Area: Parallelograms
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Transcript of Area: Parallelograms
Pre-AlgebraPre-Algebra
Area: ParallelogramsArea: Parallelograms
Lesson 10-1
Objectives: 1. to find areas of rectangles
2. to find areas of parallelograms
Pre-AlgebraPre-Algebra
Area: ParallelogramsArea: Parallelograms
Lesson 10-1
Tips: look for two segments that form a right angle when determining a base and height
Pre-AlgebraPre-Algebra
Area: ParallelogramsArea: Parallelograms
Find the area of the rectangle.
Lesson 10-1
Step 1: Change the units so that they are the same.
150 cm = 1.5 m Change 150 centimeters to meters.
Step 2: Find the area.
A = bh Use the formula for area of a rectangle.
= (4)(1.5) Replace b and h with the dimensions 4 and 1.5.
= 6 Simplify.
The area of the rectangle is 6 m2.
Pre-AlgebraPre-Algebra
Area: ParallelogramsArea: Parallelograms
Find the area of each parallelogram.
Lesson 10-1
a. b.
A = bh area formula
= (8)(2) Substitute.
= 16 Simplify.
The area is 16 m2. The area is 15 in.2.
A = bh
= (2.5)(6)
= 15
Pre-AlgebraPre-Algebra
Area: Triangles and TrapezoidsArea: Triangles and Trapezoids
Lesson 10-2
Objectives: 1. to find areas of triangles
2. to find areas of trapezoids
Pre-AlgebraPre-Algebra
Area: Triangles and TrapezoidsArea: Triangles and Trapezoids
Lesson 10-2
Tips: bases of a trapezoid are the two sides parallel to each other, even if the figure is turned
to find the area of an irregular figure, you may need to use more than one area formula. (make sure to know all the area formulas)
Pre-AlgebraPre-Algebra
Area: Triangles and TrapezoidsArea: Triangles and Trapezoids
Find the area of the triangle.
Lesson 10-2
The area is 39 in.2.
A = bh Use the formula for area of a triangle.12
= • 13 • 6 Replace b with 13 and h with 6.12
= 39 Simplify.
Pre-AlgebraPre-Algebra
Area: Triangles and TrapezoidsArea: Triangles and Trapezoids
Find the area of the figure.
Lesson 10-2
Add to find the total: 450 + 1,350 = 1,800.
Area of triangle
A = bh12
= • 45 • 2012
= 450
Area of rectangle
A = bh
= 45 • 30
= 1,350
The area of the figure is 1,800 cm2.
Pre-AlgebraPre-Algebra
Area: Triangles and TrapezoidsArea: Triangles and Trapezoids
Suppose that, through the years, a layer of silt and mud
settled in the bottom of the Erie Canal. Below is the resulting cross
section of the canal. Find the area of the trapezoidal cross section.
Lesson 10-2
The area of the cross section is 106.5 ft2.
A = h(b1 + b2) Use the formula for the area of a trapezoid.12
A = • 3(31 + 40) Replace h with 3, b1 with 31, and b2 with 40.12
= • 3(71) Simplify.12
= • 21312
= 106.5
Pre-AlgebraPre-Algebra
Area: CirclesArea: Circles
Lesson 10-3
Objectives: 1. to find areas of circles
2. to find areas of irregular figures that include parts of a circle
Pre-AlgebraPre-Algebra
Area: CirclesArea: Circles
Lesson 10-3
Tips: remember to follow the order of operations when finding the area of a circle or semi-circle. Square the radius first, and then multiply by π.
when asked to find the “exact area” of a circle (or circular shape) do not substitute 3.14 in for π, leave π as π.
make sure not to confuse the radius as the diameter…know the difference.
Pre-AlgebraPre-Algebra
Area: CirclesArea: Circles
Find the exact area of a circle with diameter 20 in.
Lesson 10-3
A = r 2
= (10)2 r = d; r = 1012
= 100 Simplify.
The area is 100 in.2.
Pre-AlgebraPre-Algebra
Area: CirclesArea: Circles
A TV station’s weather radar can detect precipitation in a circular region having a diameter of 100 mi. Find the area of the region.
Lesson 10-3
A = r 2
The area of the region is about 7,850 mi2.
= (50)2 r = d; r = 5012
= 2,500 exact area
(2,500)(3.14) Use 3.14 for .
= 7,850 approximate area
Pre-AlgebraPre-Algebra
Area: CirclesArea: Circles
A pound of grass seed covers approximately 675 ft2. Find the
area of the lawn below. Then find the number of bags of grass seed
you need to buy to cover the lawn. Grass seed comes in 3-lb bags.
Lesson 10-3
Area of region that is one fourth of a circle:
area of circle = r 2
area of quarter circle = r 2
A (3.14)(15)2 Replace with 3.14 and r with 15.
= 176.625 ft2
14
14
Pre-AlgebraPre-Algebra
Area: CirclesArea: Circles
(continued)
Lesson 10-3
Area of region that is a rectangle:
area of rectangle = bh
A = 45 • 25 Replace b with 45 and h with 25.
= 1,125
The area of the lawn is about 177 ft2 + 1,125 ft2 = 1,302 ft2.
You need to buy one 3-lb bag of grass seed.
1,302 ÷ 675 1.93 Divide to find the number of pounds of seed.
Pre-AlgebraPre-Algebra
Space FiguresSpace Figures
Lesson 10-4
Objectives: 1. to identify common space figures
2. to identify space figures from nets
Tips: a “cube” is a rectangular prism with six congruent square faces
Lateral means “on the side”. The lateral faces of a prism or pyramid are the surfaces that connect with a base.
Pre-AlgebraPre-Algebra
Space FiguresSpace Figures
Lesson 10-4
Pre-AlgebraPre-Algebra
Space FiguresSpace Figures
Lesson 10-4
Pre-AlgebraPre-Algebra
Space FiguresSpace Figures
Lesson 10-4
Pre-AlgebraPre-Algebra
Space FiguresSpace Figures
Describe the bases and name the figure.
Lesson 10-4
The bases are circles.
The figure is a cylinder.
Pre-AlgebraPre-Algebra
Space FiguresSpace Figures
Name the space figure you can form from the net.
Lesson 10-4
With two hexagonal bases and rectangular sides, you can form a hexagonal prism.
Pre-AlgebraPre-Algebra
Surface Area: Prisms and CylindersSurface Area: Prisms and Cylinders
Lesson 10-5
Objectives: 1. to find surface areas of prisms
2. to find surface areas of cylinders
Pre-AlgebraPre-Algebra
Tips: SA = surface Area, LA = lateral area, B = area of the base, p= perimeter of the base
One way to find the surface area of a space figure is to find the area of its net.
Surface Area – is measured in square units.
SA of a cylinder can also be written as SA= 2πrh + 2πr2
Surface Area: Prisms and CylindersSurface Area: Prisms and Cylinders
Lesson 10-5
Pre-AlgebraPre-Algebra
Surface Area: Prisms and CylindersSurface Area: Prisms and Cylinders
Find the surface area of the rectangular prism
using a net.
Lesson 10-5
60 + 60 + 150 + 90 + 150 + 90 = 600 Add the areas.
The surface area is 600 cm2.
Draw and label a net.Find the area of each rectangle in the net.
Pre-AlgebraPre-Algebra
Surface Area: Prisms and CylindersSurface Area: Prisms and Cylinders
Find the surface area of the rectangular prism.
Lesson 10-5
The surface area of the rectangular prism is 500 in.2.
Step 1: Find the lateral area.
L.A. = ph Use the formula for lateral area.
= (5 + 6 + 5 + 6)20 p = 5 + 6 + 5 + 6 and h = 20
= 440
Step 2: Find the surface area.S.A. = L.A. + 2B Use the formula for surface area. = 440 + 2(5 • 6) L.A. = 440 and B = 5 • 6 = 440 + 60 = 500
Pre-AlgebraPre-Algebra
Surface Area: Prisms and CylindersSurface Area: Prisms and Cylinders
Find the surface area of the cylindrical water tank.
Lesson 10-5
The surface area of the water tank is about 1,156 ft2.
Step 1: Find the lateral area.
L.A. = 2 rh Use the formula for lateral area.
2(3.14)(8)(15)
754
Step 2: Find the surface area. S.A. = L.A. + 2B Use the formula for surface area. = L.A. + 2( r 2) 754 + 2(3.14)(8)2
1,156
Pre-AlgebraPre-Algebra
Surface Area: Pyramids, Cones, and SpheresSurface Area: Pyramids, Cones, and Spheres
Lesson 10-6
Objectives:
1. to find surface areas of pyramids
2. To find surface areas of cones and spheres
Pre-AlgebraPre-Algebra
Tips: the slant height is not perpendicular to the base of a pyramid or cone. In a pyramid, it is perpendicular to the base of a triangular face. In a
cone, it is the shortest segment that joins the vertex to a point on the circle base
there are a number of analogies that one can use to remember not only the lateral-area formulas, but also the surface-area formulas, each of which involves adding lateral area and base area(s).
Surface Area: Pyramids, Cones, and SpheresSurface Area: Pyramids, Cones, and Spheres
Lesson 10-6
Pre-AlgebraPre-Algebra
Surface Area: Pyramids, Cones, and SpheresSurface Area: Pyramids, Cones, and Spheres
Find the surface area of the square pyramid.
Lesson 10-6
The surface area of the pyramid is 105 m2.
Step 1: L.A. = p Use the formula for lateral area.12
= • 20 • 8 = 80 p = 4(5) and = 8.12
Step 2: S.A. = L.A. + B Use the formula for surface area. = 80 + 52 Lateral area = 80 and B = 52. = 80 + 25 = 105
Pre-AlgebraPre-Algebra
Step 1: L.A. = r Use the formula for lateral area.
3.14(3)(7) r = 3 and = 7.
= 65.94
Surface Area: Pyramids, Cones, and SpheresSurface Area: Pyramids, Cones, and Spheres
Find the surface area of the cone.
Lesson 10-6
The surface area of the cone is about 94 m2.
Step 2: S.A. = L.A. + B Use the formula for surface area.
65.94 + 3.14(3)2 L.A. 65.94 and B = (3)2.
= 65.94 + 28.26
= 94.2
Pre-AlgebraPre-Algebra
Surface Area: Pyramids, Cones, and SpheresSurface Area: Pyramids, Cones, and Spheres
Earth has an average radius of 3,963 mi. What is Earth’s approximate surface area to the nearest 1,000 mi2? Assume that Earth is a sphere.
Lesson 10-6
The surface area of Earth is about 197,259,000 mi2.
S.A. = 4 r 2 Use the formula for surface area.
= 197,259,434.64 Multiply.
197,259,000 Round to nearest 1,000.
4(3.14)(3,963)2 r 3,963
Pre-AlgebraPre-Algebra
Volume: Prisms and CylindersVolume: Prisms and Cylinders
Lesson 10-7
Objectives: 1. to find volumes of prisms
2. to find volumes of cylinders
Pre-AlgebraPre-Algebra
Tips: read the question carefully, you can calculate both the volume and surface area of a 3-D figure, make sure to understand the concepts of both and use the correct formula.
Volume: Prisms and CylindersVolume: Prisms and Cylinders
Lesson 10-7
Pre-AlgebraPre-Algebra
Volume: Prisms and CylindersVolume: Prisms and Cylinders
Find the volume of the triangular prism.
Lesson 10-7
The volume is 1,260 cm3.
V = Bh Use the formula for volume.
= 63 • 20 B = • 9 • 14 = 63 cm212
= 1,260 Simplify.
Pre-AlgebraPre-Algebra
Volume: Prisms and CylindersVolume: Prisms and Cylinders
Find the volume of the juice can, to the nearest cubic centimeter.
Lesson 10-7
The volume is about 581 cm3.
V = Bh Use the formula for volume.
V = r 2h B = r 2
= 580.7744 Simplify.
3.14 • 3.42 • 16 Replace r with 3.4, and h with 16.
Pre-AlgebraPre-Algebra
Volume: Pyramids, Cones, and SpheresVolume: Pyramids, Cones, and Spheres
Lesson 10-9
Objectives: 1. to find volumes of pyramids and cones
2. to find volumes of spheres
Pre-AlgebraPre-Algebra
Tips: the formula for the volumes of cones and pyramids uses each figure’s height, not slant height. Remember, the height of a cone or pyramid is the length of the segment from the vertex perpendicular to the base.
the volume of a sphere is the only one (studied so far) that involves a third power.
Volume: Pyramids, Cones, and SpheresVolume: Pyramids, Cones, and Spheres
Lesson 10-9
Pre-AlgebraPre-Algebra
Volume: Pyramids, Cones, and SpheresVolume: Pyramids, Cones, and Spheres
Find the volume of the cone.
Lesson 10-9
V = Bh Use the formula for volume.13
V = r 2h B = r 213
(3.14)(2)2(12) Replace r with 2 and h with 12.13
= 50.24 Simplify.
The volume of the cone is about 50 in.3.
Pre-AlgebraPre-Algebra
Volume: Pyramids, Cones, and SpheresVolume: Pyramids, Cones, and Spheres
Find the volume of the square pyramid.
Lesson 10-9
The volume of the pyramid is 256 in.3.
V = Bh Use the formula for volume.13
V = s 2h B = s213
= (8)2(12) Replace s with 8 and h with 12.13
= 256 Simplify.
Pre-AlgebraPre-Algebra
Volume: Pyramids, Cones, and SpheresVolume: Pyramids, Cones, and Spheres
Earth has an average radius of 3,963 mi. What is Earth’s approximate volume to the nearest 1,000,000 mi3? Assume that Earth is a sphere.
Lesson 10-9
The volume of the Earth is about 260,580,000,000 mi3.
V = r 3 Use the volume formula.43
(3.14)(3,963)3 Replace r with 3,963.43
260,579,713,159 Simplify.
Additional Examples