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1 Lesson 3.1.2 Areas of Polygons
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Lesson 3.1.2Lesson 3.1.2
Areas of PolygonsAreas of Polygons
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Lesson
3.1.2Areas of PolygonsAreas of Polygons
California Standard:Measurement and Geometry 1.2Use formulas routinely for finding the perimeter and area of basic two-dimensional figures, and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
What it means for you:You’ll use formulas to find the areas of regular shapes.
Key words:• area• triangle• parallelogram• trapezoid• formula• substitution
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Areas of PolygonsAreas of PolygonsLesson
3.1.2
Area is the amount of space inside a shape. Like for perimeter, there are formulas for working out the areas of some polygons.
You’ll practice using some of them in this Lesson.
Area
Perimeter
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Square:
A = s2
s
s
l
wRectangle:
A = lw
Parallelogram:
A = bhb
h
Areas of PolygonsAreas of Polygons
Area is the Amount of Space Inside a Shape
Lesson
3.1.2
Triangles and other shapes can be a little more difficult, but there are formulas for those too — which we’ll come to next.
Area is the amount of surface covered by a shape.
Parallelograms, rectangles, and squares all have useful formulas for finding their areas.
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Areas of PolygonsAreas of Polygons
Example 1
Solution follows…
Lesson
3.1.2
Use a formula to evaluate the area of this shape.
Use the formula for the area of a rectangle.
Solution
Substitute in the values given in the question.
A = 14 in2
7 in
2 in
A = lw
A = 7 in × 2 in
Evaluate the area.
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Areas of PolygonsAreas of Polygons
Example 2
Solution follows…
Lesson
3.1.2
You can also rearrange the formulas to find a missing length:
Find the height of a parallelogram of area 42 cm2 and base length 7 cm.
Solution
Rearrange the formula for the area of a parallelogram, and substitute.
A = bh
A ÷ b = bh ÷ b = h
h = 42 ÷ 7 = 6 cm
b
hbh
Divide both sides by the base (b)
Substitute values and evaluate
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Areas of PolygonsAreas of Polygons
Guided Practice
Solution follows…
Lesson
3.1.2
1. Find the area of a square of side 2.4 m.
2. Find the length of a rectangle if it has area 30 in2, and width 5 in.
A = s2 = 2.42 = 5.76 m2
A = lw l = A ÷ w = 30 ÷ 5 = 6 in
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Areas of PolygonsAreas of Polygons
The Area of a Triangle is Half that of a Parallelogram
Lesson
3.1.2
The area of a triangle is half the area of a parallelogram that has the same base length and vertical height.
height (h)
base (b)
+ height (h)
base (b)
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Areas of PolygonsAreas of PolygonsLesson
3.1.2
In math language, the area of a triangle is given by:
Area of triangle = area of parallelogram
= (base × height)
= bh
1
2
1
2
1
2
A = bh1
2
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Areas of PolygonsAreas of Polygons
Example 3
Solution follows…
Lesson
3.1.2
Find the base length of the triangle shown if it has an area of 20 in2 and a height of 8 in.
8 in
b
Solution
Rearrange the formula for the area to give an expression for the base length of the triangle.
A = bh 1
2
2A = bh
2A ÷ h = bh ÷ h
b = 2A ÷ h
Multiply both sides by 2
Write out the formula
Divide both sides by the height (h)
Now substitute in the values and evaluate to give the base length.
b = 2A ÷ h
= 5 in
= (2 × 20) ÷ 8
Simplify
Solution continues…
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Areas of PolygonsAreas of Polygons
Guided Practice
Solution follows…
Lesson
3.1.2
3. Find the area of a triangle of base length 3 ft and height 4.5 ft.
4. Find the base length of a triangle with height 50 m and area 400 m2.
A = 0.5(3 × 4.5) = 0.5 • 13.5 = 6.75 ft2
b = 2A ÷ h = (2 • 400) ÷ 50 = 16 m
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Areas of PolygonsAreas of Polygons
Break a Trapezoid into Parts to Find its Area
Lesson
3.1.2
The most straightforward way to find the area of a trapezoid is to split it up into two triangles.
Notice that both triangles have the same height but different bases.
You then have to work out the area of both triangles and add them to find the total area.
12 height
(h)base of triangle 1 (b1)
base of triangle 2 (b2)
12
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Areas of PolygonsAreas of Polygons
So, the area of the trapezoid is the sum of the areas of each triangle.
Lesson
3.1.2
h
b1
12
b2
Area of trapezoid = area of triangle 1 + area of triangle 2
= b1h + b2h1
2
1
2
Take out the common factor of h to give:1
2
Area of trapezoid = h(b1 + b2)1
2
A = h(b1 + b2)1
2
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Areas of PolygonsAreas of Polygons
Example 4
Solution follows…
Lesson
3.1.2
Find the area of the trapezoid shown.
8 ft
30 ft
12 ft
Solution
Area of trapezoid = h(b1 + b2)1
2
Substitute in the values given in the question and evaluate.
Area of trapezoid = × 8 × (12 + 30) = × 8 × 42 = 168 ft2.1
2
1
2
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Find the areas of the trapezoids in Exercises 5–8, using the formula.
5. 6.
7. 8.
Areas of PolygonsAreas of Polygons
Guided Practice
Solution follows…
Lesson
3.1.2
3 in
10 in
5 in
4 cm
20 cm
11 cm
1.5 m
1.1 m
0.7 m
245 ft
105 ft
80 ft
0.5 • 3 • (5 + 10)=22.5 in2
0.5 • 11 • (20 + 4)= 132 cm2
0.5 • 0.7 • (1.1 + 1.5)= 0.91 m2
0.5 • 80 • (105 + 245)= 14,000 ft2
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Areas of PolygonsAreas of Polygons
Independent Practice
Solution follows…
Lesson
3.1.2
Find the area of each of the shapes in Exercises 1–2.
1. 2.
1.2 ft
1.2 ft
1 m
1 m
1.44 ft2 0.5 m2
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Areas of PolygonsAreas of Polygons
Independent Practice
Solution follows…
Lesson
3.1.2
Find the area of each of the shapes in Exercises 3–4.
3. 4.
2 in
2.3 in
7 in
2.5 in
2.3 in2 17.5 in2
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Areas of PolygonsAreas of Polygons
Independent Practice
Solution follows…
Lesson
3.1.2
Find the area of each of the shapes in Exercises 5–6.
5. 6.
20 cm
11 cm
12 cm
4.5 ft
3.1 ft
186 cm2 13.95 ft2
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Areas of PolygonsAreas of Polygons
Independent Practice
Solution follows…
Lesson
3.1.2
7. Miguel wants to know the area of his flower bed, shown below. Find the area using the correct formula.
3.1 m
2.4 m
3.72 m2
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Areas of PolygonsAreas of Polygons
Round UpRound Up
Lesson
3.1.2
Later you’ll use these formulas to find the areas of irregular shapes.
Make sure you practice all this stuff so that you’re on track for the next few Lessons.
