ACT Plane Geometry Review Let s first take a look at the common formulas you need for the ACT. Then we ll review the rules for the tested shapes. There are also some practice problems at the end of this worksheet. l w h A= r 2 C=2 r A=lw A= 12 bh b Very rarely, you will need to find volume: l h r c h b w a V=lwh V= r 2 h c 2 = a 2 +b 2 The number of degrees of arc in a circle is 360 The measure in degrees of a straight angle is 180 The sum of the degrees of the angles of a triangle is 180 On the ACT, remember that diagrams are usually drawn to scale, so you can always eyeball to determine measurements if you get stuck. Angles There are 90 degrees in a right angle. When two straight lines intersect, angles opposite each other are equal. Quadrilaterals In a square all 4 sides are equal all 4 angles are equal to 90 Area= s 2 Perimeter= 4s
In a rectangle Opposite sides are equal All four angles are equal to 90 Area= lw Perimeter= 2l+2w In a parallelogram Opposite sides are parallel and equal Area= bh Triangles In any triangle: The longest side is opposite the largest angle. The shortest side is opposite the smallest angle. Equal sides are opposite equal angles. Angles add to 180 Area=1/2(base height) The height is a perpendicular line from the base to the opposite vertex. Perimeter is the sum of the sides. In an isosceles triangle: Two sides are equal. Angles opposite the equal sides are equal In an equilateral triangle: All three sides are equal. All angles are equal to 60. Right Triangles In right triangles: Two sides intersect to form a 90 angle. The side opposite the right angle is called the hypotenuse If you know two sides of a right triangle you can find the third using the Pythagorean theorem: a 2 +b 2 =c 2 Sides appearing in the ratios of 3:4:5, 6:8:10, and 5:12:13 (also called Pythagorean triplets) appear very frequently on the ACT. 6 10 5 13 3 5 8 12 4
Remembering these will save you time because you will not have to do the Pythagorean theorem. Please note that there may be a question or two involving similar triangles or right triangles with angles in ratio of 30-60-90 or 45-45-90. Look at your bonus materials for links to online tutorials to help you review these concepts separately. Circles There are 360 in a circle. Radius The distance from the center to any point on the edge of the circle All radii in a circle are equal. Diameter The straight line distance from one point on the circle to another, passing through the center point. The longest line in the circle Equals twice the radius Chord Any line segment from one point on the circle to another The diameter is the longest chord Circumference The distance around the outside of the circle. The formula for circumference is 2 r or d. Area The amount of space within the boundaries of the circle The formula for area is r 2 Volume Volume questions are rare on the ACT, but if you do see a question involving volume on test day, use these formulas provided at the start of this worksheet. Rectangular box: V= lwh Cube: V= lwh = s 3 Cylinder: V= r 2 h
Practice Problems 1. If a rectangular floor has an area of 32 square meters and a length of 8 meters, what is the perimeter of the floor in meters? A. 20 B. 24 C. 32 D. 36 E. 64 2. In a rectangle, the width y is half the length x. What is the area of the rectangle in terms of y? F. 12 y G. 6y H. 2y 2 J. y 2 K. 4 3. If triangle ABC is an isosceles triangle and angle A is 70, which of the following can be the measure of angle B? A. 5 B. 30 C. 45 D. 70 E. 120 4. The circumference of a circle is 12π. What is its radius? F. 3 G. 6 H. 9 J. 6 K. 12 5. When a certain rectangle is divided in half, two squares are formed. If each of these squares has a perimeter of 36, what is the perimeter of the original rectangle? A. 24 B. 36 C. 48 D. 54 E. 60
6. Members of a construction crew lean a 20-foot ladder against a building. The side of the building is perpendicular to the ground, and the base of the ladder is 16 feet away from the building. How far up the building does the ladder reach? F. 10 G. 11 H. 12 J. 16 K. 20 7. A square is circumscribed about a circle of 6-inch radius, as shown below. What is the area of the square? 6 A. 144 B. 156 C. 160 D. 164 E. 270 Answer Key 1. B 2. C. (If the width y is half the length of x then x=2y. To get the area of the rectangle multiply x by y or 2y by y. 2y times y= 2y 2 ) 3. D 4. G 5. D 6. H 7. A (if the radius is 6, the diameter across the circle is 12, which is also equivalent to the side of the square. So 12 x 12 = 144).