Unit 8: Applying Formulas

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.

2121 ddA

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

CanglemAB 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 ranglemAOB

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 .

PrismsPrism – A 3-dimensional figure with two congruent, parallel faces, called bases.

Lateral Faces – Faces that are not bases

Surface Area of PrismsTHEOREM 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 CylindersTHEOREM 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.

PyramidsPyramid – 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

ConesCone – 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 ConesTHEOREM 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.

BhV31

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 ConesTHEOREM 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

31or

31

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

34 rV

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)