean Theorem

Pythagorean Theorem: a^2 + b^2 = c^2

-The Pythagorean Theorem is an equation devised to find the slope opposite the 90 degree angle or right Triangles. A^2 + B^2 = C^2 is the relationship between the sides of the triangle in regard to length.-This equation only works on right triangles- A is the height, B is the width and C is the length of the slope.

The Pythagorean Theorem Is an equation devided to find the length of the slope opposite the 90 degree angle in right triangles. A^2 + B^2 = C^2 is the relationship in length between the sides of the triangle.

For this example lets suppose that A=5 and B=12.

To solve for C: Plug in the numbers you can into the equation A^2 + B^2 = C^2. Since we know A and B we can plug those in, gic=ving you 5^2 + 12^2 = C^. This can be simplified to be 169 = C^2. Taking the square root of both 169 and C^2 you are left with the answer: C = 18.

Geometric Figures

Angles

Definition of an Angle: A shape, formed by two lines or rays diverging from a common point known as the vertex.( www.mathopenref.com )< = angle

Angle Measure

- the number of degrees of turn to rotate about the vertex (middle point) -Measured in degrees

Supplementary angles

when the sum of two
angles measure
180 degrees

Complimentary angles

when the sum of two
angles measure 90
degrees.

Congruent angles

When two angles are congruent, they are also equal !

when two angles
have the same
exact measure

Vertical angles

- vertical angles are congruent

Two nonadjacent angles
formed by two
intersecting lines

Classification of angles

Alternate- Interior- Angles

The pair of sides on the opposite of the transversal, but inside the two lines (mathisfun.com).

Alternate-exterior-Angles

The pair of angles on the opposite sides of the transversal but outside the two lines (mathisfun.com).

Corresponding angles

corresponding angles property: - when two parallel lines are cut by a transversal (intersecting line), then their corresponding angles have the same measure-if two lines in the plane are intersected by a transversal and some pair of their corresponding angles has the same measures, then the lines are parallel

The sum of angle measures in Triangles

the sum of all angles in
a triangle is 180 degrees.

Classification of Polygons

Types of Triangles

Classification by angle measure

Acute

all three interior angles are less than 90 degrees

Right

one angle is a right angle (90 degrees)

Obtuse

if one angle has a measure more than 90 degrees.

Classification by side lengths

Scalene

all three sides have a different length ( no two sides are congruent)

Equilateral

all three sides have the same length and are congruent

Isosceles

When two sides have the same length

Types of Quadrilaterals

A quadilateral means "four sides" so is any four sided shape

kite- two distinct pairs of congruent adjacent sides trapezoid- at least one pair of parallel sides isosceles trapezoid: a p[air of congruent angles along with one of the parallel sidesparallelogram- each pair of opposite sides is parallel rhombus- all of the sides are the same lengthrectangle- a right anglesquare- all sides have equal length

Circles

- a circle is a two- dimensional shape (mathisfun.com)-the set of all points that are an equal distance from the center

Regular Polygons

polygons that include
some degree of symmetry

Regular polygon- a convex polygon that is both equiangular and equilateral equilateral- a polygon with all congruent sidesequiangular- a polygon with all equal angles

How to find the measures of a Polygon

in a regular n-gon ( n = number of sides)(x = multiply)- each interior angles has a measure (n - 2) x 180 degrees/ n - each exterior angle has measure 360 degrees / n- each central angle has measure 360 degrees / n

Prisms and Pyramids

Prisms

Prism - a simple closed surface that consists of two congruent polygons which are in parallel planes (bases) and the lateral faces joining the bases, which are parallelograms * Two congruent faces (the bases)*Have to be parallel planes*other faces are bound by parallelogramstwo dimensionsal vs. three dimensional:- a prism is three dimensional and belongs on two planes in space, whereas a two dimensional figure only has one plane.

right prism

if the lateral faces of a prism are all rectangles

oblique prism

the edges between the bases are not perpendicular to the plane of the base

Pyramids

-a simple closed surface given by a polygon (its base) and a point not in the plane of the polygon, called its apex or vertex.-the pyrmaid is the union of the base with all of the triangular faces that rise from the base edges to the apex. *Polyhedron determined by polygon ( example. rectangular base = rectangular pyramid) *points not in the plane*does not have two congruent faces

A pyramid can have any polygonal region as its base !

Surface Area Polyhedrons

Surface area: The total area of the surface of a three-dimensional object

Polyhedrons are the 3-D shapes that are made up of several flat faces. Some Examples are in the Pyramids and Prisms section.

Surface Area Equations

Prism: 2B + ph

Pyramid: B + (1/2)ph

Key: B=Area of th base, p=Perimeter of the base, h=height

For this example the height(h) is 4, the length(L) is 5 and the width(w) is 3.

The first thing that needs to be done is finding the area and perimeter of this shape's base. The perimeter of the base is 2(5) + 2(3)= 16 = p The Area of the base is 5 x 3 = 15 = B Remember our equation for finding the surface area of prisms is 2B + ph. Since we know all the variables we can plug them in now. 2(15) + 16(4) - 30 + 64 - 94 u^2 = Surface area

Main topic

Volume of Polyhedrons

Volume is the measure of the space inside a three-dimentional figure.

Polyhedrons are the 3-D shapes that are made up of several flat faces. Some Examples are in the Pyramids and Prisms section.

How to find the volume of Polyhedrons

Prisms: The area of the base multiplied by the height. Equation: B x h B=Area of base h=height

Pyramids: One third of the area of the base times the height. Equation: (1/3) x B x h B=Area of base h=height

For This Example h=5, w=4, L=6

The first thing we have to do in order to solve this problem is to fing the area of the base. To do that we need to multiply Lxh. 6x4=24. 24 is now B. Remember the equation to find the volume of pyramids is (1/3)xBxh=V. we can now plug in B and h and proceed to find the volume. (1/3)x24x5 is what our equation looks like now. multiplying that out gives us 40. 40 u^3 is the volume of this pyramid

Area and Perimeter

Area

TRAPIZOID
To find the area of a trapizoid you take the sum of the bases and then multiply by the height of the trapizoid, and then you divivde the number by two. Viola!

RECTANGLE
To find the area if a rectangle you multiply length times width

CIRCLE
To find the area of a circle you use the formula
Area=pie*diameter or if you just have the radius then the formula is Area=pie*radius*radius

Perimeter

TRIANGLE
To find the perimeter of a triangle you simply add up all of the sides if all of the sides lengths are given. If on the other hand the triangle has a missing sign you then use the pythagorean Theorem which looks like this
A2+B2=C2

CIRCLE
To find the perimeter of a circle you use the formula
Circumfrence= Pie*diameter or Circumfrence=Pie*Radius *radius

SQUARE
To find the peimeter of a square all you have to do is multiply what ever number is shown for the sides by four because all of the sides on a square are the same size.