Download All of Unit 4

Unit 4 Reference Sheet
Circle: The set of points in a plane that are
fixed distance from a given point
called the center of the circle.
Chord: A segment whose endpoints
both lie on the same circle.
Name:
Radius: A segment whose endpoints are the center
of a circle and a point on the circle.
chord
Diameter: A segment that has endpoints
on a circle and that passes
through the center of the circle.
Secant: A line that intersects a
circle at two points.
Tangent: A line that is in the same plane as a
circle and intersects the circle at
exactly one point. The radius is
perpendicular to the tangent at the point
of tangency.
Internal Tangent: A tangent that is
common to two circles and
intersects the segment joining the
centers of the circles.
External Tangent: A tangent that is
common to two circles and does
not intersect the segment joining
the centers of the circles.
Inscribed Angle: An angle whose vertex is on a
circle and whose sides contain chords of
the circle. The angle measure of the
inscribed angle is ½ of the intercepted arc.
Central Angle: An angle whose vertex is
the center of a circle.
Angle Between 2 Chords: The angle
between 2 chords is equal to ½ the
sum of the two intercepted arcs.
Angle Between 2 Secants: The angle between 2
chords is equal to ½ the difference of the
two intercepted arcs.
M. Winking (Unit 4 -00)
p.87
Sec 4.1 – Circles & Volume
The Language of Circles
1.
Name:
Using the Pythagorean Theorem to find the value of x in each of the diagrams below:
1.
2.
x=
x=
Converse of the Pythagorean Theorem. Which of the following are right triangles?
3.
11
4.
5.
6.
17
17
15
35
7.
5
12
4
15
37
6
8
8
8
Right Triangle?
(circle one)
YES
(circle one)
NO
YES
Right Triangle?
Right Triangle?
Right Triangle?
(circle one)
(circle one)
YES
NO
YES
NO
______8. G
______9. A
A. Diameter
B. Radius
______10. DE
C. Center
______11. GC
D. Secant
______12. JB
E. Chord
______13. HJ
F. Point of tangency
______14. HI
G. Common external tangent
______15. AB
H. Common internal tangent
(circle one)
NO
Determine if AB is tangent to the circle centered at point C. Explain your reasoning.
16.
17.
M. Winking
Unit 4-1
page 88
Right Triangle?
YES
NO
AB and AD are tangent to the circle centered at point C. Find the value of x.
18.
19.
x=
x=
Given the center of the circle is point A, find the requested measure.
20. mEF =
21. mCE =
22. mCDF =
23. mDE =
24. mBC =
25. mFB =
26. mFBE =
27. mDFC =
28. mDFB =
29. mBEC =
Determine the measure of BC .
30.
31.
32.
Find the requested measure for each circle.
33. FC =
34. mBG  _________
M. Winking
Unit 4-1
page 89
Sec 4.2 – Circles & Volume
Inscribed Angles
1.
Central Angle: An angle whose vertex is the center
of the circle.
Name:
Inscribed Angle: An angle whose vertex is on a circle and
whose sides contain chords of the circle
Central
Angle
Inscribed
Angle
Inscribed Angle Properties: Consider the following diagram an inscribed angle of the circle center at A.
D
C
D
D
A
B
Consider the inscribed angle
∡𝐶𝐵𝐷 which intercepts arc
̂ that measures 70˚.
𝐷𝐶
C
A
C
B
Since the central angle
̂
∡𝐶𝐴𝐷 intercepts arc 𝐷𝐶
then 𝑚∡𝐶𝐴𝐷 = 70°.
D
A
B
Triangle ∆DAB is isosceles
because the legs are radii of
the circle. The measure of
angle 𝑚∡𝐷𝐴𝐵 = 110° since it
forms a linear pair with ∡𝐶𝐴𝐷.
C
A
The based angles of ∆DAB
must be congruent and the
interior angles of triangle
must sum to 180˚. So,
110 + 𝑥 + 𝑥 = 180
In a similar fashion using addition or subtraction, it can be shown this idea extends to any inscribed angle.
“An inscribed angle’s measure is exactly half of the arc measure that it intercepts.”
Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.)
1.
2.
3.
A
A
x=
A
x=
M. Winking
x=
Unit 4-2
page 90
B
Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.)
4.
5.
6.
A
A
A
x=
x=
7.
x=
8.
9.
A
A
A
x=
x=
10.
x=
11.
12.
A
A
A
x=
x=
M. Winking
x=
Unit 4-2
page 91
Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.)
13.
14.
15.
A
A
x=
A
x=
x=
17.
16.
18.
A
A
A
x=
x=
19.
x=
20.
21.
A
A
x=
A
x=
M. Winking
x=
Unit 4-2
page 92
1.
Sec 4.2a – Circles & Volume
Tangent Circle Construction
Name:
[Creating a Tangent To a Circle] Construct a line tangent to circle with center A and passing through point C.
Step I: First draw a segment with end points A & C.
Step 3: Create a circle
centered at the midpoint
of segment ̅̅̅̅
𝐴𝐶 and with
a radius from the
midpoint to point A.
Step 2: Create
a perpendicular
bisector to
segment ̅̅̅̅
𝐴𝐶
Step 4: Draw a line
that passes through
point C and either of
the intersections of the
original circle and the
newly created circle
(point E in the
diagram).
Construct a tangent line to circle with center A that passes through point C.
C
●
●
M. Winking
Unit 4-a2
page 93
A
Sec 4.3 – Circles & Volume
Angles of Circles
1.
Name:
Tangent Line Angles
Consider the tangent line ⃡𝐷𝐶 and
̂
the ray 𝐶𝐵 which intercepts arc 𝐵𝐶
and has a measure of x˚.
̅̅̅̅ to
Draw an auxiliary segment ̅̅̅̅
𝐴𝐵 and 𝐴𝐶
create an isosceles triangles. We know
that 𝑚∡𝐴 = 𝑥° as a central angle and the
interior angles of ∆ABC sum to 180˚.
So, 𝑚∡𝐵 + 𝑚∡𝐶 = 180° − 𝑥
Also, 𝑚∡𝐵 = 𝑚∡𝐶 because they are the
base angles of an isosceles triangle. So,
𝑚∡𝐵 + 𝑚∡𝐵 = 180° − 𝑥 which simplifies:
2 ∙ 𝑚∡𝐵 = 180° − 𝑥 or
𝑚∡𝐵 =
̅̅̅̅ must be perpendicular to
Finally, 𝐴𝐶
⃡𝐷𝐶 since it is tangent of the circle. The
angle ∡𝐷𝐶𝐵 & ∡𝐴𝐶𝐵 must sum to 90˚.
So, we can find
180° − 𝑥
𝑚∡𝐷𝐶𝐵 = 90 −
2
which simplifies:
𝑥
𝑚∡𝐷𝐶𝐵 =
2
180°−𝑥
2
“The measure of an angle formed by a tangent and a chord drawn to the
point of tangency is exactly ½ the measure of the intercepted arc.”
Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume CE is tangent to the circle.)
1.
2.
3.
x=
x=
M. Winking
x=
Unit 4-3
page 94
Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume CE is tangent to the circle.)
4.
5.
6.
(You may assume DF is a diameter.)
x=
x=
x=
Intersecting Chords Interior Angles
Consider the intersecting
chords ̅̅̅̅
𝐵𝐸 and ̅̅̅̅
𝐹𝐶 that
̂
intercept the arcs 𝐶𝐵
̂.
and 𝐹𝐸
Draw an auxiliary segment
̅̅̅̅
𝐵𝐹 to create the inscribed
angles that we know are half
of the intercepted arc. So,
𝑚∡𝐹𝐵𝐸 =
𝑧°
2
and 𝑚∡𝐵𝐹𝐶 =
𝑦°
2
Since triangles interior angles
sum to 180˚. So we can
subtract the 2 angles of
triangle ∆DBF to find angle
𝑚∡𝐵𝐷𝐹 = 180° − 𝑦°
− 𝑧°2
2
Finally, since ∡𝐹𝐷𝐸 forms a
linear pair with ∡𝐵𝐷𝐹 we can
subtract from 180˚ to find:
𝑚∡𝐹𝐷𝐸 = 180° − (180° − 𝑦°2 − 𝑧°2)
which simplifies to
𝑦°+𝑧°
𝑚∡𝐹𝐷𝐸 = 2
“The measure of an angle formed by two intersecting chords of the same
circle is exactly ½ the measure of the sum of the two intercepted arcs.”
Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.)
7.
8.
x=
x=
M. Winking
Unit 4-3
page 95
Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.)
9.
10.
(You may assume BE is a diameter.)
x=
x=
11.
12.
x=
x=
13.
14.
x=
x=
M. Winking
Unit 4-3
page 96
Secant Lines Exterior Angle
Consider the rays 𝐵𝐷 and 𝐵𝐹
̂ and 𝐹𝐷
̂
which intercepts arc 𝐶𝐸
which measure a˚ and b˚
respectively.
Draw an auxiliary segment ̅̅̅̅
𝐸𝐷 . We
𝑏°
know that 𝑚∡𝐷𝐸𝐹 = and
2
𝑎°
𝑚∡𝐶𝐷𝐸 = because each is an
2
inscribed angle.
𝑏°
Finally, 𝑚∡𝐷𝐸𝐵 = 180° − since the two
2
angles at point E forma linear pair.
𝑏°
𝑎°
Furthermore, 𝑚∡𝐵 = 180° − (180° − 2 ) − 2
since a triangle’s interior angles sum to 180˚ and
that would simplify to 𝑚∡𝐵 =
(𝑏−𝑎)°
2
.
“The measure of an angle formed on the exterior of a circle by two intersecting secants of
the same circle is exactly ½ the measure of the difference of the two intercepted arcs.”
Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.)
15.
16.
x=
17.
x=
18.
x=
x=
M. Winking
Unit 4-3
page 97
Find the most appropriate value for ‘x’ in each of the diagrams below. (Assume point ‘A’ is the center of the circle.)
19.
20.
You may assume DB is
tangent to the circle.
x=
x=
21.
22.
You may assume EC and DE
are tangent to the circle.
You may assume DB is
tangent to the circle.
x=
x=
23.
24.
x=
x=
M. Winking
Unit 4-3
page 98
1.
Sec 4.4 – Circles & Volume
Circle Segments
Name:
Intersecting Chords
Consider the intersecting
chords ̅̅̅̅
𝐷𝐶 and ̅̅̅̅
𝐸𝐹 that
intersect at point B.
Draw an auxiliary segment ̅̅̅̅
𝐷𝐸 and ̅̅̅̅
𝐶𝐹
to create triangles ∆DBE and ∆FBC. We
know that ∡𝐷𝐸𝐵 ≅ ∡𝐹𝐶𝐵 because they
are both inscribed angles that intercept
̂ . Similarly, we know
the same arc 𝐹𝐷
∡𝐸𝐷𝐵 ≅ ∡𝐶𝐹𝐵. Then, by AA we know
∆𝐷𝐵𝐸~∆𝐹𝐵𝐶
Using proportions of similar triangles:
𝒙𝟏 𝒚𝟏
=
𝒚𝟐 𝒙𝟐
We can cross-multiply to give us the
following statement:
𝒙𝟏 ∙ 𝒙𝟐 = 𝒚𝟏 ∙ 𝒚𝟐
Part1 Part2
Part1 Part2
“If two chords intersect then the product of the measures of the two subdivided
parts of one chord are equal to the product of the parts of the other chord.”
Find the most appropriate value for ‘x’ in each of the diagrams below.
1.
2.
x=
x=
3.
4.
x=
x=
M. Winking
Unit 4-4
page 99
Find the most appropriate value for ‘x’ in each of the diagrams below
5.
6.
x=
x=
Segments of Secants
Consider the intersecting
̅̅̅̅ and
segments of secants 𝐸𝐶
̅̅̅̅ that intersect at point C.
𝐴𝐶
̅̅̅̅ and 𝐵𝐸
̅̅̅̅
Draw an auxiliary segment 𝐴𝐷
to create triangles ∆ADC and ∆EBC. We
know that ∡𝐶𝐴𝐷 ≅ ∡𝐶𝐸𝐵 because they
are both inscribed angles that intercept
̂ . Reflexively, we also
the same arc 𝐵𝐷
know ∡𝐶 ≅ ∡𝐶. Then, by AA we know
∆𝐴𝐷𝐶~∆𝐸𝐵𝐶
Using proportions of similar triangles:
𝒙𝟐 + 𝒙𝟏 𝒚𝟏
=
𝒚𝟏 + 𝒚𝟐 𝒙𝟏
We can cross-multiply to give us the
following statement:
(𝒙𝟐 + 𝒙𝟏 ) ∙ 𝒙𝟏 = (𝒚𝟏 + 𝒚𝟐 ) ∙ 𝒚𝟏
Whole
External
Whole
External
“If 2 secants intersect the same circle on the exterior of the circle then the product of the ‘whole’ and
the ‘external’ segment measures is equal to the same product of the other secant’s portions.
Find the most appropriate value for ‘x’ in each of the diagrams below.
7.
8.
x=
x=
M. Winking
Unit 4-4
page 100
Find the most appropriate value for ‘x’ in each of the diagrams below.
9.
10.
x=
x=
11.
12.
x=
x=
Segments of Secants and Tangents
Consider the intersecting
̅̅̅̅ and
segment of a secant 𝐴𝐶
̅̅̅̅ that
segment of a tangent 𝐴𝐹
intersect at point A.
Draw an auxiliary segment ̅̅̅̅
𝐵𝐷 and ̅̅̅̅
𝐶𝐷
to create triangles ∆ADC and ∆ABD. We
know that ∡𝐵𝐷𝐴 ≅ ∡𝐴𝐶𝐷 because they
are both have a measure of half of the
̂ . Reflexively, we also
intercepted arc 𝐵𝐷
know ∡𝐴 ≅ ∡𝐴. Then, by AA we know
∆𝐴𝐷𝐶~∆𝐴𝐵𝐷
Using proportions of similar triangles:
𝒙𝟏
𝒚𝟏
=
𝒚𝟏 + 𝒚𝟐 𝒙𝟏
We can cross-multiply to give us the
following statement:
(𝒚𝟏 + 𝒚𝟐 ) ∙ 𝒚𝟏 = 𝒙𝟏 ∙ 𝒙𝟏
Whole
M. Winking
Unit 4-4
page 101
External Tangent Tangent
Find the most appropriate value for ‘x’ in each of the diagrams below.
13.
14.
x=
x=
𝒂 ∙ 𝒃 = 𝒄 ∙ 𝒅
Part1 Part2
Part1 Part2
(𝒂 + 𝒃) ∙ 𝒂 = 𝒄𝟐
(𝒂 + 𝒃) ∙ 𝒂 = (𝒄 + 𝒅) ∙ 𝒄
Whole
External
Whole
External
Whole
External
Tangent 2
Find the most appropriate value for ‘x’ in each of the diagrams below.
13.
14.
x=
x=
M. Winking
Unit 4-4
page 102
1.
1.
Sec 4.5 – Circles & Volume
Circumference, Perimeter, Arc Length
Name:
Cut a piece of string that perfectly fits around the outside edge of the circle. How many diameters long
is the piece of string (use a marker to mark each diameter on the string)? Tape the string to this page.
Tape String Here:
2.
A.
Find the Circumference of the following circles (assuming point A is the center):
C.
E.
2a. C =
B.
2c. C =
D.
2b. C =
2e. C =
2d. C =
M. Winking
Unit 4-5
page 103
3. Find the Radius of each circle given the following information:
A.
B.
3b. r =
3a. r =
̂ . (Assuming point A is the center)
4. Find the Arc Length of arc 𝑩𝑪
B.
A.
4b.
4a.
D.
C.
4c.
4d.
M. Winking
Unit 4-5
page 104
̂.
5. Find the Arc Length of arc 𝑩𝑪
(Assuming point A is the center)
A.
B.
(Assume EF is tangent to the circle.)
5b.
5a.
6. Find the most appropriate value for x in each diagram.
A.
B.
6b.
6a.
M. Winking
Unit 4-5
page 105
The Babylonian Degree method of measuring angles. Around
1500 B.C. the Babylonians are credited with first dividing the circle
up in to 360̊. They used a base 60 (sexagesimal) system to count
(i.e. they had 60 symbols to represent their numbers where as we
only have 10 (a centesimal system of 0 through 9)). So, the number
360 was convenient as a multiple of 60. Additionally, according to
Otto Neugebauer, an expert on ancient mathematics, there is
evidence to support that the division of the circle in to 360 parts may
have originated from astronomical events such as the division of the
days of a year. So, that the earth moved approximately a degree a day
around the sun. However, this would cause problems as years passed to keep the seasons
accurately aligned in the calendar as there are 365.242 actual days in a year. Some ancient
Persian calendars did actually use 360 days in their year further supporting this idea.
The transition to Radian measure of angle: Around 1700 in the United Kingdom,
mathematician Roger Cotes saw some advantages in some situations to
measuring angles using a radian system. A radian system simply put,
drops a unit circle (a circle with a radius of 1) on to an angle such that
the center is at the vertex and the length of the intercepted arc is the
radian measure. So, a full circle of 360̊ is equivallent to 2π∙(1)
radians. In the example at the right, an angle of 50̊ is shown. Then, a
circle that has a radius of 1 cm is drawn with its center at the vertex.
1 cm
 50 
Arc Length  
  2  1 cm   0.873 cm
 360 
Finally, the intercepted arc length is determined to be approximately 0.873 or more precisely
Similarly, it can be demonstrated the basically that 180˚ is equivalent to π radians.
6. Using the ratio of 180˚: π convert the following degree measures to radians.
a. 30˚
b. 80˚
c. 225˚
d. 360˚
7. Using the ratio of 180˚: π convert the following radian measures to degrees.
a.
𝜋
4
rads
b.
3𝜋
10
rads
M. Winking
c.
Unit 4-5
5𝜋
8
d. 0.763
rads
page 106
rads
5𝜋
18
radians.
̂ using similar circles or a fraction of the circumference.
8. Find the Arc Length of 𝑩𝑪
A.
B.
8b.
8a.
9. Solve the following.
A. Determine the perimeter
of the rhombus shown.
B. Find an expression that would
represent the perimeter of the
triangle.
9a.
9b.
M. Winking
C. Given the perimeter of the
rectangle shown below is 32 cm2
and the length of one side is 6 cm,
determine the area of the rectangle.
9c.
Unit 4-5
page 107
10. Find the perimeter of each compound figure below.
A.
Assume the compound figure
includes a semicircle.
B.
(Assume all adjacent sides are perpendicular.)
10b.
10a.
C.
Assume the compound figure
includes three semicircles.
Assume the compound figure includes
a rectangle and 2 sectors centered at
point A and C respectively.
D.
10d.
10c.
M. Winking
Unit 4-5
page 108
Mathematically, we can determine the value of pi using the Pythagorean Theorem.
Sec 4.6 – Circles & Volume
Circumference, Perimeter, Arc Length
Problem
Hint
1. Find the area of the rectangle and write down the
formula for finding the area of a rectangle.
1.
2. Find the area of the right triangle and write down
the formula for finding the area of a right triangle.
3. Find the area of the acute triangle and write down
the formula for finding the area of an acute triangle.
4. Find the area of the parallelogram and write down
the formula for finding the area of an acute triangle.
M. Winking
Unit 4-6
page 109
Name:
Formula
Problem
5. Find the area of the
trapezoid and write down
the formula for finding the
area of a rectangle.
Hint
Formula
Make a Copy
4
6
Rotate Copy
8
6
Creates a Parallelogram exactly twice the size of the trapezoid
6. Find the area of the circle
below.
1. Find the area and perimeter of each of the following shapes.
3 cm
11.5 cm
8 cm
4 cm
Area:
Area:
Perimeter
r:
Perimeter
r:
7 in
6 in
1 in
Perimeter
r:
Area:
9 in
Area:
Perimeter
r:
Area:
Perimeter
r:
Area:
Perimeter
r:
2. Solve the following area problems.
Determine the area of
the rhombus shown.
Area:
Find an expression that would
represent the area of the triangle.
Find the length of the radius given
2
the area of the circle is 531 cm .
x =
Area:
3. Find the following sector areas (shaded regions) using fractional parts.
Sector
Area
:
Sector
Area
M. Winking
Sector
Area
:
Unit 4-6
page 111
:
4. Find the area of each of the shaded regions.
Area:
x =
Area:
5. Solve the following problems.
Find the area of the shaded region,
given that AC is tangent to the
circle at point B and 𝒎∡𝑨𝑩𝑬 = 𝟕𝟎°
Sector
Area
:
Find the central angle of a sector
2
that has an area of 71 cm and a
radius of 7 cm.
Sector
Area
M. Winking
Find the radius of a sector that
2
has an area of 92 cm and a
central angle of 130˚.
Sector
Area
:
Unit 4-6
page 112
:
6. Find the area of the following compound figures (assume all curved shapes are semicircles).
Area:
x =
Area:
7. Find the area of the following shaded regions
4 cm
Area:
Area:
M. Winking
Unit 4-6
page 113
1.
Sec 4.7 – Circles & Volume
Nets & Surface Areas
1. Sketch a NET of each of the following solids:
A.
B.
C.
D.
M. Winking
Unit 4-7
page 114
Name:
2. Sketch a NET of each of the following solids:
3. Find the surface area of the following solids (figures may not be drawn to scale).
Surface
Area
Surface
Area
:
M. Winking
Unit 4-7
page 115
:
4. Find the surface area of the following solids (figures may not be drawn to scale).
Surface
Area
:
:
12 cm
Surface
Area
4 cm
Surface
Area
Surface
Area
:
M. Winking
Unit 4-7
page 116
:
5. Find the surface area of the following solids (figures may not be drawn to scale).
Surface
Area
Surface
Area
:
Surface
Area
:
Surface
Area
M. Winking
Unit 4-7
page 117
:
:
6. Solve the following problems. (figures may not be drawn to scale).
Find the Surface Area of a baseball given that
its largest circumference is 23.5 cm.
Determine the Surface Area of the following
rectangular prism with a missing portion.
Surface
Area
Surface
Area
:
Determine the amount of surface area that is water
on our planet in square miles. You may assume the
earth is spherical, has a diameter of 7918 miles, and
that water covers 71% of the Earth’s surface.
Surface
Area
Given the circumference of the base of a cone
is 31.4cm and the slant height is 13 cm, find
the surface area of the cone.
Surface
Area
:
M. Winking
:
Unit 4-7
page 118a
:
1.
Sec 4.8 – Circles & Volume
Volume of Pyramids & Cones
Name:
1. Find the Volume of the following solids (figures may not be drawn to scale).
Volume:
Volume:
Volume:
Volume:
M. Winking
Unit 4-8
page 118b
12 cm
2. Find the Volume of the following solids (figures may not be drawn to scale).
4 cm
Volume:
Volume:
Volume:
Volume:
M. Winking
Unit 4-8
page 119
3. Find the volume of the following solids (figures may not be drawn to scale).
Using a micrometer find the volume of 6 pennies stacked
directly on top of each other (which is a cylinder). Show
measurements to the nearest hundredth of a millimeter.
Using a micrometer find the volume of 6 pennies stacked
on top of each other but so that they are slanted. Show
measurements to the nearest hundredth of a millimeter.
Volume:
Volume:
Find volume of the oblique rectangular prism.
Find volume of the sphere.
Volume:
Volume:
M. Winking
Unit 4-8
page 120
4. Find the volume of the following solids (figures may not be drawn to scale).
The solid below shows a gas tank for a tractor trailer truck.
It is in the shape of a cylinder with a radius of 9 inches and
a height of 60 inches. How many gallons of fuel will it hold
if there are 231 cubic inches in one gallon?
A sphere is inscribed in a cube with a volume of 27
cm3. What is the volume of the sphere?
Gallons:
Volume:
A snowman is created from two spherical snow balls. Given the
circumference of each sphere determine the volume of the snowman.
Find volume of the regular hexagonal prism.
Volume:
Volume:
M. Winking
Unit 4-8
page 121
1.
Sec 4.9 – Circles & Volume
Volume of Pyramids & Cones Name: UsingCavalieri’sPrinciplewecanshowthatthevolumeofapyramidisexactly⅓thevolumeofaprismwiththesame
Baseandheight.
Considerasquarebased
pyramidinscribedincube.
Next, translatethepeakofthe
pyramid.Cavalieri’sPrinciple
wouldsuggestthatthevolumeof
theobliquepyramidisthesameas
theoriginalpyramid.
Next,wecancreate2moreoblique
pyramidswiththesamevolumeof
theoriginalwiththeremaining
spaceinthecube.
Inthisdiagram,wecanseethe3obliquepyramidsofequalvolumepulledoutfromthecube.
So,thisdemonstratesapyramidinscribedinacubehasexactly⅓thevolumethecube.
Thisideacanbeextendedtoanypyramidorcone.
M.Winking
Unit4‐9page122
1. FindtheVolumeofthefollowingsolids(figuresmaynotbedrawntoscale).
Volume:
Volume:
Volume:
Volume:
M.Winking
Unit4‐9page123
2. FindtheVolumeofthefollowingsolids(figuresmaynotbedrawntoscale).
Volume:
Volume:
Findthevolumeoftheregularoctahedron.
Findthevolumeoftheirregularsolid.Thebase
hasanareaof80cm2andaheightof9cm.
Volume:
Volume:
M.Winking
Unit4‐9page124
3. FindtheVolumeofthefollowingsolids(figuresmaynotbedrawntoscale).
Volume:
Volume:
ConsidertriangleABCwithverticesatA(0,0),
B(4,6),andC(0,6)plottedandacoordinategrid.
Determinethevolumeofthesolidcreatedby
rotatingthetrianglearoundthey‐axis.
Volume:
M.Winking
Unit4‐9page125
UsingCavalieri’sPrinciplewecanshowthatthevolumeofaspherecanbefoundby
∙
First,considera
hemispherewitha
radiusofR.Createa
cylinderthathasabase
withthesameradiusR
andaheightequalto
theradiusR.Then,
removeaconefromthe
cylinderthathasthe
samebaseandheight.
Next,consideracross
sectionthatisparallel
tothebaseandcuts
throughbothsolids
usingthesameplane.
Cavalieri’sPrinciplesuggestsifthe2crosssectionshavethesameareathenthe2solidsmusthavethesamevolume.
Theareaofthecrosssectionofthesphereis:
∙ UsingthePythagoreantheoremweknow:
or
So,withsimplesubstitution:
∙
Theareaofthecrosssectionofthesecondsolid is:
∙ ∙
Usingsimilartrianglesweknowthath=bandthen,using
simplesubstitution
∙ ∙
VolumeofHemisphere=VolumeofCylinder–VolumeofCone= ∙
∙
WealsoknowthatR=b=h.So,VolumeofHemisphere=
∙
∙
∙
∙
Tofindthevolumeofacompletesphere,wecanjustdoublethehemisphere:VolumeofSphere=
M.Winking
Unit4‐9page126
∙
∙