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Math: Geometry and Trigonometry

Area, volume, angles, similarity and right-triangle trig at PSAT scope, where the whole circles testing point, radians and the unit circle are gone.

19 topics · 73 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Area and Perimeter of Plane Figures

  1. How it worksHow do you decide whether a problem wants the boundary or the surface inside it?
  2. Worked exampleFind the area of a trapezoid whose parallel sides measure 9 and 15 with a height of 6.
  3. Common mistakeWhy isn't the area of a parallelogram its base times its slanted side?
Your turneasy1 of 3
Find the area of a trapezoid whose parallel sides measure 15 and 22 with a height of 14.

Composite Figures and Shaded Regions

  1. How it worksHow do you cut an L-shaped figure into rectangles you already know how to handle?
  2. Worked exampleFind the shaded area left when a circle of radius 3 is punched out of an 8 by 8 square.
  3. Worked exampleFind the perimeter of an L-shaped room with outer sides 12 and 9 and a 4 by 5 notch cut away.
  4. Going furtherWhat changes about a shaded-area answer when the inner shape slides around but keeps its size?
Your turneasy1 of 3
A 24 by 19 rectangle has a 7 by 3 rectangle removed from one corner. Find the area that remains.

Surface Area and Volume of Solids

  1. How it worksHow does a net unfold a solid so its surface area becomes flat shapes you can just add up?
  2. Worked exampleFind the volume of a cone with radius 5 and height 12.
  3. Worked exampleFind the height of a cylinder of radius 4 whose volume comes to 200 π.
  4. Common mistakeWhy isn't the surface area of a cube of volume 64 also equal to 64?
Your turneasy1 of 3
Find the volume of a cone with radius 9 and height 14. Use π = 3.14159.

The Scale Factor Rule: k, k Squared, and k Cubed

  1. How it worksHow does a scale factor of k multiply lengths by k, areas by k squared and volumes by k cubed?
  2. How it worksHow does the squaring come from area needing two length measurements multiplied together?
  3. Worked exampleFind the new volume when a solid of volume 54 gets scaled up by a factor of 3.
  4. Common mistakeWhy isn't a model built at 1 to 10 scale one tenth the weight of the real object?
  5. Worked exampleFind the scale factor when a photo's area grows from 24 square inches to 216 square inches.
Your turneasy1 of 3
A solid of volume 142 is scaled up by a factor of 5. Find the new volume.

Selecting and Applying the Right Formula

  1. How it worksHow do the words of a problem tell you whether to reach for area, surface area or volume?
  2. Worked exampleFind the paint needed for the curved side of a cylindrical tank 10 tall with radius 3.
  3. Common mistakeWhy isn't a pyramid's volume formula the same as the one for a prism on the same base?
Your turnmedium1 of 3
Find the area of the curved side of a cylindrical tank of height 14 and radius 9. Use π = 3.14159.

Congruence of Triangles

  1. How it worksHow does each congruence shortcut pin a triangle down with only three pieces of information?
  2. Worked exampleFind x when triangle ABC is congruent to triangle DEF, side AB is 3x - 4 and side DE is 17.
  3. Common mistakeWhy doesn't matching two sides and a non-included angle guarantee two triangles are congruent?
Your turneasy1 of 3
Triangle ABC is congruent to triangle DEF. Side AB measures 6x - 8 and side DE measures 38. Find x.

Similarity of Triangles

  1. How it worksHow do two pairs of equal angles force the side ratios of two triangles to match?
  2. Worked exampleFind the height of a flagpole casting a 24 foot shadow while a 6 foot person casts a 4 foot shadow.
  3. Worked exampleFind the missing sides of a triangle similar to one with sides 6, 8 and 10 whose shortest side is 15.
  4. Going furtherWhat happens to a similarity ratio when the two triangles are drawn at different orientations?
Your turneasy1 of 3
A flagpole casts a shadow 69 feet long at the same moment a 6 foot person casts a shadow 9 feet long. Find the flagpole's height in feet.

Dilation: Lengths Scale, Angles Stay

  1. How it worksHow does a dilation multiply every length by k while leaving every angle exactly alone?
  2. Common mistakeWhy isn't an angle measured in the kind of unit a scale factor is allowed to multiply?
  3. Worked exampleFind the perimeter and the angle measures of a 5, 12, 13 triangle after a dilation by 2.5.
Your turnhard1 of 3
A right triangle with legs 9 and 12 is dilated by a factor of 6. Find the perimeter of the image.

Sufficiency: Which Statement Satisfies the Theorem

  1. How it worksHow do you test a candidate statement by asking whether it locks the figure into one shape?
  2. How it worksHow does a sufficiency question differ from an ordinary solve-for-x question?
  3. Worked exampleFind which single extra fact proves two triangles similar when only one pair of angles is known equal.
  4. Common mistakeWhy isn't a true statement about a figure automatically enough to satisfy a theorem?
Your turnmedium1 of 3
Two triangles are known to share one pair of equal angles, each measuring 62 degrees. Which single extra fact proves the triangles similar?

The Triangle Angle Sum Theorem

  1. How it worksHow does tearing off a triangle's three corners and lining them up prove the 180 degree rule?
  2. Worked exampleFind the third angle of a triangle whose other two angles measure 47 and 68 degrees.
  3. Going furtherWhat happens to the angle sum when a fourth vertex turns a triangle into a quadrilateral?
Your turneasy1 of 3
Find the third angle in degrees of a triangle whose other two angles measure 62 and 69 degrees.

Vertical Angles

  1. How it worksHow does the straight-line pair on each side force vertical angles to match?
  2. Worked exampleFind all four angles at a crossing when one measures 4x + 10 and the one beside it measures 6x.
  3. Common mistakeWhy aren't neighbouring angles at a crossing equal the way vertical ones are?
Your turnmedium1 of 3
Two lines cross. One angle measures 6x + 41 degrees and the angle beside it measures 11x degrees. Find x.

Parallel Lines Cut by a Transversal

  1. How it worksHow do corresponding, alternate interior and same-side interior angles each earn their rule?
  2. How it worksHow does sliding one parallel line along the transversal show why corresponding angles agree?
  3. Worked exampleFind every angle when a transversal cuts two parallel lines and one of them measures 118 degrees.
  4. Common mistakeWhy doesn't the alternate interior angle rule survive when the two lines are not parallel?
  5. Ties togetherHow do the transversal rules combine with the triangle angle sum to pin down an exterior angle?
Your turneasy1 of 3
A transversal cuts two parallel lines and one interior angle measures 117 degrees. Find the measure in degrees of its same-side interior partner.

The Pythagorean Theorem

  1. How it worksHow does rearranging four copies of a right triangle inside a square prove the a squared plus b squared rule?
  2. Worked exampleFind the hypotenuse of a right triangle with legs of 9 and 40.
  3. Worked exampleFind the long diagonal of a rectangular box measuring 3 by 4 by 12.
  4. Common mistakeWhy doesn't adding the two legs directly give you the hypotenuse?
Your turneasy1 of 3
Find the hypotenuse of a right triangle with legs of 15 and 27.

Special Right Triangles: 30-60-90 and 45-45-90

  1. How it worksHow does slicing an equilateral triangle in half produce the 30-60-90 side pattern?
  2. Worked exampleFind both missing sides of a 30-60-90 triangle whose shorter leg measures 7.
  3. Worked exampleFind the diagonal of a square of side 11 using the 45-45-90 pattern.
  4. Going furtherWhat happens to the side ratios when a 30-60-90 triangle is scaled to a hypotenuse of 100?
Your turnmedium1 of 3
A 30-60-90 triangle has a hypotenuse of 29. Find its longer leg.

Right Triangle Trigonometry

  1. How it worksHow do sine, cosine and tangent each pick out a different pair of sides?
  2. How it worksHow do you choose which ratio to use from what a problem gives you and what it asks for?
  3. Worked exampleFind the height a 30 foot ramp reaches when it rises at 12 degrees.
  4. Worked exampleFind the angle a 20 foot ladder makes with the ground when its base sits 6 feet from the wall.
  5. Common mistakeWhy isn't tangent ever the ratio that involves the hypotenuse?
Your turnmedium1 of 3
A ramp 48 feet long rises at an angle of 18 degrees to the horizontal. Find the height in feet that it reaches.

The Entire Circles Testing Point Is SAT-Only

  1. How it worksHow does the framework carve circles out of the PSAT while keeping them on the SAT?
  2. How it worksHow much of the SAT geometry topic map survives once only the PSAT subset is highlighted?
  3. Common mistakeWhy isn't College Board's own PSAT math web page a reliable guide to what the PSAT tests?
  4. Going furtherWhat still counts as fair game about circles on the PSAT once that testing point is removed?
  5. Ties togetherHow does losing circles push PSAT geometry down to roughly 12.5 percent of the math section?
Your turneasy1 of 3
A PSAT/NMSQT student has 43 days left. Which geometry skill is on the SAT but NOT on the PSAT?

Radians and the Unit Circle Are Not on the PSAT

  1. How it worksHow does the unit circle sit inside the circles testing point that PSAT scope deletes?
  2. Common mistakeWhy isn't the reference sheet's line about 2 π radians a sign that radians get tested?
  3. Going furtherWhat happens to trigonometry on the PSAT once the unit circle is taken away from it?
Your turnmedium1 of 3
The reference sheet states that a circle has 2 π radians of arc. A student with 43 days left takes that as a signal to revise radians. What is the right reading?

What PSAT Scope Drops from Right Triangle Trigonometry

  1. How it worksHow does a right triangle scaling up keep the ratio of two of its sides pinned in place?
  2. How it worksHow does the sine of an angle come out equal to the cosine of its complement?
  3. Common mistakeWhy isn't the complementary angle relationship something a PSAT question will ask you to use?
  4. Ties togetherHow does the similarity behind the trig ratios connect to the scale factor rule for lengths?
Your turnmedium1 of 3
A PSAT student with 43 days left is pruning a right-triangle trigonometry list. Which item can be crossed off as SAT-only?

The Math Reference Sheet

  1. How it worksHow does the reference sheet cover circle, triangle and volume formulas and then stop?
  2. Worked exampleFind the volume of a sphere of radius 6 using only what the reference sheet hands you.
  3. Common mistakeWhy isn't the slope formula on the reference sheet when half the math section leans on it?
  4. Going furtherWhat has to be memorised because the sheet leaves it out, from slope to the quadratic formula?
Your turneasy1 of 3
Find the volume of a sphere of radius 11 using the reference sheet's formula. Use π = 3.14159.

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