Volume, Density, Radians, Arc Length
Volume
Density
Volume of General Prism=_________________
Volume of Cylinder=_______________
Volume of Sphere=______________________
Volume of Cone=____________________
Volume of Pyramid=_____________________
**Watch out for hidden right triangles- (Pythagorean theorem, 45-45- rt triangle, 30-60-rt triangle
Trigonometry (SOH, CAH , TOA))
A
E
B
F
D
C
To change degrees to radians:
To change radians to degrees:
Length of an Arc s = Ɵr, when the angle
Area of Sector =
1 2
θ r , when the angle,
2
θ,
θ is measured in radians.
is measured in radians.
Ratio of Areas & Ratio of Volumes
*scale factor can be ratio of sides, radii, heights, diagonals (or simplified ratio of the
corresponding segments of the similar shapes).
Area Ratio
area1 ⎛ side1 ⎞
=⎜
⎟
area 2 ⎝ side2 ⎠
2
ratio area = (scale factor)2
The ratio of the areas of similar objects is the square of the scale factor.
Volume Ratio
volume1 ⎛ side1 ⎞
=⎜
⎟
volume2 ⎝ side2 ⎠
3
ratio volume = (scale factor)3
The ratio of the volumes of similar objects is the cube of the scale factor.
EX1: The ratio of the surface areas of two similar triangular prisms is 9:16.
What is the ratio of their volumes?
EX2: A cylindrical pot with an 8 inch diameter holds 416 cubic inches of water.
How much water will fill a similar pot with a diameter of 10 inches?
Density
1. A cup full of water has a mass of 1000g. The cup itself has a mass of 214.6g. If the cup has a diameter of
8cm and a height of 10cm, what is the approximate density of the water.
Volume
1. Find the volume of the following shape:
2. The base of a triangular prism is a right triangle. The lengths of a leg
of the base is 6 cm and the hypotenuse of the base is 10 cm. The height of the prism is 8 cm. What is
the volume of the prism?
3. For a pyramid, if the area of the base remains the same and the altitude is doubled, the volume will
(1) be multiplied by 4
(3) remain the same
(2) triple
(4) double
4. If the volume of a cube is 125 cm3, what is the surface area of the cube?
5. What is the volume of a sphere with a diameter of 18 feet?
6. If the volume of the pyramid is 84 yd3 and the area of the base is 36 yd2,
what is the height of the pyramid?
7. The Volume of the pyramid is
8. If a cube has a side of 8 cm and a rectangular prism has the dimensions of 5 x 6 x 7 cm, what is the
difference in their volumes?
9. The volume of a rectangular pond is 15,360 cubic meters. Its length, width, and depth are in the ratio of
10:6:4. Find the number of meters in each of the three dimensions of the pool.
10. A can of tennis balls has an inside diameter of 7 cm and a height of 20 cm. If the diameter of a tennis ball
is 6 cm, what is the packing efficiency of the tennis ball can? That is, the three balls represent what
percentage of the volume of the can?
11. Find the volume.
Leave answers in terms of
π .
12. Find the volume.
Round to the nearest tenth.
13. The volume of a sphere is 972π cm3. Find the surface area of the sphere.
14. Find the volume.
Radians , Degrees, Arc length, Area of Sector
1. Convert the degree measures into radians. Leave answers as exact values in most reduced form.
a) 90°
b) 30°
c) 300°
d) 270°
2. Convert the following radian measures into degrees.
5π
9π
4π
a)
b)
c)
3
20
15
d)
7π
6
3. Find the radius of a circle in which a central angle of 5 radians intercepts an arc length of 62.5 feet?
4. Find the measure (in radians) of a central angle that intercepts an arc of length 16 cm
in a circle of radius 8 cm.
b) A = 6π cm2, Ɵ = a) r = 10 cm, Ɵ = 2 rad. π
3
rad. c) r = 3 cm, Ɵ = 300° θ
θ
r
θ
r
r