3.3 Problem Solving in Geometry

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

straight angle

180°

right angle

90°

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.

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°

Groups

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