The topic of geometry is quite a common one to appear on your ASVAB test. In this section, we detail all the types of question you need to know to succeed in this area.
Posted in 4th December, 2014 | Mathematics KnowledgeGeometry
Geometry
In this last section of our ASVAB Mathematics Knowledge study guide, we will take a look at geometry. Geometry, for those of you who may be unaware, is a branch of mathematics that studies shape, size, and proportions in space. Luckily, questions involving geometry invariably involve analyzing shapes, which means you can extract the answer just by being able to “read” the data contained in the diagram. For the remaining questions, you will need to have knowledge of the various geometrical terminologies. Thankfully, this isn’t a difficult task as the terms become more and more familiar as you study more and more geometrical questions.
We will begin this section by looking at this terminology, before going on to look at some sample questions toward the end.
Geometrical Terminology
There are certain terms you need to become familiar with before continuing. Review the following terminology and make sure you have some basic understanding of them:
- A circle is composed of 360 degrees. Each degree is, in turn, subdivided into 60 minutes, with each minute, in turn, subdivided further into 60 seconds.
- The diameter of a circle is a straight line that passes through the center while touching the wall at either point.
- The radius of a circle is defined as half the diameter – the distance from the center of a circle to the edge of that circle.
- The internal angles of a triangle always add up to 180 degrees. A triangle has three angles – so if one angle is 70 degrees and a second is 40 degrees, the third angle, by deduction, has to be 70 degrees as well.
- A straight line has an angle of 180 degrees, but a right-angle has an angle of 90 degrees. The edge of a table, for example, is 90 degrees, while the stretch joining these table edges has an angle of 180 degrees. The side of the triangle directly opposite the right-angle is known as the hypotenuse.
- An acute angle is described as any angle between 0 and 90 degrees.
- An obtuse angle is any angle between 90 but less than 180 degrees.
- If two angles sum to 90 degrees, they are referred to as complementary angles.
- If two angles sum to 180 degrees, they are described as supplementary angles.
- If all three angles in a triangle are 60 degrees, it is referred to as an equilateral triangle.
- If two sides of a triangle have the same length, the third side of the triangle will have equal angles with the two sides. This is known as an isosceles triangle.
There is one other main concept you need to know. This is called Pythagoras theorem. This is a mathematical theorem that allows you to calculate the length of a side of one a triangle, once you know the lengths of the other two sides. Let’s take a look at an example to see how this works:
In this example, we know the values of two sides of the triangle, but we are missing the length of the hypotenuse. Let’s not forget that the hypotenuse is the name of the side of the triangle directly opposite a right-angle. We employ the following formula to calculate the third side of the triangle:
- a^2 + b^2 = c^2
- 9^2 + 4^2 = c^2
- 81 + 16 = c^2
- 97 = c^2
- √9 7= c
- c = 9.9cm
Always remember to use the c^2 to reflect the side of the hypotenuse of each right-angled triangle.
In terms of circles, you should be able to calculate the circumference of a circle. We can see one such example of this below:
Q. If a circle has a radius of 14 feet, what is the approximate circumference of the circle?
- a) 66ft
- b) 77ft
- c) 88ft
- d) 99ft
In this example, note we’re given the value of the radius. The radius is defined as only one half the diameter of the entire circle so, in this sense, D = 2R. If the radius of the circle is 14 feet, then the diameter of the circle is 28 feet. Why do we need the value of the diameter?
If we wish to know the circumference of a circle, we must first know its diameter. After all, the equation used to calculate circumference is π.d. Thus, the equation reads as follows:
- Radius of Circle = 14 feet
- Therefore, Diameter of Circle = 28 feet
- Equation to Calculate Circumference = π.d
- The value of π is 3.142, so the circumference is 3.142 x 28 feet = 88ft!
If you wish to calculate the area of the circle, you would use the simple equation πr^2. If you do not have a calculator at the ready, you can think of π either representing the value of 22⁄7 or 3.14.
Calculating Volume
The volume of something is the amount of “stuff” that can fill a three-dimensional space. So, take a standard box. The amount of space occupying the inside of the box is the same as its volume. You may be asked to calculate volume in the math exam, particularly in reference to boxes and cylinders.
The formula for calculating the area of a box is V = l.w.h.
In other words, you need to multiply the length, the width, and the height together. If a standard box were to measure 6 feet long, 5 feet wide, and 3 feet deep, the total area would be 6 x 5 x 3, which is 90ft^3. Why is it measured in cubic feet (3)? This is because you had to multiply three dimensions together to attain the correct answer.
The formula for calculating the area of a cylinder is V = πr^2h
A cylinder is defined by the presence of two circles at the base of the shape. This is the shape we find pipes in. To calculate the volume, we use π (which has a constant value), the radius of the circle, as well as the height of the cylinder. If a cylinder has a radius of 5 inches, and a height of 9 inches, the volume of the cylinder would be (3.14).(5).(9), which when multiplied out gives 141.4 cubic inches.
Let’s take a look at another example of how to calculate the volume of a cylinder:
In the final section, we take a look back over this study guide and offer some useful pointers about how you should structure your future study.
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