Mth 97 Fall 2013 Chapter 3
3.1 Perimeter, Circumference, and Area of Rectangles and Triangles
The Perimeter of a Polygon is the ______of the lengths of its sides.
Perimeters of Common Quadrilaterals – Theorem 3.1Description of Figure / Formula / Drawing
Square with sides of length s / P = /
Rectangle with side lengths
of a and b / P = /
Parallelogram with side
lengths of a and b / P = /
Rhombus with sides of length s / P = /
Kite with side lengths
of a and b / P = /
Find the following perimeters:
Square with sides of length 2.2 cm Kite with side lengths of 3.5 cm and 5.7 cm
History of π: For any circle the ratio of the circumference (distance around a circle or perimeter) to the
diameter, ______, is constant and is represented by the Greek letter π. Today π (read” pi”) has been refined from Archimedes’ approximation of between and to over a trillion decimal places and is identified as an IRRATIONAL number.
Postulate 3.1 ( Circumference of a Circle )
The circumference of a circle is the product of π and the diameter of the circle.
C = dπ = 2πr
Find the perimeter of the following figures.
7.4 cm
Read and discuss Postulate 3.2 on page 109 and Postulate 3.3 given below.
- Any two triangles whose corresponding angles and sides are congruent have the same area.
Area of Rectangles and TrianglesDefinition or Theorem / Formula / Drawing(s)
Dfn:
The Area of Rectangle is the ______of its length and width. / A = lw /
Thm 3.2:
The Area of a Right Triangle is half the product of the lengths of its ______. / A = ½ab /
Thm. 3.3: The Area of a Triangle is half the product of the length of one side and the ______or ______to that side. / A = ½bh /
Find the area of each polygon.
10cm 6.2 cm
------8.5 cm
13 cm
Converting Units of Area Using Dimensional Analysis
How many square centimeters (cm2) are in a square meter (m2)? Remember the exponent on the unit of measure tells you how many times to use the conversion factor.
How many square inches are in a rectangle with An Olympic size swimming pool of uniform depth a length of 3 feet and a width of 2 feet? measures 50 m by 21 m. The bottom of the pool is to be treated with a sealer and 1 gallon of sealer 2 ft covers 300 square feet of surface. How many gallons of sealer will be required for the job?
3 ft
6 mi2 = ______km2
Do problem 1 of ICA 4
3.2 More Area Formulas
Description / Area Formula / DrawingThm 3.4 / Area of a Parallelogram is equal to the product of the length of one side and the ______to that side. / A = bh /
Thm 3.5 / Area of a Trapezoid is equal
to half the product of the height and the ______of the length of its bases. / A = ½ h(b1 + b2) /
Thm 3.6 / Area of a Regular Polygon is equal to half the product of the perimeter of the polygon and the perpendicular distance from its center to one of its sides, called the
______. / A = ½ Ph
where h is the length of the apothum /
Thm 3.7 / Area of a Circle is equal to the product of π and the square of its ______. / A = πr2 /
Find the area of the following figures.
a) b)
16 cm 10 cm
20 cm 20 cm
c) d) Do # 16 on page 129
10 cm
Do the rest of ICA 4.
3.3 The Pythagorean Theorem and Right Triangles
Thm 3.8 - Pythagorean TheoremThe sum of the ______of the lengths of the legs (a and b) of a right triangle is equal to the square of the length of the hypotenuse (c). / a2 + b2 = c2 /
Find the missing lengths in each right triangle. Give the exact answer and the answer rounded to hundredths, if the exact answer is irrational. .
a) b) c) x + 2
8 cm c 8 in y x
12 in x + 1
10 cm
d) Find the area of a square that has a diagonal 10 inches long.
Use the Pythagorean Theorem to test the lengths below to determine if they are the sides of a right triangle. a) 11, 19, 15 b) 7, 16.8, 18.2
Special Right Triangles
Theorem 3.9 - 30-60-90 triangles/ In a 30°- 60° right triangle, the length of the leg opposite the 30° angle is half the length of the hypotenuse. The length of the leg opposite the 60° angle is times the length of the hypotenuse.
Show that the dimensions for the triangle above are the dimensions of a right triangle.
Theorem 3.10 – 45 – 45 – 90 triangles/ In a 45° - 45° right triangle the length of the hypotenuse is times the length of a leg.
Show that the dimensions for the triangle above are the dimensions of a right triangle.
Find the missing side lengths.
a) b) c)
y 9 cm 10 ft y y 20 cm
x x x
x = x = x =
y = y = y =
3.4 Surface Area
The surface area of a right prism is the sum of the area of its lateral faces and bases.
Find the surface area of a rectangular prism whose length is 8 cm, width is 3 cm and height is 5 cm.
A formula you’ve probably used before is: SA = 2lw + 2wh + 2lh
Let’s look at the problem in the previous example by drawing a net (unfold the box) for the prism. Examine your sketch, looking for an easier way to find the prism’s surface area. SA = 2(area of the base) + (______)(height)
SA = This method works for all ______prisms.
The surface area of a right pyramid is the sum of the area of its base and lateral faces.
Find the surface area of a right regular triangular pyramid if the sides of the base are 4 cm and slant height is 5 cm. SA = Base area + lateral area 1 + lateral area 2 + lateral area 3
Draw a net for this pyramid. Is there an easier way to find surface area involving the perimeter of the base?
SA = Base area + 1/2 (perimeter of base)(slant height) =
The surface area of a right circular cylinder is the same as that of a prism except the perimeter becomes the ______of the circular base. SA = 2(area of base) + C (height)
12 cm Find the surface area of the right cylinder to the left using this formula.
7 cm
l The surface area of a right circular cone is the same as a pyramid except the perimeter h becomes the ______of the circular base. A = area of the base + ½ C (slant height)
Find the surface area of a right circular cone whose slant height is 8 cm and whose base has a radius of 6 cm using the above formula.
The formula for the surface area of a sphere is 4π times the square of its radius. SA = 4 π r2 or 4(area of a circle) Find the surface area of a sphere having a diameter of 6.7 m.
6.7 m
3.5 Volume
Read and discuss Postulate 3.4, the Volume Postulate, on page 159.
Remember: ______units are used in measuring perimeter and circumference.
______units are used in measuring area and surface area.
______units are used in measuring volume.
The Volume of a Right Prism is the product of the area of its base and its height of the prism. V = A h
Name each right (lateral faces must be ______) prism and then find the volume of each prism.
8 cm 5 in
4 cm 5 in
15 cm 12cm 5 in
______
V = A h or V = _____h V = A h = ______h V = A h =______h
The Volume of a Right Cylinder is the product of the area of its base and its height of the cylinder. 6 cm V = A h =
10 cm
The Volume of a Right Pyramid is ______the product of the area of its base and its height.
Right Regular Pentagonal Pyramid
10 cm V = ⅓Ah =
5 cm
4 cm
The Volume of a Right Cone is one-______the product of the area of its base and its height.
Right Circular Cone V = ⅓Ah =
6 in
r = 3 in
The Volume of a Sphere is 4/3 π times the cube of its radius. Find the volume of the sphere below.
r = 2 cm
8