Math ACT Prep

Must Know Geometry Formulas

• Area of a Rectangle: A = lw

• Area of a Triangle:

  

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• Area of a Circle: A = π r ²

• Circumference of a Circle: C = π d or C = 2π r

• Diameter and Radius of a circle: d = 2 r

• Volume of a Rectangular Prism: V = Bh , where B is the area of the base

• Degrees in a:

  • Right Angle: 90°

  • Straight Line: 180°

  • Triangle: 180°

  • Circle: 180°

• Parallel Lines and Angles – When a line intersects two parallel lines

  • Two kinds of angles are formed: big angles and small angles.

  • Each big angle is equal to the other big angles.

  • Each small angle is equal to the other small angles.

  • Any big angle plus any small angle is 180°.

• Pythagorean Theorem: a ² + b ² = c ² , where c is the hypotenuse of a right triangle.

• SOHCAHTOA

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Nice to Know (But Can be Solved with Strategy) Geometry Formulas

• Area of a Square: A = s ² (based on Area of a Rectangle formula)

• Area of a Parallelogram: A = bh (break into 2 triangles or a rectangle and 2 triangles)

• Area of a Trapezoid:

  

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  (break into 2 triangles and a rectangle)

• Volume of a Cube: V = s ³ (based on volume of a rectangular prism)

• Volume of a Rectangular Solid: V = lwh (based on volume of a rectangular prism)

• Volume of a Cylinder: V = π r ² h (based on volume of a rectangular prism)

• Special Right Triangles:

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(use the Pythagorean theorem)

• Sum of angles in an n -sided polygon: ( n – 2)180° (break polygon into triangles)

• Angle measure of each angle in a regular n -sided polygon: 
  
  

  

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  (break polygon into triangles)

• Surface area of a rectangular solid: S = 2( lw + lh + wh ) (add areas of all faces)

• Surface area of a cube: S = 6 s ² (add areas of all faces)

• Surface area of a right circular cylinder: S = 2π r ² + 2π rh (add areas of all faces)

Nice to Know (But Rarely Tested) Geometry Formulas

• Reciprocal Trigonometric Functions:

  

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• Law of sines:

  

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• (sometimes provided in a question)

• Surface area of a sphere: S = 4π r ² (sometimes provided in a question)

• Volume of a sphere:

  

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  (sometimes provided in a question)

Coordinate Geometry Formulas

Some ACT Coordinate Geometry questions are really just geometry questions in disguise, but other questions will require formulas specific to this area of study.

Must Know Coordinate Geometry Formulas

• Slope:

  

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• Slope-intercept form of a line: y = mx + b , where m is the slope and b is the y -intercept

Nice to Know (But Can be Solved with Strategy) Coordinate Geometry Formulas

• Distance:

  

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  (make the distance the hypotenuse of a right triangle and use the Pythagorean theorem)

• Midpoint:

  

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  (the average of the x -coordinates is the midpoint’s x ; same with the y -coordinates)

• Standard form: Ax + By = C (can always rearrange into slope-intercept form)

  • Slope:

    

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  • y -intercept:

    

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Nice to Know (But Rarely Tested) Coordinate Geometry Formulas

• Circle centered at (0,0) = x ² + y ² = r ² , where r is the radius

• Circle centered at ( h , k ) = ( x – h ) ² + ( y – k ) ² = r ² , where r is the radius

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(sometimes given in a question)

Algebra Formulas

There are a couple of formulas that are useful when specifically working with quadratics in the form ax ² + bx + c = 0:

Quadratic formula:

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Discriminant: D = b ² – 4 ac (the expression under the radical in the quadratic formula)

• If D > 0, there will be two distinct, real solutions.

• If D = 0, there will be one distinct real solution.

• If D < 0, there will be no real solutions. Instead, there will be two complex solutions.

The sum of the roots:

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The product of the roots:

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The midpoint of the roots/the
x -coordinate of the vertex:

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Problem Solving Formulas

Most questions testing the following formulas can be solved with careful reading and strategy. However, if you’re the sort that prefers to memorize, these can be helpful to know.

• Arithmetic sequence: n th term = Original Term + ( n – 1) d , where d is the constant difference between terms

• Direct Variation:

  

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  or y = kx , where k is a constant

• Inverse Variation: x ₁ y ₁ = x ₂ y ₂ or

  

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  , where k is a constant

• Geometric sequence: n th term = Original Term ´ r ^{( n – 1)} , where r is the constant ratio between terms

• Group Formula: Total = Group 1 + Group 2 – Both + Neither

Rectangles

Area

Area=lw

• l is the length of the rectangle

• w is the width of the rectangle

Perimeter

Perimeter=2l+2w

Rectangular Solid

Volume

Volume=lwh

• h is the height of the figure

Parallelogram

An easy way to get the area of a parallelogram is to drop down two right angles for heights and transform it into a rectangle.

• Then solve for h using the pythagorean theorem

Area

Area=lh

• (This is the same as a rectangle’s lw. In this case the height is the equivalent of the width)

Triangles

Area

Area=12bh

• b is the length of the base of triangle (the edge of one side)

• h is the height of the triangle

  • The height is the same as a side of the 90 degree angle in a right triangle. For non-right triangles, the height will drop down through the interior of the triangle, as shown in the diagram.

Pythagorean Theorem

a2+b2=c2

• In a right triangle, the two smaller sides (a and b) are each squared. Their sum is the equal to the square of the hypotenuse (c, longest side of the triangle)

Properties of Special Right Triangle: Isosceles Triangle

• An isosceles triangle has two sides that are equal in length and two equal angles opposite those sides.

• An isosceles right triangle always has a 90 degree angle and two 45 degree angles.

• The side lengths are determined by the formula: x, x, x√2, with the hypotenuse (side opposite 90 degrees) having a length of one of the smaller sides * √2.

  • E.g., An isosceles right triangle may have side lengths of 12, 12, and 12√2.

Properties of Special Right Triangle: 30, 60, 90 Degree Triangle

• A 30, 60, 90 triangle describes the degree measures of its three angles.

• The side lengths are determined by the formula: x, x√3, and 2x.

  • The side opposite 30 degrees is the smallest, with a measurement of x.

  • The side opposite 60 degrees is the middle length, with a measurement of x√3.

  • The side opposite 90 degree is the hypotenuse, with a length of 2x.

  • For example, a 30-60-90 triangle may have side lengths of 5, 5√3, and 10.

Trapezoids

Area

• Take the average of the length of the parallel sides and multiply that by the height.

Area=[(parallelsidea+parallelsideb)2]h

• Often, you are given enough information to drop down two 90 angles to make a rectangle and two right triangles. You’ll need this for the height anyway, so you can simply find the areas of each triangle and add it to the area of the rectangle, if you would rather not memorize the trapezoid formula.

• Trapezoids and the need for a trapezoid formula will be at most one question on the test. Keep this as a minimum priority if you're feeling overwhelmed.

Circles

Area

Area=πr2

• π is a constant that can, for the purposes of the ACT, be written as 3.14 (or 3.14159)

  • Especially useful to know if you don’t have a calculator that has a π feature or if you're not using a calculator on the test.

• r is the radius of the circle (any line drawn from the center point straight to the edge of the circle).

Area of a Sector

• Given a radius and a degree measure of an arc from the center, find the area of that sector of the circle.

• Use the formula for the area multiplied by the angle of the arc divided by the total angle measure of the circle.

Areaofanarc=(πr2)(degreemeasureofcenterofarc360)

Circumference

Circumference=2πr

or

Circumference=πd

• d is the diameter of the circle. It is a line that bisects the circle through the midpoint and touches two ends of the circle on opposite sides. It is twice the radius.

Length of an Arc

• Given a radius and a degree measure of an arc from the center, find the length of the arc.

• Use the formula for the circumference multiplied by the angle of the arc divided by the total angle measure of the circle (360).

Circumferenceofanarc=(2πr)(degreemeasurecenterofarc360)

• Example: A 60 degree arc has 16 of the total circle's circumference because 60360=16

An alternative to memorizing the “formulas” for arcs is to just stop and think about arc circumferences and arc areas logically.

• If you know the formulas for the area/circumference of a circle and you know how many degrees are in a circle, put the two together.

  • If the arc spans 90 degrees of the circle, it must be 14th the total area/circumference of the circle, because 36090=4.

  • If the arc is at a 45 degree angle, then it is 18th the circle, because 36045=8.

• The concept is exactly the same as the formula, but it may help you to think of it this way instead of as a “formula” to memorize.

Equation of a Circle

• Useful to get a quick point on the ACT, but don’t worry about memorizing it if you feel overwhelmed; it will only ever be worth one point.

• Given a radius and a center point of a circle (h,k)

(x−h)2+(y−k)2=r2

Cylinder

Volume=πr2h

Trigonometry

Almost all the trigonometry on the ACT can be boiled down to a few basic concepts

SOH, CAH, TOA

Sine, cosine, and tangent are graph functions

• The sine, cosine, or tangent of an angle (theta, written as Θ) is found using the sides of a triangle according to the mnemonic device SOH, CAH, TOA.

Sine - SOH

Sine‌Θ=oppositehypotenuse

• Opposite = the side of the triangle directly opposite the angle Θ

• Hypotenuse = the longest side of the triangle

Sometimes the ACT will make you manipulate this equation by giving you the sine and the hypotenuse, but not the measure of the opposite side. Manipulate it as you would any algebraic equation:

 SineΘ=oppositehypotenuse → hypotenuse*sinΘ=opposite

Cosine - CAH

CosineΘ=adjacenthypotenuse

• Adjacent = the side of the triangle nearest the angle Θ (that creates the angle) that is not the hypotenuse

• Hypotenuse = the longest side of the triangle

Tangent - TOA

Tangent‌Θ=oppositeadjacent

• Opposite = the side of the triangle directly opposite the angle Θ

• Adjacent = the side of the triangle nearest the angle Θ (that creates the angle) that is not the hypotenuse

Cosecant, Secant, Cotangent

• Cosecant is the reciprocal of sine

  • Cosecant‌Θ=hypotenuseopposite

• Secant is the reciprocal of cosine

  • Secant‌Θ=hypotenuseadjacent

• Cotangent is the reciprocal of tangent

  • Cotangent‌Θ=adjacentopposite

Useful Formulas to Know
Sin2Θ+Cos2Θ=1

SinΘCosΘ=TanΘ