UNIT 4 Notes
11-1 – 11-3 Formulas:
Area Formulas: measured in squared units.
Parallelogram: A = bh
Area of an Equilateral Triangle: A = 1/4s2
Trapezoid: A = 1/2h(b1 + b2)
Rhombus: A = 1/2d1d2 d1 & d2 are the lengths of the diagonals
Circle: A = πr2
Circumference of a Circle: c = 2πr
Regular Polygon: A = 1/2Pa P = Perimeter, a = apothem
Triangle: A = 1/2bh
ORA = 1/2ab * SinC for the triangle below
12-1 Three Dimensional Figures:
Polyhedron: a solid with all flat surfaces that enclose a single region of space.
Face: each flat surface of a polyhedron.
Edges: line segments where faces intersect.
Prism: a polyhedron with two parallel congruent faces.
Pyramid: a polyhedron with all but one of the faces intersecting at one vertex.
Cylinder: a solid with congruent circular bases in a pair of parallel planes.
Cone: a solid with a circular base and a vertex.
Sphere: a set of points in space that are a given distance from a given point.
Cross Section: when a plane intersects a solid figure.
3-Dimensional Figures
Cone Cylinder Sphere
Triangular Pyramid Rectangular Pyramid Square Pyramid
Pentagonal Prism Square Prism Trapezoidal Prism
Rectangular Prism
12-2 Surface Area:
Surface Area: the sum of the areas of each face of a solid.
UNIT 4 Notes
12-3 Surface Areas of Prisms:
Prism: a polyhedron with two parallel congruent faces.
Bases: are congruent faces in parallel planes.
Lateral Faces: faces that are NOT bases.
Lateral Areas: is the sum of the areas of the lateral faces.
Lateral Area of a Prism: L = Ph
L = Lateral Area, P = Perimeter of the base, h = height
Surface Area of a Prism: T = L + 2B
T = Surface Area of a Prism, L = Lateral Area, B = area of the base.
12-4 Surface Areas of Cylinders:
Cylinder: a solid with congruent circular bases in a pair of parallel planes.
Lateral Area of a Cylinder: L = 2πrh
L = Lateral Area, r = radius of the base, h = height
Surface Area of a Cylinder: T = 2πrh + 2πr2
T = Surface Area, r = radius of the base, h = height
12-5 Surface Areas of Pyramids:
Pyramid: a polyhedron with all but one of the faces intersecting at one vertex.
Slant Height: the height of each lateral face.
Lateral Area of a Pyramid: L = 1/2Pl
L = Lateral Area, P = Perimeter, l = slant height
Surface Area of a Pyramid: T = 1/2Pl + B
T = Surface Area, P = Perimeter of the base, l = slant height, B = area of the base
12-6 Surface Areas of Cones:
Cone: a solid with a circular base and a vertex.
Slant Height: the height of each lateral face.
Lateral Area of a Cone: L = πrl
L = Lateral Area, r = radius, l = slant height
Surface Area of a Cone: T = πrl + πr2
T = Surface Area, r = radius, l = slant height
12-7 Surface Areas of Spheres:
Sphere: a set of points in space that are a given distance from a given point.
Great Circle: when a plane intersects a sphere through the center.
Hemisphere: The 2 halves that are formed when a sphere is intersected by a plane through the center.
Surface Area of a Sphere: T = 4πr2
T = Surface Area, r = radius
Surface Area of a Hemisphere: T = ½(4πr2) + πr2
T = Surface Area, r = radius
13-1 Volumes of Prisms & Cylinders:
Volume: of a figure is the measure of the amount of space that a figure encloses. It is measure in cubic units.
Volume of a Prism: V = Bh
V = Volume, B = the area of the base, h = height
Volume of a Cylinder: V = πr2h
V = Volume, r = radius of the base, h = height
13-2 Volumes of Pyramids & Cones:
Volume of a Pyramid: V = 1/3Bh
V = Volume, B = the area of the base, h = height
Volume of a Cone: V = 1/3πr2h
V = Volume, r = radius of the base, h = height
13-3 Volumes of Spheres:
Volume of a Sphere: V = 4/3πr3
V = Volume, B = the area of the base, h = height
Cross Sections ofThree Dimensional Figures:
Cross Section: when a plane intersects a solid figure.
The vertical cross section of a cylinder is a RECTANGLE.
The horizontal cross section of a cylinder is a CIRCLE.
The vertical cross section of a sphere is a CIRCLE.
The horizontal cross section of a sphere is a CIRCLE.
The vertical cross section of a cone is a TRIANGLE.
The horizontal cross section of a cone is a CIRCLE.
A CYLINDER is formed when you rotate a rectangle.
A CONE is formed when you rotate an equilateral or isosceles triangle.
A SPHERE is formed when you rotate a circle.
Density:
Density: A measure of how much matter is in a certain volume.
The measure of the relative heaviness of objects.
Describes how compact or crowded the material appears to be.
A ratio of mass to volume.
Examples:
A rock is more dense than a crumpled piece of paper.
A Styrofoam cup is less dense than a ceramic cup.
Density Formula:
d = m/V
d = Density, m = mass, V = Volume
Population Density = total area population _
land area in square units
Unit 4
Formulas
Area Formulas: measured in squared units.
Parallelogram: A = bh
Area of an Equilateral Triangle: A = 1/4 s2
Trapezoid: A = 1/2h(b1 + b2)
Rhombus: A = 1/2d1d2 d1 & d2 are the lengths of the diagonals
Circle: A = πr2
Circumference of a Circle: c = 2πr
Regular Polygon: A = 1/2Pa P = Perimeter, a = apothem
Triangle: A = 1/2bh
ORA = 1/2ab * SinC for the triangle below
Lateral Area of a Prism: L = Ph L = Lateral Area, P = Perimeter of the base, h = height
Surface Area of a Prism: T = L + 2B T = Surface Area of a Prism, L = Lateral Area, B = area of the base.
Lateral Area of a Cylinder: L = 2πrh L = Lateral Area, r = radius of the base, h = height
Surface Area of a Cylinder: T = 2πrh + 2πr2T = Surface Area, r = radius of the base, h = height
Lateral Area of a Pyramid: L = 1/2Pl L = Lateral Area, P = Perimeter, l = slant height
Surface Area of a Pyramid: T = 1/2Pl + BT = Surface Area, P = Perimeter of the base, l = slant height, B = area of the base
Lateral Area of a Cone: L = πrl L = Lateral Area, r = radius, l = slant height
Surface Area of a Cone: T = πrl + πr2T = Surface Area, r = radius, l = slant height
Surface Area of a Sphere: T = 4πr2T = Surface Area, r = radius
Surface Area of a Hemisphere: T = ½(4πr2) + πr2T = Surface Area, r = radius
Volume of a Prism: V = BhV = Volume, B = the area of the base, h = height
Volume of a Cylinder: V = πr2hV = Volume, r = radius of the base, h = height
Volume of a Pyramid: V = 1/3BhV = Volume, B = the area of the base, h = height
Volume of a Cone: V = 1/3πr2hV = Volume, r = radius of the base, h = height
Volume of a Sphere: V = 4/3πr3V = Volume, B = the area of the base, h = height
Density Formula:
d = m/V
d = Density, m = mass, V = Volume
Population Density = total area population _
land area in square units