Algebra & Geometry Practice Problems

Try the algebra and geometry practice problems on this page to help you get ready for your exam.

Algebra and Geometry Quiz

Instructions: There are 20 questions in total in this algebra and geometry quiz. Select your answer and then click on the “Show Answer” button. You will then see the answers and explanations after each question. Once you have studied each solution, click on the “Next” button to see the next question. Your score will be given at the end of the quiz.

1. Expand the polynomial: (x – 5)(3x + 8)

 
 
 
 

2. If 3/4x − 2 = 4, x = ?

 
 
 
 

3. If x – 15 > 0 and y = x – 15, then y > ?

 
 
 
 

4. X is 5 times Y, and Y is 5 more than 4 times Z. Which of the following describes the relationship between X and Z?

 
 
 
 

5. The two lines represented by the equations below intersect in the xy-plane. What is the y-coordinate at the point of intersection?
2x – 3y = 37
x = 5

 
 
 
 

6. Which of the following is equivalent to (x2 × x4)3?

 
 
 
 

7. A field has an area of exactly 49 square feet. Which one of the following could be the length (L) and width (W) of the field?

 
 
 
 

8. Factor using the greatest common factor: 18xy – 24x2y – 48y2x2

 
 
 
 

9. √3 = ?

 
 
 
 

10. What number is next in this sequence? 5, 10, 20, 40

 
 
 
 

11. If f1(x) = x2 + x, what is the value of f1(5) ?

 
 
 
 

12. How many 2 item permutations can be made from a four-item set?

 
 
 
 

13. Express 73 as a logarithmic function.

 
 
 
 

14. Which one of the following is a solution to the following ordered pairs of equations?
–3x – 1 = y
x + 7 = y

 
 
 
 

15. Becky is making a patchwork quilt that is going to be 6 feet long and 5 feet wide. What will the surface area of the quilt be in square feet?

 
 
 
 

16. A fence needs to be put around a field that is 12 yards long and 9 yards wide. What figure below best represents the perimeter of this field in yards?

 
 
 
 

17. A circular fish pond is being designing for your local park. The pond has an area of about 78.5 square feet. What is the approximate diameter (in feet) of the pond?

 
 
 
 

18. A tank that holds dye is 5 feet wide, 8 feet long, and 3 feet high. How many cubic feet of dye can the tank hold when it is completely full?

 
 
 
 

19. A cylindrical tank has a 5 meter radius and is 21 meters in height. What is the volume of the tank?

 
 
 
 

20. A confection company manufactures three different sizes of ice cream cones. The large cones are 6 inches high and have a 1.5 inch radius, the medium cones are 5 inches high and have a 1 inch radius, and the small cones are 4 inches high and have a 0.5 inch radius. What is the difference between the volume in cubic inches of the large cone and the medium cone?

 
 
 
 


Geometry concepts

In addition to the geometry concepts included in the quiz above, you should also review the following concepts and formulas.

Triangle laws

algebra and geometry practice problems image 1

A carpenter is making a special triangular-shaped corner shelf for a custom order. The customer lives in a 300-year-old house, so the walls are not completely straight and the corners are not completely square. He needs to make a triangular shelf that will have one 44° angle and one 47° angle. What is the measurement in degrees of the third angle of this shelf?

A)  45°                            

B)  45.5°             

C)  89°                            

D)  90°

Tip:

The sum of the angles in a triangle is 180 degrees.

Answer:

The correct answer is C. we know from the concept stated above that the sum of the angles in a triangle is 180 degrees. So, subtract the measurements of the other two angles to solve:

180° – 47° – 44° = 89°

Degrees in a circle

algebra and geometry practice problems image 2

A real-estate developer has recently purchased a circular-shaped tower. The first floor of the building has been divided into 5 pie-shaped segments that join at the center of the circle. The first segment measures 82° along the outside edge. The second segment has a measurement of 79°, the third has a measurement of 46° and the fourth has a measurement of 85°. What is the measurement in degrees of outside edge the fifth segment?

A)  48                              

B)  49                              

C)  58                              

D)  68

Tip:

A circle has 360 degrees in total.

Answer:

The correct answer is D. From the tip after the question, we can see that a circle has 360 degrees. So, subtract to solve:

360 – 82 – 79 – 46 – 85 = 68

Area of a circle

algebra and geometry practice problems image 3

A building project has a circular tower. The floor of the tower, which has a 12-foot radius, needs to be filled in with concrete. In order to do this, the area of the floor of the tower needs to be calculated. What is the approximate area of the floor of the tower in square feet?

A)  452.16                       

B)  376.80                       

C)  226.08                      

D)  37.68

Tip:

area of a circle ≈ 3.14 × (radius)2

Answer:

The correct answer is A. From the formula, we can see that the area of a circle ≈ 3.14 × (radius)2. So, put in 12 feet for the radius to solve: 

3.14 × (12 × 12) = 3.14 × 144 = 452.16

Circumference of a circle

algebra and geometry practice problems image 4

A technician measures the wear on tractor tires. In order to determine the rate of wear, the circumference of each tire must be determined first. The tire currently being measured has a diameter of 46.5 inches. What is the circumference?

A)  23.500 inches

B)  73.005 inches

C)  146.01 inches           

D)  292.02 inches

Tip:

circumference of a circle ≈ 3.14 × diameter

Answer:

The correct answer is C. From the formula, we know that the circumference of a circle ≈ 3.14 × diameter. The problem states that the diameter of the tractor tire is 46.5 inches, so use that in the formula to solve:

3.14 × 46.5 = 146.01 inches

Cubic inches

beaker image

A beaker is cylindrical and measures 18 inches high and 12 inches in diameter. However, the volume has to be converted from cubic inches to gallons for a report. What is the approximate volume of the beaker in terms of gallons?

A)  2.9 gallons                

B)  8.8 gallons                

C)  10.4 gallons  

D)  8,138.88 gallons

Tip:

Cylinder volume ≈ 3.14 × radius2 × height

Answer:

The correct answer is B. Step 1 – Find the volume in terms of cubic inches. Remember that radius is half of diameter. Here we have a diameter of 12, so the radius is 6.

Cylinder volume ≈ 3.14 × radius2 × height ≈

3.14 × 62 × 18 ≈ 3.14 × 36 × 18 ≈ 2034.72

Step 2 – Convert the volume in cubic inches to gallons. 1 gallon = 231 cubic inches, so divide by 231 to convert to gallons:

2034.72 ÷ 231 = 8.8 gallons

Cubic volume

cube image

The volume of a cube-shaped object needs to be calculated. The cube has a side length of 9 feet. However, a report is asking for the volume of the object in terms of cubic inches. Which figure below should be used?

A)  729 cubic inches                                           

B)  1,728 cubic inches                                       

D)  1,259,712 cubic inches

C)  139,968 cubic inches

Tips:

volume of a cube = (length of side)3

1 cubic foot = 1,728 cubic inches

Answer:

The correct answer is D. Step 1 – First we need to calculate the volume in terms of cubic feet. The volume of a cube = (length of side)3. The length of the side is 9 feet, so the volume is 9 × 9 × 9 = 729 cubic feet.

Step 2 – We have to convert the result from Step 1 to cubic inches. From the formula, we can see that 1 cubic foot = 1,728 cubic inches, so multiply to solve: 729 × 1,728 = 1,259,712 cubic inches

Need more algebra and geometry practice problems?

Try our pre-algebra quiz, which covers pre algebra practice problems, such as arithmetic reasoning, basic statistics, and probability

What algebra practice problems do I need?

Your algebra test is likely to cover the following skills. So be sure to try the algebra practice problems above to help improve your skills.

Simplifying rational algebraic expressions, including expressions with a single variable on one side of an equation and expressions with a single variable on both sides of an equation

Algebraic and polynomial equations, including factoring polynomials and expanding polynomials

Roots, radicals, and exponent laws

Linear and quadratic equations and inequalities

Evaluating systems of equations

Algebraic and logarithmic functions

Venn diagrams and graphing

Equivalent expressions for mathematical equations 

Choosing step-by-step solutions in narrative form 

Problem solving techniques for real-life problems  

 Equations to calculate journey time    

 Expressing part of an algebraic equation as a fraction of the entire equation

Working with maps, including calculating map scale and actual distance    

 Correcting erroneous calculations or determining that the equation provided is true   

Other Algebraic Concepts, such as transformations, matrices and absolute value

What do I need for the geometry test?

These geometry skills are usually assessed on most standardized math exams:

Coordinate geometry, including the midpoint formula, the distance formula, slope and slope-intercept

Triangle laws, including the Pythagorean theorem, 30° – 60° – 90° triangles, and triangle area and similarity

The rules of angles , including angles and parallel lines and angles within lines imposed on rectangles

Area and perimeter of squares and rectangles 

Working with circles, such as understanding how to calculate circumference, radius, diameter, area, arcs, and radians

Volume of boxes, cones, cylinders, and pyramids 

Understanding hybrid shapes and the relationships between square figures of different sizes

Need more help with algebra?
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