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2-Dimensional Geometry Somerset Math Club 2016-2017
A
B
C
D
E
G
F
H
Area Formulas
1.
Circles: A=
2
a.
Circular Sectors: A=
2
*
/
360
(in degrees) 2.
Triangles:
=
ℎ
/
2
3.
Rectangles/Parallelograms/Squares: A=b*h a.
Trapezoids: A=(
1
2
)∗
ℎ
/
2
Challenge: Can you think of a way to explain why this is the area formula for a trapezoid?
Hint: Try rearranging parts of the trapezoid.
Challenge Question
ABCD is a trapezoid with AB parallel to DC. E and G and midpoints of AD and BC, respectively. If the area of trapezoid ABCD is 100, what is the area of the quadrilateral, EFGH?
Cir cles
The
2-Dimensional Geometry Somerset Math Club 2016-2017
re are 360 degrees in a full circle. However, radians can also be used instead of degrees. There are 2
radians in 360 degrees because there are 2
radiuses in a circumference. A radian is the central angle that is formed by a sector of the circumference of length r. Convert the following angles from degrees to radians in terms of
: 1.
60° 2.
90° 3.
180° 4.
360° Convert the following angles from radians to degrees:
1.
/
3
2.
/
2
3.
4.
2
5.
360 Why do we use radians?
First, a degree is an arbitrary definition: we define 360 degrees to be the number degrees in a full circle, but any number could have been defined as the number of degrees in a circle. The radian is a measurement defined in terms of a property of a circle, making it less arbitrary. It is also a dimensionless unit, so it can be squared or square rooted without issues with units.
Second, using radians give a much more easily used formula for the area of a sector of a circle or the length of an arc on a circle. The formula for the length of an arc with a central angle
2-Dimensional Geometry Somerset Math Club 2016-2017
b
c
in degrees, which can be proven using conversion from radians to degrees, is r
while the area of a circular sector with that central angle is r
2
∗
/
2
.
Overlapping/Tangent Circles
If the radius of both circle A and circle C is 6 cm, what is the area that the two circles have in common (the green area)? If the radius of circle A is 6 cm and the radius of circle C is 3 cm, what is the area that the two circles have in common? If the radius of the smallest circle is 1 cm, the radius of the 2nd smallest circle is 2 cm, and the radius of the largest circle is 3 cm, what is the area of the triangle formed by connecting the centers of the circles?
Right Triangles
The area of a right triangle is ab/2. This is based on the area of a rectangle. By the Pythagorean Theorem, a
2
+ b
2
= c
2
In an acute triangle, a
2
+ b
2
> c
2
while in an obtuse triangle, a
2
+ b
2
< c
2
The following are the trigonometric functions in a right
2-Dimensional Geometry Somerset Math Club 2016-2017
a
b
c
triangle.
sin
=
ℎ
sin is the abbreviation for sine.
cos
=
ℎ
cos is the abbreviation for cosine.
tan
=
tan is the abbreviation for tangent.
Using trigonometry, we can also find a formula for the area of non-right triangles. Given that the angle between sides a and b is
, the area of the triangle is a*b*sin
/2. The formula for the area of any triangle, given its three side lengths, is
√
(
−
)(
−
)(
−
)
where s=
+
+
2
Triangle Problems
If a triangle with an area of 84 has side lengths of 13, 14, and 15, what are the three angles of the triangle? (Use a calculator)
What is the area of a triangle with side lengths 5, 7, and 9?
If two angles of a triangle are equal, what type of triangle is it? If all three angles of a triangle are equal, what type of triangle is it?
What is tangent in terms of sine and cosine? What is sin 30? What is cos 30? If the angles of a triangle are 30 degrees, 60 degrees, and 90 degrees, what is the ratio of the lengths of the sides of the triangle? If the angles of a triangle are 45 degrees, 45 degrees, and 90 degrees, what is the ratio of the lengths of the sides of the triangle?

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