TOPIC 18
More Areas of Geometric Shapes
We have worked with the areas of squares, rectangles, parallelograms, and trapezoids on the
cycle problems. Here is a miscellany of problems to practice as well as other useful area
formulas. There is space to add notes with the explanations about why these methods work.
First, let’s write a list of the area formulas or these shapes.
Area of a rectangle = _________________________
Area of a parallelogram = ________________________
Area of a square = ______________________
Area of a triangle = __________________________
(Why is this the area of a triangle?)
Area of a trapezoid = _________________________
Let’s derive it again. Draw a trapezoid whose bases are labelled as b1 and b2. The
height is h. Draw one of the diagonals, and use the triangles to derive a formula for the
area of the trapezoid.
TOPIC 18: More Areas and Volumes of Geometric Shapes
I.
page 2
More area examples.
Find the areas. You might need to add some auxiliary lines (such as altitudes) to subdivide the
regions. Use the area formulas you know, the information about right triangles and special right
triangles, and other clever and creative problem solving.
1.
2.
3.
4.
5.
6.
TOPIC 18: More Areas and Volumes of Geometric Shapes
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7.
If the width of a rectangle is increased by 10% and its length increases by 25%, how is
the area of the rectangle changed?
8.
If the width of a rectangle is increased by 50% and the length is decreased by 25%, how
is the area of the rectangle affected?
9.
h1 = 8
h2 = _____
10.
A paved walk which is 3 feet wide surrounds a rectangular grass plot which is 30 feet
long and 18 feet wide. Find the area of the paved walk.
TOPIC 18: More Areas and Volumes of Geometric Shapes
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11.
Find the area of a parallelogram with sides 12 inches and 18 inches long if one of its
angles measures 1200.
12.
The sides of a parallelogram are 6 inches and 8 inches long. Find the length of the longer
altitude if a shorter altitude is 4 inches long.
II.
Quadrilaterals with perpendicular diagonals
 Kites
 Rhombuses
 Squares
 Others
Because the diagonals are perpendicular, what is the shape of the four triangles which fill
each quadrilateral?
Write the area of each triangle, using lengths of parts of diagonals. Then add the four
areas and simplify.
TOPIC 18: More Areas and Volumes of Geometric Shapes
13.
14.
If the sides of a square measure 6 cm, what is the length
of the diagonals? Use the new formula above, as well as
the usual formula, to calculate the area of a square.
For the kite, BC = CD = 5
and BD = 6. AC = 14
Find the area of the kite.
15.
The sides of the rhombus measure 5 inches. The
shorter diagonal measures 6 inches. Find the area
of the rhombus.
page 5
TOPIC 18: More Areas and Volumes of Geometric Shapes
page 6
16.
Find the area of a rhombus with sides 13 cm long and long diagonal 10 cm long.
17.
The shorter diagonal of a rhombus has the same length as a side of the rhombus. Find the
area of the rhombus if the longer diagonal is 12 inches long.
18.
AC = 6
DE = 8
Area of the kite =
19.
TOPIC 18: More Areas and Volumes of Geometric Shapes
20.
GJ = 20
JH = 24
page 7
JI = 13
Area of the kite =
III.
Miscellaneous areas and ratios of areas.
21.
ABCD is a trapezoid with AB || DC . Find the area of the trapezoid given that AB = 25,
BC = 20, CD = 4 and DA = 13.
22.
An isosceles trapezoid with bases of lengths 12 and 16 inscribed in a circle of radius 10.
The center of the circle lies in the interior of the trapezoid. Find the area of the trapezoid.
TOPIC 18: More Areas and Volumes of Geometric Shapes
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23.
Two sides of a triangle are 6 cm and 10 cm long. Find the ratio of the areas of the two
triangles into which the bisector of the angle determined by these sides divides the
triangle. (Hint: An altitude of a triangle can be inside, on, or outside the triangle. An
altitude can serve as the altitude of more than one triangle, as long as it is perpendicular
to each of the respectively bases.)
IV.
Areas of Equilateral triangles
Draw an altitude. If the side has length, s, calculate
the measure of the height. Multiple the base and
height, divide by 2, and then simplify.
Your result should only have the variable, s.
24.
Find the area of an equilateral triangle whose sides measure 8 cm.
TOPIC 18: More Areas and Volumes of Geometric Shapes
25.
Find the area of an equilateral triangle whose perimeter is 12 feet.
26.
Find the length of a side of an equilateral triangle whose area is 25 3 ft2.
V.
Area of triangles (alternate methods)
A.
Using trigonometry (SAS)
We will derive an area formula for the triangle determined by SAS.
page 9
TOPIC 18: More Areas and Volumes of Geometric Shapes
B.
If three sides of a triangle are given (SSS), then Heron’s formula can be used:
A=
C.
page 10
s ( s  a )( s  b)( s  c) , where s = semi-perimeter of the triangle.
Using coordinates of vertices and matrices
We would agree that the area o this triangle is
1
(5)(7) = 17.
2
If we look at the vertices of the triangle, we
can use a kind of determinant to calculate the
area in a clever way.
Look at the coordinates of each of the vertices
of the triangle. Start anywhere, and travel
counterclockwise around the triangle. Record
the (x , y) coordinates. Using the starting point at the beginning and the end.
0

1 5
Area = 
2 7

0
0
0 
. To evaluate this determinant, multiply all of the pairs on the “NW5

0
SE” diagonal, and sum those products. Then multiply all of the pairs on the “SW-NE”
diagonal and sum those products. Finally, take the first sum and subtract the second sum.
This will be the area of the triangle!
TOPIC 18: More Areas and Volumes of Geometric Shapes
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Try this determinant method on this polygon. Check your answer by calculating the area using
the subtractive method.
Exercises with areas of triangles, using these new methods.
27.
In a triangle ABC, b = 4, m<A = 380, and c = 11. Find the area.
28.
The three sides of a triangle measure 5, 6, and 9. Find the area.
TOPIC 18: More Areas and Volumes of Geometric Shapes
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29.
The three sides of a triangle measure 12, 15, and 30. Find the area.
(Put your seat belts on!!)
30.
Find the area of a triangle whose coordinates are (0,0) , (4, 0), and (0, 3). You can use a
traditional method, or you can use the ideas with matrices.
31.
Find the area of a triangle whose coordinates are (0 , 0), (6 , 0), and (2 , 3).
32.
Find the area of a triangle whose coordinates are (3 , 1), (6 , 2), and (1 , 4).
33.
Find the area of a triangle whose coordinates are (4, -3), (10, 5), and (-1, 3)
TOPIC 18: More Areas and Volumes of Geometric Shapes
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34.
Find the area of a polygon whose coordinates (in counterclockwise order) are (0, 0),
(2, -1), (4, 2), (3, 4), (1, 4), and (3 , 2).
VI.
Regular polygons.
Defn: The apothem of a regular polygon
is ….
We will derive the area of a regular polygon, using a hexagon as an example.
35.
Find the area of a regular hexagon whose sides measure 6.
TOPIC 18: More Areas and Volumes of Geometric Shapes
36.
Find the area of a regular octagon whose sides measure 4.
VII.
Subtractive methods.
page 14
For some irregular areas, you can surround the figure with another shape whose area is
easier to calculate. Then you can subtract off pieces of shapes that are outside the desired
area.
37.
38.
TOPIC 18: More Areas and Volumes of Geometric Shapes
page 15
VIII. Comparing the perimeters and areas of similar figures.
39.
Suppose that two triangles are similar, and the a1 = 6 and a2 = 12. Then the ratio of all
corresponding parts of the triangles is 1:2. Find the ratio of the perimeters and the ratio
of the areas of the two triangles.
40.
Suppose that two rectangles are similar. The dimensions of one rectangle are 4 by 9 and
the dimensions of the other are 12 by 27. Find the ratio of the perimeters and the ratio of
the areas.
41.
What is the ratio of:
a)
DE
AB
b)
perimeter DCE
perimeter ABC
c)
42.
area DCE
area ABC
area DCE
area ACB
TOPIC 18: More Areas and Volumes of Geometric Shapes
IX.
page 16
Moving from regular polygons to circles
AREAS OF POLYGONS (apothem = 1) # of Sides m<1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 60 45 36 30 25.71429 22.5 20 18 16.36364 15 13.84615 12.85714 12 11.25 10.58824 10 9.473684 9 8.571429 8.181818 7.826087 7.5 7.2 1/2 side 1.732047
0.999999
0.726542
0.57735
0.481574
0.414213
0.36397
0.324919
0.293626
0.267949
0.246478
0.228243
0.212556
0.198912
0.186932
0.176327
0.16687
0.158384
0.150726
0.143778
0.137447
0.131652
0.126329
side perimeter area 3.464095 10.39228 5.196142
1.999997 7.999989 3.999995
1.453083 7.265417 3.632709
1.154699 6.928196 3.464098
0.963148 6.742038 3.371019
0.828426 6.627411 3.313705
0.72794 6.551458 3.275729
0.649839 6.498388 3.249194
0.587252 6.459777 3.229889
0.535898 6.430775 3.215387
0.492955 6.408419 3.204209
0.456487 6.390812 3.195406
0.425113 6.376691 3.188346
0.397824
6.36519 3.182595
0.373864 6.355696 3.177848
0.352654 6.347766 3.173883
0.333741 6.341073 3.170537
0.316769 6.335372 3.167686
0.301451 6.330476 3.165238
0.287556
6.32624 3.16312
0.274893 6.322549 3.161275
0.263305 6.319314 3.159657
0.252659 6.316464 3.158232
convert to radians 1.047197
0.785398
0.628318
0.523598
0.448799
0.392699
0.349066
0.314159
0.285599
0.261799
0.241661
0.224399
0.209439
0.196349
0.184799
0.174533
0.165347
0.15708
0.1496
0.1428
0.136591
0.1309
0.125664
Take the formula for a regular polygon:
A=
1
ap
2
As a regular polygon becomes a circle, the apothem becomes the length of the radius, and
the perimeter becomes the circumference.
1
1
ap  r (2 r )   r2
2
2
(Notice that the perimeter above of a many sided regular polygon approaches 6.28 with a radius
of 1, so this provides the c = 2  r)
A=
TOPIC 18: More Areas and Volumes of Geometric Shapes
X.
page 17
Parts of a circle.
Defn: An arc of a circle is a part of the circle. (It is a length)
Defn: A sector of a circle is a two dimensional part of a circle bounded by two radii and
an arc.
Length of the arc =
Length of the arc =

360
2 r (degrees)

2 r (radians)
2
sector
arc
= r
Area of a sector =
Area of a sector =

360
 r 2 (degrees)

 r 2 (radians)
2
1
=  r2
2
43.
Find the length of an arc whose central angle measures 1200 in a circle with radius 12
inches.
44.
Find the length of an arc in a circle whose measure is 1.5 radians and whose radius has
measure 8cm.
TOPIC 18: More Areas and Volumes of Geometric Shapes
45.
Find the area of a sector whose arc measures 600and whose radius is 20 meters.
46.
Find the area of a sector whose arc measures 2100 and whose radius is 3 inches.
47.
Find the circumference of a circle whose area is 49  .
page 18