Math Fancy Formulae
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Math
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Fancy Formulae
Prerequisite: None Math Fancy Formulae
Course Description
There are no cheat sheets on the ACT. This course covers those formulae you must memorize and provides some application practice. We’ll work on distance, midpoint, slope intercept, circle problems, Pythagorean theorem, quadrilaterals, and solids.
Math Fancy Formulae
Content Review PYTHAGOREAN THEOREM PROBLEMS Identify: If you see a RIGHT triangle, and you have to solve for one of the sides, then it’s probably a Pythagorean theorem question. If you need to find the length of any DIAGONAL line, you will probably need the Pythagorean theorem. Setup: Use the Pythagorean theorem: a2 + b2 = c2 Make Sure: Make sure you know which angle is the right angle and which side is the long side (hypotenuse). Also, check that it actually is a right triangle (just because it looks like a right triangle doesn’t mean it is!). Execute: Just do the math.
Math Fancy Formulae
Content Review CIRCLE PROBLEMS Identify: Look for problems with a circle diagram, radius, or diameter. Set Up: If the question involves CIRCUMFERENCE: 2πr = πd If the question involves AREA: πr2 Make Sure: When calculating area, be sure you use the radius, NOT the diameter. Execute: Just do the math. QUADRILATERAL PROBLEMS Identify: Look for shapes with 4 sides. Set Up: Use the following area equations: Rectangles
Perimeter = 2(Base) + 2(Height) Area = (Base) × (Height) Sum of Angles = 360°
Trapezoids
ONE pair of parallel sides Sum of Angles = 360° Area = (Base1 + Base2) × (Height)
Parallelograms
TWO pairs of parallel sides Area = (Base) × (Height) Sum of Angles = 360°
Make Sure: Be careful when selecting the base and height of a parallelogram. They must be perpendicular to each other. Execute: Just solve the equation. Math Fancy Formulae
Content Review CIRCLE PROBLEMS Identify: Look for problems with a circle diagram, radius, or diameter. Set Up: If the question involves CIRCUMFERENCE: 2πr = πd If the question involves AREA: πr2 Make Sure: When calculating area, be sure you use the radius, NOT the diameter. Execute: Just do the math. QUADRILATERAL PROBLEMS Identify: Look for shapes with 4 sides. Set Up: Use the following area equations: Rectangles
Perimeter = 2(Base) + 2(Height) Area = (Base) × (Height) Sum of Angles = 360°
Trapezoids
ONE pair of parallel sides Sum of Angles = 360° Area = (Base1 + Base2) × (Height)
Parallelograms
TWO pairs of parallel sides Area = (Base) × (Height) Sum of Angles = 360°
Make Sure: Be careful when selecting the base and height of a parallelogram. They must be perpendicular to each other. Execute: Just solve the equation. Math Fancy Formulae
Content Review SOLIDS PROBLEMS Identify: Look for shapes with 3 dimensions. Set Up: Use the Volume or Surface Area formulas below. For shapes other than rectangles, the necessary formulas will be in the question. Rectangular Solids
Cylinders
Volume = (Length) x (Width) x (Height)
Volume = (Area of Base) x (Height)
Surface Area = 2(Area of Side 1) + 2(Area of Side 2) + 2(Area of Side 3)
Surface Area = 2(Area of Base) + [(Circumference of Base) x (Height)] Content Review
Make Sure: For Surface Area questions, be sure to add ALL of the individual sides of the solid. Execute: Just solve the equation.
Math Fancy Formulae
Content Review DISTANCE AND MIDPOINT PROBLEMS Identify: You are asked to find the distance between two points, or the midpoint. Set Up: Use the Distance and Midpoint Formulas: Distance Formula: Midpoint Formula: Make Sure: Keep your ordered pairs straight! Be careful plugging in the coordinates into the formulas. Execute: Just solve the equation.
Math Fancy Formulae
Content Review SLOPE PROBLEMS Identify: You will be given TWO POINTS on a line in the xy-coordinate plane and you will be asked to find the slope. Set Up: Use the slope formula:
Rise = y - y Slope = Run x - x 2
1
2
1
Make Sure: Don’t mix up the x and y coordinates. And remember that the y-values go on TOP of the fraction. Execute: Do the math! SLOPE-INTERCEPT PROBLEMS Identify: You will be given the EQUATION OF A LINE in the xy-coordinate plane. Set Up: Use the slope-intercept form to identify the slope (m) and y-intercept (b). y = mx + b Make Sure: If the problem asks for a PARALLEL line, slopes are equal. If the problem asks for a PERPENDICULAR line, slopes are negative reciprocals. Execute: Solve for the question being asked. Math Fancy Formulae
Review Questions Which of the following is the length, in inches, of the remaining side of a right triangle if one side is 5 inches long and the hypotenuse is 12 inches long? A. 7 B. 119 C. 11 D. 13 E. 119
Math Fancy Formulae
Review Questions If the area of a given right triangle is 30 square meters and one leg is 5 meters in length, how long, in meters, is the hypotenuse? (Note: the hypotenuse is not a leg of the triangle.) A. 22 B. 34 C. 61 D. 6 E. 13
Math Fancy Formulae
Review Questions The circumference of each tire on a motorcycle is 60 inches. About how many revolutions does one of these motorcycle tires make traveling 300 feet (3,600 inches) without slipping? F. 6 G. 18 H. 60 J. 360 K. 864
Math Fancy Formulae
Review Questions A square has sides that are the same length as the radius of a circle. If the circle has an area of 64π square units, how many units long is the perimeter of the square? A. 16 B. 32 C. 64 D. 128 E. 256
Math Fancy Formulae
Review Questions A certain circle has an area of 4π square inches. How many inches long is its radius? A. 1 2
B. 2 C. 4 D. r 4
E. π
Math Fancy Formulae
Review Questions In the diagram below, both circles have the same center. Let A be the area of the smaller circle. If the area of the shaded region is also equal to A, then what is the total area of the larger circle, in terms of A? A. A B. 2A C. 3A D. 4A E. 5A
Math Fancy Formulae
Review Questions The length of a rectangle is (p + 4) units and its width is (p + 6) units. Which of the following expresses the remaining area of the rectangle, in square units, if a square, p units in length, is removed from the interior of the rectangle? F. 10 G. 24 H. 2p+10 J. 10p+24 K. p2+10p+24
Math Fancy Formulae
Review Questions The surface area of a cube is given by the formula A = 6s2, where s is the length of one of the edges of the cube. If s is doubled, then the new cube’s surface area is what percent of the original surface area? A. 25% B. 50% C. 200% D. 400% E. 800%
Math Fancy Formulae
Review Questions In the diagram below, ΔABE is an isosceles right triangle with an area of 2. If the sides of the parallelogram BCDE all have equal length, what is the area of BCDE? A. 2 B. 2 2 C. 4 D. 4 2 E. 8
Math Fancy Formulae
Review Questions The measure of each interior angle of a regular n-sided polygon is 180 n^n - 2h . A regular hexagon is shown below. What is the measure of the designated angle? F. 120° G. 156° H. 210°
?
J. 240° E. 300°
Math Fancy Formulae
Review Questions What is the radius of a circle with the equation (x – 2)2 + (y + 4)2 – 5 = 0? F. –5 G. 0 H. 5 J. 5 K. 25
Math Fancy Formulae
Review Questions Josh installed a ladder on level ground next to his above-ground pool. The pool is a right circular cylinder with a diameter of 20 feet and a height of 5 feet. A diagram of the pool and its ladder is shown below.
To the nearest cubic foot, what is the volume of water that will be in the tank when it is filled with water to a depth of 3.5 feet? (Note: The volume of a cylinder is given by πr2h, where r is the radius and h is the height.) F. 99 G. 198 H. 396 J. 1,099 K. 1,245
Math Fancy Formulae
Review Questions The length of the base of a rectangular prism is 8 cm, and the width is 6 cm. If the height of the
rectangular prism is 4 cm, what is its volume in cubic cm? F. 192 G. 124 H. 86 J. 48 K. 24
Math Fancy Formulae
Review Questions What is the radius of the largest sphere that can be inscribed inside a cube of volume 64? A. 1 B. 2 C. 4 D. 8 E. 16
Math Fancy Formulae
Review Questions In the standard (x, y) coordinate plane, (15, 3) is halfway between (8a, a + 2) and (–2a, a – 6). What is the value of a ? F. 2 G. 3 H. 4 J. 5 K. 6
Math Fancy Formulae
Review Questions What is the distance, in coordinate units, between the points M (–5, 3) and N (–2, –1) in the standard (x, y) coordinate plane? A.
5
B. 7 C. 3 D. 5 E. 7
Math Fancy Formulae
Review Questions In the standard (x,y) coordinate plan, if the distance between points (9, b) and (b, 2) is 5 coordinate units, which of the following could be the value of b ? A. 5 B. 3 C. –3 D. –5 E. –6
Math Fancy Formulae
Review Questions In the (x, y) coordinate plane, what is the y-intercept of the line 8x – 2y = 8 ? F. –4 G. –2 H. 0 J. 4 K. 8
Math Fancy Formulae
Review Questions Which of the following lines has the smallest slope? A. y=2x+1 B. y=3x–5 C. y=x+2 D. 4y=2x+3 E. –2y=3x
Math Fancy Formulae
Review Questions If two lines in the standard (x, y) coordinate plane are perpendicular and the slope of one of the lines is 4, what is the slope of the other line? A. –4 B. –1 C. - 1 4
D. 1 4
E. 4
Math Fancy Formulae
Review Questions If 9x – 3y = 10 and 3x – 2y = 6 are equations that represent adjacent sides LM and LP of quadrilateral LMNP, what are the (x, y) coordinates of vertex L? F. f- 2, 28 p 3
G. f 2 , - 8 p 9
3
H. f 4 ,2 p 9
J. f 2 , - 4 p 3
3
K. f2, - 8 p 3
Math Fancy Formulae
Review Questions In the standard coordinate plane, if l is a line that is perpendicular to the y-axis, then what is the slope of line l? A. –1 B. 0 C. 1 D. Undefined E. Cannot be determined from the given information
Math Fancy Formulae
Math
1 2
0 1 2
/ 0 1 2
0 1
Fancy Formulae
Prerequisite: None Math Fancy Formulae
Course Description
There are no cheat sheets on the ACT. This course covers those formulae you must memorize and provides some application practice. We’ll work on distance, midpoint, slope intercept, circle problems, Pythagorean theorem, quadrilaterals, and solids.
Math Fancy Formulae
Content Review PYTHAGOREAN THEOREM PROBLEMS Identify: If you see a RIGHT triangle, and you have to solve for one of the sides, then it’s probably a Pythagorean theorem question. If you need to find the length of any DIAGONAL line, you will probably need the Pythagorean theorem. Setup: Use the Pythagorean theorem: a2 + b2 = c2 Make Sure: Make sure you know which angle is the right angle and which side is the long side (hypotenuse). Also, check that it actually is a right triangle (just because it looks like a right triangle doesn’t mean it is!). Execute: Just do the math.
Math Fancy Formulae
Content Review CIRCLE PROBLEMS Identify: Look for problems with a circle diagram, radius, or diameter. Set Up: If the question involves CIRCUMFERENCE: 2πr = πd If the question involves AREA: πr2 Make Sure: When calculating area, be sure you use the radius, NOT the diameter. Execute: Just do the math. QUADRILATERAL PROBLEMS Identify: Look for shapes with 4 sides. Set Up: Use the following area equations: Rectangles
Perimeter = 2(Base) + 2(Height) Area = (Base) × (Height) Sum of Angles = 360°
Trapezoids
ONE pair of parallel sides Sum of Angles = 360° Area = (Base1 + Base2) × (Height)
Parallelograms
TWO pairs of parallel sides Area = (Base) × (Height) Sum of Angles = 360°
Make Sure: Be careful when selecting the base and height of a parallelogram. They must be perpendicular to each other. Execute: Just solve the equation. Math Fancy Formulae
Content Review CIRCLE PROBLEMS Identify: Look for problems with a circle diagram, radius, or diameter. Set Up: If the question involves CIRCUMFERENCE: 2πr = πd If the question involves AREA: πr2 Make Sure: When calculating area, be sure you use the radius, NOT the diameter. Execute: Just do the math. QUADRILATERAL PROBLEMS Identify: Look for shapes with 4 sides. Set Up: Use the following area equations: Rectangles
Perimeter = 2(Base) + 2(Height) Area = (Base) × (Height) Sum of Angles = 360°
Trapezoids
ONE pair of parallel sides Sum of Angles = 360° Area = (Base1 + Base2) × (Height)
Parallelograms
TWO pairs of parallel sides Area = (Base) × (Height) Sum of Angles = 360°
Make Sure: Be careful when selecting the base and height of a parallelogram. They must be perpendicular to each other. Execute: Just solve the equation. Math Fancy Formulae
Content Review SOLIDS PROBLEMS Identify: Look for shapes with 3 dimensions. Set Up: Use the Volume or Surface Area formulas below. For shapes other than rectangles, the necessary formulas will be in the question. Rectangular Solids
Cylinders
Volume = (Length) x (Width) x (Height)
Volume = (Area of Base) x (Height)
Surface Area = 2(Area of Side 1) + 2(Area of Side 2) + 2(Area of Side 3)
Surface Area = 2(Area of Base) + [(Circumference of Base) x (Height)] Content Review
Make Sure: For Surface Area questions, be sure to add ALL of the individual sides of the solid. Execute: Just solve the equation.
Math Fancy Formulae
Content Review DISTANCE AND MIDPOINT PROBLEMS Identify: You are asked to find the distance between two points, or the midpoint. Set Up: Use the Distance and Midpoint Formulas: Distance Formula: Midpoint Formula: Make Sure: Keep your ordered pairs straight! Be careful plugging in the coordinates into the formulas. Execute: Just solve the equation.
Math Fancy Formulae
Content Review SLOPE PROBLEMS Identify: You will be given TWO POINTS on a line in the xy-coordinate plane and you will be asked to find the slope. Set Up: Use the slope formula:
Rise = y - y Slope = Run x - x 2
1
2
1
Make Sure: Don’t mix up the x and y coordinates. And remember that the y-values go on TOP of the fraction. Execute: Do the math! SLOPE-INTERCEPT PROBLEMS Identify: You will be given the EQUATION OF A LINE in the xy-coordinate plane. Set Up: Use the slope-intercept form to identify the slope (m) and y-intercept (b). y = mx + b Make Sure: If the problem asks for a PARALLEL line, slopes are equal. If the problem asks for a PERPENDICULAR line, slopes are negative reciprocals. Execute: Solve for the question being asked. Math Fancy Formulae
Review Questions Which of the following is the length, in inches, of the remaining side of a right triangle if one side is 5 inches long and the hypotenuse is 12 inches long? A. 7 B. 119 C. 11 D. 13 E. 119
Math Fancy Formulae
Review Questions If the area of a given right triangle is 30 square meters and one leg is 5 meters in length, how long, in meters, is the hypotenuse? (Note: the hypotenuse is not a leg of the triangle.) A. 22 B. 34 C. 61 D. 6 E. 13
Math Fancy Formulae
Review Questions The circumference of each tire on a motorcycle is 60 inches. About how many revolutions does one of these motorcycle tires make traveling 300 feet (3,600 inches) without slipping? F. 6 G. 18 H. 60 J. 360 K. 864
Math Fancy Formulae
Review Questions A square has sides that are the same length as the radius of a circle. If the circle has an area of 64π square units, how many units long is the perimeter of the square? A. 16 B. 32 C. 64 D. 128 E. 256
Math Fancy Formulae
Review Questions A certain circle has an area of 4π square inches. How many inches long is its radius? A. 1 2
B. 2 C. 4 D. r 4
E. π
Math Fancy Formulae
Review Questions In the diagram below, both circles have the same center. Let A be the area of the smaller circle. If the area of the shaded region is also equal to A, then what is the total area of the larger circle, in terms of A? A. A B. 2A C. 3A D. 4A E. 5A
Math Fancy Formulae
Review Questions The length of a rectangle is (p + 4) units and its width is (p + 6) units. Which of the following expresses the remaining area of the rectangle, in square units, if a square, p units in length, is removed from the interior of the rectangle? F. 10 G. 24 H. 2p+10 J. 10p+24 K. p2+10p+24
Math Fancy Formulae
Review Questions The surface area of a cube is given by the formula A = 6s2, where s is the length of one of the edges of the cube. If s is doubled, then the new cube’s surface area is what percent of the original surface area? A. 25% B. 50% C. 200% D. 400% E. 800%
Math Fancy Formulae
Review Questions In the diagram below, ΔABE is an isosceles right triangle with an area of 2. If the sides of the parallelogram BCDE all have equal length, what is the area of BCDE? A. 2 B. 2 2 C. 4 D. 4 2 E. 8
Math Fancy Formulae
Review Questions The measure of each interior angle of a regular n-sided polygon is 180 n^n - 2h . A regular hexagon is shown below. What is the measure of the designated angle? F. 120° G. 156° H. 210°
?
J. 240° E. 300°
Math Fancy Formulae
Review Questions What is the radius of a circle with the equation (x – 2)2 + (y + 4)2 – 5 = 0? F. –5 G. 0 H. 5 J. 5 K. 25
Math Fancy Formulae
Review Questions Josh installed a ladder on level ground next to his above-ground pool. The pool is a right circular cylinder with a diameter of 20 feet and a height of 5 feet. A diagram of the pool and its ladder is shown below.
To the nearest cubic foot, what is the volume of water that will be in the tank when it is filled with water to a depth of 3.5 feet? (Note: The volume of a cylinder is given by πr2h, where r is the radius and h is the height.) F. 99 G. 198 H. 396 J. 1,099 K. 1,245
Math Fancy Formulae
Review Questions The length of the base of a rectangular prism is 8 cm, and the width is 6 cm. If the height of the
rectangular prism is 4 cm, what is its volume in cubic cm? F. 192 G. 124 H. 86 J. 48 K. 24
Math Fancy Formulae
Review Questions What is the radius of the largest sphere that can be inscribed inside a cube of volume 64? A. 1 B. 2 C. 4 D. 8 E. 16
Math Fancy Formulae
Review Questions In the standard (x, y) coordinate plane, (15, 3) is halfway between (8a, a + 2) and (–2a, a – 6). What is the value of a ? F. 2 G. 3 H. 4 J. 5 K. 6
Math Fancy Formulae
Review Questions What is the distance, in coordinate units, between the points M (–5, 3) and N (–2, –1) in the standard (x, y) coordinate plane? A.
5
B. 7 C. 3 D. 5 E. 7
Math Fancy Formulae
Review Questions In the standard (x,y) coordinate plan, if the distance between points (9, b) and (b, 2) is 5 coordinate units, which of the following could be the value of b ? A. 5 B. 3 C. –3 D. –5 E. –6
Math Fancy Formulae
Review Questions In the (x, y) coordinate plane, what is the y-intercept of the line 8x – 2y = 8 ? F. –4 G. –2 H. 0 J. 4 K. 8
Math Fancy Formulae
Review Questions Which of the following lines has the smallest slope? A. y=2x+1 B. y=3x–5 C. y=x+2 D. 4y=2x+3 E. –2y=3x
Math Fancy Formulae
Review Questions If two lines in the standard (x, y) coordinate plane are perpendicular and the slope of one of the lines is 4, what is the slope of the other line? A. –4 B. –1 C. - 1 4
D. 1 4
E. 4
Math Fancy Formulae
Review Questions If 9x – 3y = 10 and 3x – 2y = 6 are equations that represent adjacent sides LM and LP of quadrilateral LMNP, what are the (x, y) coordinates of vertex L? F. f- 2, 28 p 3
G. f 2 , - 8 p 9
3
H. f 4 ,2 p 9
J. f 2 , - 4 p 3
3
K. f2, - 8 p 3
Math Fancy Formulae
Review Questions In the standard coordinate plane, if l is a line that is perpendicular to the y-axis, then what is the slope of line l? A. –1 B. 0 C. 1 D. Undefined E. Cannot be determined from the given information
Math Fancy Formulae
