Unit 8: Applying Formulas Sections: 10-3, 10-5, 10-6, 10-7 11-2, 11-4, 11-5, and 11-6.
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Transcript of Unit 8: Applying Formulas Sections: 10-3, 10-5, 10-6, 10-7 11-2, 11-4, 11-5, and 11-6.
Unit 8: Applying Formulas
Sections: 10-3, 10-5, 10-6, 10-7
11-2, 11-4, 11-5, and 11-6
Learning Target:
• I will be able to identify the appropriate formula for a figure and find the area, volume, surface area, or perimeter.
Area of Parallelograms
THEOREM 10.1 – Area of a RectangleThe area of a rectangle is the product of its base and height.
bhA
THEOREM 10.2 – Area of a ParallelogramThe area of a parallelogram is the product of a base and a corresponding height.
bhA
Area of TriangleA triangle is half of a parallelogram.
THEOREM 10.3 – Area of a TriangleThe area of a triangle is half the product of a base and the corresponding height.
bhA21
Area of Trapezoid
THEOREM 10.4 – Area of a TrapezoidThe area of a trapezoid is half the product of the height and the sum of the bases.
2121
bbhA
Area of Rhombus and Kite
THEOREM 10.5 – Area of a Rhombus or a KiteThe area of a rhombus or kite is half the product of the lengths of the diagonals.
2121ddA
Regular Polygons
radius – the distance from the center to a vertex
We can center any regular polygon inside of a circle:
Regular polygons:-all sides congruent -all angles congruent -convex
apothem – perpendicular distance from the center to a side
THEOREM 10.6 – Area of a Regular Polygon
The area of a regular polygon is half the product of the apothem and the perimeter
Areas of Regular Polygons
apA21
Areas of Regular Polygons
Regular decagon:
Words…Circle: the set of all points equidistant from a given point called the center (name a circle by its center)
Pradius
diameterCentral angle – an angle whose vertex is the center of the circle.
Arcs…
Semicircle – an arc that is half the circle
Minor arc – smaller than a semicircle
Major arc – greater than a semicircle
Minor Arc: Defined as the same as the measure of its corresponding central angle.
Major Arc: This is 360° minus the degree measure of the minor arc that has the same endpoints as the major arc..
Arc Length…
THEOREM 10.9: Circumference of a Circle
The circumference of a circle is times the diameter.
dC
THEOREM 10.10: Arc Length
Canglem
AB 360
)( arc ofLength
Area…
THEOREM 10.11: Area of a Circle
The area of a circle is the product of and the square of the radius.
2rA
THEOREM 10.12: Area of a Sector of a Circle
2
360
)(sector of Area r
anglemAOB
Arc Length…
Find the length of each arc shown in red. Leave your answer in terms of .
Area…
Find the area of sector ZOM. Leave your answer in terms of .
Prisms
Prism – A 3-dimensional figure with two congruent, parallel faces, called bases.
Lateral Faces – Faces that are not bases
Surface Area of Prisms
THEOREM 11.1 – Surface Area of PrismThe surface area of a prism is the sum of the lateral area and the area of the two bases.
2L.A.S.A. B
4 in.12 in
.
RegularPentagon
Surface Area of Cylinder
THEOREM 11.2 – Surface Area of CylinderThe surface area of a cylinder is the sum of the lateral area and the area of the two bases.
22S.A.or 2L.A.S.A. rChB
Volume of Prisms
4 cm
6 cm
12 cm
THEOREM 11.6 – Volume of PrismThe volume of a prism is the product of the area of a base and the height of the prism. Area of the base times the height. BhV
Volume of Cylinders
THEOREM 11.7 – Volume of CylinderThe volume of a cylinder is the product of the area of the base and the height of the cylinder.
V Bh or V r2h
Find the volume of the following cylinder in terms of π.
Pyramids
Pyramid – A 3-D figure with one face (the base) that is any polygon and the other faces (the lateral faces) are triangles that meet at a common vertex.
Regular Pyramid – Base is a regular polygon.
Slant Height (l ) – The length of an altitude of a lateral face of the pyramid.
How are the height and slant height related to the edge of the base of a pyramid???
Surface Area of Pyramids
THEOREM 11.3 – Surface Area of Regular PyramidThe surface area of a regular pyramid is the sum of the lateral area and the area of the base.
L.A.S.A. B
Cones
Cone – A 3-D figure with a circular base and a curved lateral surface
How are the radius, height, and slant height related???
Surface Area of Cones
THEOREM 11.4 – Surface Area of ConeThe surface area of a cone is the sum of the lateral area and the area of the base.
2S.A.or L.A.S.A. rrB
Find the surface area of the following cone.
Volume of Pyramids
THEOREM 11.8 – Volume of PyramidThe volume of a pyramid is one third the product of the area of a base and the height of the pyramid.
BhV3
1
The Pyramid arena in Memphis, TN has a base of area 300,000 ft2. Its height is 321 ft. What is the volume of the pyramid?
Volume of Cones
THEOREM 11.9 – Volume of ConeThe volume of a cone is one third the product of the area of the base and the height of the cone.
hrVBhV 2
3
1or
3
1
Find the volume of the following cone in terms of π.
THEOREM 11.10 – Surface Area of a SphereThe surface area of a sphere is four times the product of pi and the square of the radius of the sphere. 24 rSA
Surface Area and Volume of a Sphere
THEOREM 11.11– Volume of a SphereThe volume of a sphere is four thirds the product of pi and the cube of the radius of the sphere.
3
3
4rV
Four Types of Rigid Motion
A rigid motion is the action of taking an object and moving it to a different location without altering its shape or size.
1. Translation (slide)2. Reflection (about a line or an axis)3. Dilation (little/big)4. Rotation (clockwise or counterclockwise)