Download Volumes, Perimeters, Circles, Spheres, Triangles, Angles, Rays and

Volumes, Perimeters, Circles, Spheres, Triangles, Angles, Rays and Segments
This week’s lesson deals with figures such as rectangles, parallelograms, triangles, circles and spheres.
Rays and Segments
A line extends infinitely in both directions. If it has one endpoint, it is called a ray. The notation for ray is
, where A is the starting point and B is some other point it passes through. If the line has two
endpoints, it is called a segment, and is denoted .
Two rays meeting at one point form an angle. An angle below 90° is acute, at 90° is right, and above 90°
is obtuse. If the two rays form a line, we call it a straight angle, and it has an angle of 180°.
Two angles are complementary is they sum to 90°. Two angles are supplementary is they sum to 180°.
Triangles:
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Any three-sided planar figure is called a triangle. The sum of the interior angles is always 180°
If all three sides are equal in length (equivalently, the three interior angles are all the same, 60°),
the triangle is called equilateral.
If two of the sides (equivalently, two of the angles) are the same, the triangle is called isosceles.
If all three sides are different, it is called scalene.
If one of the angles meets at 90°, it is called a right triangle. The other two angles then sum to 90°
as well, and are called complementary. Equivalently, the sum of two complementary angles is 90°.
If the lengths a and b are given for the two legs of a right triangle, the hypotenuse c is found by the
Pythagorean Formula, = √
+ .
If one interior angle is above 90°, then the triangle is obtuse. Otherwise, it is acute.
If two triangles are proportional, they are called similar. You can solve for unknown sides if you
are given some information about the lengths by setting up simple proportions.
The area of any triangle is = baseheight, where base and height are at 90 degrees.
Example: Look at the picture below.
Note that triangles ABC and DBE are similar. Thus, 9 is to 7 as (x + 10) is to 10. As a ratio, we have
= . Solving for x, we cross multiply to get 90 = 7! + 10. This gives 90 = 7! + 70, or 20 = 7!.
Thus, ! =
≈ 2.857.
Both triangles in this case are right triangles. Thus, the hypotenuse of ABC is the distance from B to C.
The legs are 9 and 12.857 (the distance from A to B is 2.857 + 10). Thus,
= (9
+ 12.857
= √246.302449 ≈ 15.69.
= '
Four-sided figures
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Any four sided figure is called a quadrilateral.
The sum of the interior angles is 360°.
If two of the sides are parallel, it is called a trapezoid.
If both pairs of opposite lines are parallel, it is called a parallelogram.
If a parallelogram has all sides of equal length, it is a rhombus.
If a rhombus also has all angles at 90°, it is a square.
If a parallelogram has angles of 90°, it is a rectangle.
The area of a parallelogram is base times height, where height is always at 90° to the base, not necessarily
along the “slant” side. The area of a square is side times side, or side-squared, or = , .
Circles
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A circle is the set of points equidistant from a common point, called the center.
A line from the center to a point on the circle is a radius.
Any line that connects two points on a circle is a chord.
A chord that goes through the center is a diameter. Equivalently, the length of a diameter is twice
the radius.
The distance (perimeter) around a circle is the circumference.
The ratio of a circle’s circumference to its diameter is -, or about 3.14159… . In symbols, we
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have - = /, or that ' = -0, or ' = 2-1.
The area of a circle is = -1 .
Three-dimensional figures
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Any box-shaped figure that is composed of rectangular or square faces is called a parallelepiped,
or simply a “rectangular box”. The volume of any such shape is V = length times width time
height, or 2 = 34ℎ. If all lengths are the same, the box is a cube, and the volume is 2 = , 6 , where
s is the side length.
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The volume of a sphere is 2 = 6 -1 6 , and the surface area of a sphere is 8 = 4-1 . A
hemisphere is half a sphere.
Example: Look at this picture of a hemisphere of radius 6:
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The volume is half of a sphere, so 2 = 96 -66 : = 6 -216 ≈ 452.39. The surface area of the
hemisphere “dome” is 8 = 4-6
= 2-36 ≈ 226.19. The total surface area would include the
base, which is a circle with area = -6
= 26- ≈ 113.09, so the total surface area is 226.19 +
113.09 = 339.29.
When handling areas and volumes, be sure all units are the same.
Example: a rectangle has length 5 feet and height 24 inches. The area would be A = (5)(2) = 10 square
feet. Be sure to convert the 24 inches to 2 feet first.
Example: A box is 10 feet by 8 feet by 4 inches. The volume is V = (10)(8)(1/3)=26.667. We had to
convert the 4 inches to 1/3 of a foot.
Pyramids and Cones
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A pyramid has a flat base (of any shape) and sides that meet at a single point, called the apex.
If the base is a circle, then we have a cone.
The volume of a pyramid is 2 = heightareaofbase.
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Example: Look at the shape below:
Its volume is 2 = 107
=
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= 163.333units 6 .
Be aware of shapes that are subtracted out from bigger shapes. For example, the pyramid above came
from a rectangular solid that was 7 by 7 by 10, or 490 cubic units of volume. If the pyramid itself has
volume 23.333 cubic units, then the remaining material would have a volume of 490 – 163.333 = 326.67
cubic units.
Be aware also that many shapes are the sum (or difference) of more basic shapes. Thus, there is no need to
develop formulas for all these shapes. Mentally decompose these shapes into smaller shapes and use the
formulas above.