Cracking the SAT Subject Test in Math 2, 2nd Edition What is the area of the parallelogram? (A) 0 073 (B) 4 (C) 4 619 (D) 5 819 (E) 6 928 50 The pyramid above has a square base If each edge of the pyr[.]
What is the area of the parallelogram? (A) 0.073 (B) (C) 4.619 (D) 5.819 (E) 6.928 50 The pyramid above has a square base If each edge of the pyramid is s, then what is the height of the pyramid? (A) (B) s (C) s (D) (E) Summary ○ Plane geometry is not directly tested on the SAT Subject Test in Math 2 However, many solid and coordinate geometry questions test your plane geometry knowledge, so make sure you work on and memorize any concepts you may have forgotten! ○ Some rules about lines and angles: • A 90° angle is formed by two lines perpendicular to each other • There are 180° in a line • When two straight lines intersect, the angles created opposite each other are equal The adjacent angles (two angles beside each other along the same straight line) have a sum of 180° ○ When parallel lines are transected by a third line, big angles and small angles are created All the big angles are equal, all the small angles are equal, and a big plus a small equals 180° ○ Some rules about triangles • The sum of the angles in a triangle is 180° • The longest side of a triangle is across from the largest angle The smallest side of a triangle is across from the smallest angle Equal sides are across from equal angles • Isosceles triangles have at least two equal sides and two equal angles Equilateral triangles have three equal sides and three equal angles • The third side rule states that the length of any side of a triangle must be between the sum and the difference of the other two sides • The area of a triangle is A = bh The height must be perpendicular to the base ○ The area of a triangle may also be found using A = absinθ, where a and b are the measures of two side lengths and θ is the angle between those sides ○ Right triangles are triangles with one 90° angle The Pythagorean Theorem states that, in a right triangle, a2 + b2 = c2, where c is the hypotenuse of the triangle and a and b are the two legs ○ Special right triangles are helpful in simplifying the math They often provide an easier route to the correct solution than using the Pythagorean Theorem, so look closely for opportunities to use them The following is a list of special right triangles: • There are some Pythagorean triplets that are helpful to have in your back pocket They are 3:4:5, 5:12:13, 7:24:25, and any multiples of these as well • The sides of a 45°-45°-90° triangle have a very specific ratio: x: x: x , where x is the length of each leg • The sides of a 30°-60°-90° triangle have a very specific ratio: x: x :2x, where x is the length of the shorter leg ○ Similar triangles have the same angle measures The corresponding sides and heights of similar triangles are proportional ○ Quadrilaterals are four-sided figures The sum of the angles in a quadrilateral is 360° • Parallelograms have two sets of equal, parallel lines The area of a parallelogram is A = bh, where the base is perpendicular to the height • You can also find the area of a parallelogram using A = absinq, where a and b are the measures of two adjacent sides and q is the measure of an angle of the parallelogram • Rectangles are parallelograms with four right angles • Squares are rectangles with four equal sides • Trapezoids are four-sided figures whose top and bottom are parallel but different in length The area of a trapezoid is , where b1 is one base and b2 is the other The sum of the angles of an n-sided polygon is (n 2) ì 180 Here are some things to remember about a circle: ã A circle contains a total of 360 ã The radius is the distance from the center of the circle to any point on the circle • The diameter is a straight line drawn from one point on a circle through the center to another Its length is twice the radius • The circumference of the circle is the distance around the circle You can think of it as the perimeter of the circle Its formula is C = πd You may also know it as C = 2πr • The formula for area of a circle is A =πr2 • A sector is a slice of pie of the circle The part of the circumference that the sector contains is called an arc and is in the same proportion to the circumference as the angle of the sector is to 360° • A central angle is an angle whose vertex is the center of the circle An inscribed angle has its vertex on the circle and its two endpoints on the circle Its angle is half of what the central angle is to those same two endpoints • Any angle inscribed in a semicircle is a right angle • A line tangent to the circle touches the circle in only one place and is always perpendicular to the radius drawn to the point of tangency ○ For the purposes of the SAT Subject Test in Math 2, prisms are 3dimensional figures with two equal, parallel bases The bases can be any shape from plane geometry ○ The volume of a prism is the area of the base, often referred to as B, times the height, h ○ Let’s talk rectangular prisms: • The formula for the volume of a rectangular prism is V = lwh • The formula for the surface area of a rectangular solid is SA = 2lw + 2wh + 2lh Think about painting the outside of the figure Find the area of each side • The Super Pythagorean Theorem, which is helpful in solving questions about the diagonal of a rectangular prism, is a2 + b2 + c2 = d2 ○ Let’s talk cubes Remember that a cube is just a rectangular prism whose length, width, and height are equal If you forget a formula, just use the rectangular prism formula! • The volume of a cube is V = s3 • The surface area of a rectangular solid is SA = 6s2 ○ Let’s talk cylinders A cylinder is a prism whose bases are circles • The volume of a cylinder is V = πr2h • The surface area of a rectangular solid is SA = 2πr2 + 2πrh If you forget this, remember that you’re just painting the outside So you’ll need the area of two circles and the area of the other piece, which, when rolled out (like a roll of paper towels), is a rectangle whose sides are the circumference of the circle and the height ○ A cone is similar to a cylinder except that one of its bases is merely a point • The formula for the volume of a cone is V = πr2h, where the height must be perpendicular to the base • The formula for the surface area of a cone is SA = πrl + πr2, where l is the slant height ○ A sphere is a hollow ball • The formula for the volume of a sphere is V = πr3 • The formula for the surface area of a cone is SA = 4πr2 ○ Pyramids are like cones, but the base is a plane geometry shape The formula for the volume of a pyramid is V = Bh ○ Inscribed figures always have a line or curve that connects the inner figure to the outer figure ○ Questions about solids produced by rotation usually test your ability to visualize the solid created by the rotation of a flat shape ... A line tangent to the circle touches the circle in only one place and is always perpendicular to the radius drawn to the point of tangency ○ For the purposes of the SAT Subject Test in Math 2, prisms are 3dimensional figures with two equal, parallel bases... A circle contains a total of 360° • The radius is the distance from the center of the circle to any point on the circle • The diameter is a straight line drawn from one point on a circle through the center to another... Special right triangles are helpful in simplifying the math They often provide an easier route to the correct solution than using the Pythagorean Theorem, so look closely for opportunities to use them The following is a list of special right triangles: