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Perimeter (Circumference), Area, and Volume
Ex. Find the area:
26m
18m 18m 21.1m
37m
A = ½h(a + b)A = ½(18)(26 + 37)A = ½(18)(63)A = 9(63)A = 567
The area is 567 m2
Ex. A rectangle has a width of 46cm and a perimeter of 208cm. What is its length?
x = length
width = 46 cm
length = x
P = 2L + 2W 208 = 2(x) + 2(46) 208 = 2x + 92208 – 92 = 2x + 92 – 92 116 = 2x 116 = 2x 2 2 58 = xThe length is 58 cm
Circles
diameter – distance across circle through the center
radius – distance from center to edge of circle (half the diameter)
Note: π is approximately 3.14
d
r
Ex. Find the area and circumference(a) in terms of π (b) rounded to the nearest whole number
9m
A = πr2
A = π(9)2
A = π81 A = 81π(a) area is 81π m2
A = 81π A ≈ 81(3.14)A ≈ 254.34(b) area is approximately 254 m2
C = 2πrC = 2π(9)C = 18π(a) circumference is 18π m
C = 18πC ≈ 18(3.14)C ≈ 56.52 m(b) circumference is
approximately 57 m
Ex. A cylinder with radius 3 inches and height 4 inches has its radius tripled. How many times greater is the volume of the larger cylinder than the smaller cylinder?
Ex. A cylinder with radius 3 inches and height 4 inches has its radius tripled. How many times greater is the volume of the larger cylinder than the smaller cylinder?
V1 = πr2h
V1 = π(3)2(4)
V1 = π(9) (4)
V1 = 36π in3
V2 = πr2h
V2 = π(9)2(4)
V2 = π(81) (4)
V2 = 324π in3
9 times
4in3in 9in
4in
936
324
V
V
1
2
1
9
Ex. A water reservoir is shaped like a rectangular solid with a base that is 50 yards by 30 yards, and a vertical height of 20 yards. At the start of a three-month period of no rain, the reservoir was completely full. At the end of this period, the height of the water was down to 6 yards. How much water was used in the three-month period?
Ex. A water reservoir is shaped like a rectangular solid with a base that is 50 yards by 30 yards, and a vertical height of 20 yards. At the start of a three-month period of no rain, the reservoir was completely full. At the end of this period, the height of the water was down to 6 yards. How much water was used in the three-month period?
Vstart = lwhVstart = (50)(30)(20)Vstart = 30,000 yd3
Vend = lwhVend = (50)(30)(6)Vend = 9,000 yd3
Vstart – Vend = 30,000 – 9,000 = 21,000 yd3
20yd
6yd
30yd
50yd
Angles of a Triangle
C
A B
The sum of the interior angles of a triangle is 180°.
A° + B° + C° = 180°
Ex. One angle of a triangle is three times as large as another. The measure of the third angle is 40°more than that of the smallest angle. Find the measure of each angle.
Ex. One angle of a triangle is three times as large as another. The measure of the third angle is 40°more than that of the smallest angle. Find the measure of each angle.
3x
x + 40 x
A° + B° + C° = 180°
x + 3x + x + 40 = 180 5x + 40 = 180 5x + 40 – 40 = 180 – 40 5x = 140 5x = 140 5 5 x = 28x = 28°3x = 3(28) = 84°x + 40 = 28 + 40 = 68°
28°, 84°and 68°40x3
x32
x1
complementary angles – 2 angles whose sum is 90°
If one angle is x B complementary angle is 90 – x
A
A + B = 90°
supplementary angles – 2 angles whose sum is 180°
If one angle is xB A supplementary angle is 180 – x
A + B = 180°
angle comp supp 50° 40° 130° 17° 73° 163° x° (90 – x)° (180 – x)°
Ex. Find the measure of an angle whose supplement measures 39° more than twice its complement.
angle = xcomp = 90 – xsupp = 180 – x
180 – x = 2(90 – x) + 39 180 – x = 180 – 2x + 39 180 – x = 219 – 2x 180 – x + 2x = 219 – 2x + 2x 180 + x = 219180 + x – 180 = 219 – 180 x = 39
The angle is 39°