SAT Geometry and Trigonometry: Every Question Type and Method

The digital SAT's Math section has four content domains, and Geometry and Trigonometry is one of the two smallest: about 15% of the section, or 5 to 7 of its 40 scored questions. The College Board splits it into four skills:

• Area and volume: perimeter, area, surface area and volume, shaded regions, and figures scaled up or down.
• Lines, angles and triangles: angles at crossing and parallel lines, triangle angles, and similar and congruent triangles.
• Right triangles and trigonometry: the Pythagorean theorem, special right triangles, and sine, cosine and tangent.
• Circles: central and inscribed angles, arcs and sectors, radians, and circle equations.

This SAT geometry review covers every question type in seven groups: the four skills, with the last three each split in two. The table below gives each group's method, and the formulas after it show what the SAT's reference sheet gives you and what it leaves out. Each section after that teaches the method, says when Desmos is faster, and gives you an SAT-style question to try. On most geometry questions, the formula is the easy part: you earn the points by labeling the figure, a radius or a diameter, a leg or the hypotenuse, before you calculate.

How to answer an SAT geometry question

The same four moves work on every question in this guide:

1. Read the last line first. Is it asking for an angle, a length, an area, a volume, or an arc's measure or its length?
2. Label the figure, or sketch one: put every number in its place, and add what the words give you, like right angles, equal sides and radii.
3. Name the fact or formula, then pick the faster route. The setup is yours, by hand. When the arithmetic is awkward, type the whole expression into Desmos: it's built into the digital SAT for the whole Math section.
4. Answer the question in the form it asks for, keeping $\pi$ and square roots exact unless it asks for a decimal.

Each section below turns these moves into the steps for its question type. The Desmos course teaches the calculator itself.

If geometry questions are new to you, start with lesson 1.

SAT geometry formulas: what's on the reference sheet and what isn't

During the Math section, the SAT's reference sheet is one click away. Every formula on it is geometry:

The sheet leaves out the rest, so learn these, each in the lesson that uses it:

The sheet can't tell you which formula a question needs, or which length is the radius: that's what the methods below are for.

Area and volume questions name a figure or a solid, or describe one in a story, and ask how far around it is, how much it covers, how much it takes to wrap it or how much it holds. Others shade part of a figure, or scale one up and ask what happens to its area or volume. Decide what's measured, and label each length's job, before any numbers go in.

How to answer area and volume questions

1. Decide what's measured: around (perimeter or circumference), inside (area), wrap (surface area) or fill (volume). Units back it up: square units for area, cubic for volume.
2. Label each length by its job: a radius or a diameter, a perpendicular height or a slanted side.
3. Write one expression, the formula before the numbers. For a shaded region, take the whole minus the hole; for a scaled figure, multiply lengths by $k$, areas by $k^2$ and volumes by $k^3$.
4. Do the arithmetic in your head when it's quick, or type the whole expression into Desmos when it isn't. Keep $\pi$ when the choices have it: Desmos exact answers SAT shows how to get an exact answer back from Desmos.

Perimeter and area

The reference sheet gives the rectangle, triangle and circle, but not a parallelogram, $A=bh$, or a trapezoid, $A=\frac12(b_1+b_2)h$, the average base times the height. A height is always measured at $90^\circ$ to the base, never along a slanted side. See SAT perimeter and area.

Shaded and composite regions

Add pieces that don't overlap, and subtract with the whole minus the hole. Find any missing length first: a frame's width comes off both ends of each side. Desmos does the arithmetic in one line, like 18(12)-14(8), but it can't see the shape. See area of shaded region.

Surface area and volume

Ask “wrap it or fill it?” The sheet gives the volumes, but no surface area: add up the faces, and leave out a face two solids share. A cone or pyramid takes a third of the prism or cylinder with the same base and height, and its volume uses the perpendicular height, never the slant. See SAT surface area and volume.

Scale factors

When every length is multiplied by $k$, perimeters scale by $k$, areas by $k^2$ and volumes by $k^3$, because each direction brings its own $k$. Given an area or volume ratio, take the square or cube root to find $k$ first. See area and volume scale factors.

Mistakes to avoid with area and volume

Halve the diameter before you square it: $\pi(9)^2=81\pi$. Area uses the perpendicular height, $8.7$, not the slanted side: $14.6\times8.7=127.02$. And area scales by $k^2$: $450\times\frac1{25}=18$.

Angle questions give crossing lines, parallel lines cut by a transversal, or a triangle, with angle measures as numbers or expressions like $(3x+7)^\circ$, and ask for $x$ or an angle. Every one comes down to one question: are these angles equal, or do they add up to a known total?

How to answer angle questions

1. Circle what it asks for: $x$, or an angle, which may not be the one in your equation.
2. Pick the fact. Equal: vertical angles, corresponding or alternate angles at parallel lines, and the angles across from equal sides. A total: $180^\circ$ on a straight line and in a triangle, $360^\circ$ around a point.
3. Write one equation and solve it. Desmos can check the algebra, but only once you've named the fact: working by hand is usually quicker, as when not to use Desmos SAT explains.
4. Go back for the angle: put $x$ into the expression the question asks about.

Lines and angles

Vertical angles, across from each other where two lines cross, are equal; neighbors on a straight line add up to $180^\circ$. Where a transversal cuts parallel lines, the small angles all match, the big angles all match, and a small one plus a big one makes $180^\circ$. Lines are parallel only when the question says so or marks them. See SAT line and angle relationships.

Triangle angles

A triangle's angles add up to $180^\circ$, and an exterior angle equals the two far angles added together. In an isosceles triangle, equal sides face equal angles. A polygon with $n$ sides splits into $n-2$ triangles, so its angles add up to $(n-2)180^\circ$. See SAT triangle angle theorems.

Mistakes to avoid with angles

Vertical angles are equal: $3x+7=76$, so $x=23$. $18$ is $x$: the angle is $5(18)-4=86^\circ$. And figures aren't always drawn to scale, so trust what's stated and marked, not how wide an angle looks.

Similar triangle questions show the same shape twice at two sizes: one inside another, two triangles across parallel lines, or a person and a pole with their shadows. They ask for a missing side. Congruence questions give a few matching sides and angles and ask what must be true, or which extra fact would be enough. Either way, match the parts first: sides pair by their endpoints, not by where they sit on the page.

How to answer similar triangle questions

1. Find the equal angles: a shared angle, vertical angles, angles at parallel lines, right angles, or the letter order in “$\triangle ABC\sim\triangle DEF$”. Two pairs make the triangles similar.
2. Match the parts: write the vertices in matching order and pair each side by its endpoints. In a nested figure, a whole side matches a whole side.
3. Scale in one direction: $k$ is a side over its match, the same way in every ratio. Angles don't scale. With friendly numbers, finish by hand; with awkward decimals, type the whole proportion into Desmos, as solve SAT equations with Desmos shows.
4. Check the size: a $k$ greater than $1$ gives the longer side.

Similar triangles

Sides scale and angles stay. When $DE$ is parallel to $BC$ inside triangle $ABC$, the small triangle $ADE$ matches the whole triangle $ABC$, so $AD$ pairs with all of $AB$. See similar triangles proportions.

Congruent triangles

Congruent triangles match in size and shape. SSS, SAS, ASA and AAS prove it, where the angle in SAS sits between the two sides. SSA doesn't, because two different triangles can fit it, and equal angles alone only make triangles similar. To find which extra fact is enough, add each choice to the givens and look for a pattern. See SAT triangle congruence.

Mistakes to avoid with similar and congruent triangles

Angles don't scale, so $\angle P=37^\circ$. Pick one direction, big over small or small over big, and keep it in every ratio. And sides in the same ratio prove similar triangles; congruent ones need the ratio to be $1$.

Right triangle questions give two sides and ask for the third, or one side and an angle of $30^\circ$, $45^\circ$ or $60^\circ$. Often the right triangle is hidden: a ladder against a wall, a rectangle's diagonal, half of a square or of an equilateral triangle, two points on a grid. Find the right angle first: the hypotenuse is the side across from it.

How to answer right triangle questions

1. Find the right angle: a corner mark, a stated $90^\circ$, perpendicular lines, something upright on level ground, or a rectangle's corner. The side across from it is the hypotenuse, the longest side.
2. Pick the tool. Two sides known: the Pythagorean theorem, $a^2+b^2=c^2$, adding for the hypotenuse and subtracting for a leg. A $45^\circ$ angle, or a $30^\circ$ or $60^\circ$: the special right triangle ratio. Both are on the reference sheet.
3. Name each side and scale: in a 30-60-90 triangle, $x:x\sqrt3:2x$, the short leg is across from $30^\circ$ and the long leg across from $60^\circ$; in a 45-45-90 triangle, $x:x:x\sqrt2$.
4. Keep it exact and answer the question: keep $\sqrt2$ and $\sqrt3$ to the end, and reread the last line.

The Pythagorean theorem

The sheet gives you $a^2+b^2=c^2$; your job is to find the right angle and put the side across from it alone as $c$. Take the positive square root, and keep a radical exact. For a distance between points, the legs are the changes in $x$ and $y$. With messy decimal coordinates, define the points in Desmos and type distance(A,B), as Desmos distance midpoint SAT shows. See Pythagorean theorem SAT.

Special right triangles

The hard part is spotting them: a square's diagonal makes two 45-45-90 triangles, and an equilateral triangle's height makes two 30-60-90 triangles. Match the side you know to its place, find $x$, then build the side you need. By hand is quicker here than trig or Desmos. See special right triangles SAT.

Altitudes and inscribed circles

When an altitude drops from the right angle to the hypotenuse, it makes three similar triangles: the altitude squared is the product of the two pieces, and a leg squared is the whole hypotenuse times its own piece. See altitude to hypotenuse. A circle inscribed in a right triangle has radius $r=\frac{a+b-c}{2}$: add the legs, subtract the hypotenuse, and halve. See right triangle inradius SAT.

Mistakes to avoid with right triangles

For a leg, subtract: $25^2-7^2=576$, so the leg is $24$. The long leg is $x\sqrt3=5\sqrt3$; the side twice the short leg is the hypotenuse. And a diagonal is a 45-45-90 hypotenuse, so the square's side is $18$ and its perimeter $72$.

SAT trigonometry is mostly right triangles: a question gives an acute angle and one side and asks for another side, or gives one ratio, like $\tan(P)$, and asks for another. Some use the fact that the two acute angles add up to $90^\circ$, so the sine of one is the cosine of the other, and some give a point $(x,y)$ and ask for an angle's sine or cosine, which is the same triangle drawn from the origin. Name the sides from your angle before you pick a ratio.

How to answer trigonometry questions

1. Find the right angle and the hypotenuse across from it.
2. Name the sides from your angle: the opposite leg is across from it, and the adjacent leg is the leg that touches it. The hypotenuse touches it too, so name it first.
3. Pick the ratio with the side you know and the side you need: $\sin=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan=\frac{\text{opposite}}{\text{adjacent}}$. Write the equation before you calculate.
4. Pick the route: at $30^\circ$, $45^\circ$ or $60^\circ$, use the special triangle's exact sides. At any other angle, when you need a decimal, rearrange once and type the whole expression into Desmos in Degrees mode.

Sine, cosine and tangent

The reference sheet doesn't list the trig ratios, so learn SOHCAHTOA. Given one ratio, label the two sides it names as $3k$ and $4k$, say, and find the third with the Pythagorean theorem. A right triangle's two acute angles add up to $90^\circ$, so the sine of one is the cosine of the other: the side across from one angle is next to the other. For an angle drawn from the origin through a point $(x,y)$, $\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$, with $r=\sqrt{x^2+y^2}$. See SAT trigonometry.

Trig in Desmos

A degree sign means Degrees mode, which is how Desmos opens on the SAT, so type the angle straight in. For a cable that meets the ground at $41^\circ$ and reaches $18$ meters up a pole, $\sin41^\circ=\frac{18}{c}$, so type 18/sin(41) and read $27.4$. In radians, the same keys give $-113.5$. The SAT Desmos tutorial shows when to switch.

Mistakes to avoid with trigonometry

$\cos37^\circ=\frac{12}{c}$, so $c=\frac{12}{\cos37^\circ}$, longer than $12$, as a hypotenuse must be. In Degrees, $\sin41^\circ\approx0.656$. And $\sin30^\circ=\frac12$ while $\cos30^\circ=\frac{\sqrt3}{2}$: sine and cosine trade only between the two angles that add up to $90^\circ$.

Circle angle questions show angles with their vertex at the center, on the circle, or outside it where tangent lines meet, and ask for an angle or an arc. Arc and sector questions ask how long part of the circle is, or how much area a slice covers, and some switch between degrees and radians. Where the angle's vertex sits decides the arc, and the arc decides what fraction of the circle you take.

How to answer circle angle and arc questions

1. Mark what's asked: an angle, an arc's measure in degrees, an arc's length, or a sector's area.
2. Find the vertex. At the center, the angle equals its arc; on the circle, it's half its arc. A radius meets a tangent line at $90^\circ$, and two tangents from one point are equal. Work this part by hand: Desmos can't tell which arc an angle cuts off.
3. Find the fraction of the circle: the arc's degrees over $360$, or its radians over $2\pi$.
4. Take that fraction of the whole: of the circumference, $2\pi r$, for an arc's length, or of the area, $\pi r^2$, for a sector. When the numbers don't cancel quickly, type only the number in front of $\pi$ into Desmos, then put the $\pi$ back, as in Desmos exact answers SAT.

Central, inscribed and tangent angles

A central angle equals its arc, and an inscribed angle on the same arc is half as big. Trace each side of the angle out to the circle to see which arc it cuts off. A line from the center through the middle of a chord that isn't a diameter meets it at $90^\circ$. See SAT circle theorems.

Arc length, sector area and radians

The reference sheet gives the whole circle, $2\pi r$ and $\pi r^2$, and the $360^\circ$ in it, so build an arc or a sector as a fraction of those. To convert, $180^\circ=\pi$ radians: multiply by $\frac{\pi}{180}$ to get radians, and by $\frac{180}{\pi}$ to get degrees. With an angle in radians, $s=r\theta$. See SAT arc length sector area radians.

Mistakes to avoid with circle angles and arcs

Only an inscribed angle is half its arc, so the arc is $124^\circ$. An arc's length is its fraction of the circumference: $\frac{80}{360}\cdot45=10$. And $s=r\theta$ needs radians: $60^\circ=\frac{\pi}{3}$, so $s=\frac{5\pi}{3}$.

Circle equation questions give an equation and ask for the center or the radius, give a center and a point and ask for the equation, or ask whether a point is inside, on or outside a circle. Harder ones give an expanded equation, like $x^2+y^2-4x+10y=7$, or the two ends of a diameter. Every circle equation holds two facts: where the center is, and how far the circle reaches.

How to answer circle equation questions

1. Read what it gives: a center and a radius or a point, the ends of a diameter (the center is their midpoint), or an expanded equation.
2. Pick the route. In standard form, read it by hand. Graph an expanded equation in Desmos exactly as written: when its highest and lowest points land on whole numbers, they give the center and the radius, as circle equations Desmos shows. When they don't, or the question wants an exact standard form or a radius like $\sqrt{30}$, complete the square.
3. Complete the square: group the $x$-terms and the $y$-terms, halve each coefficient and square it, and add both numbers to both sides.
4. Read the answer: the center's signs are opposite the ones in the parentheses, and the right side is $r^2$, so $r$ is its square root.

Circle equations

In $(x-h)^2+(y-k)^2=r^2$, ask what makes each parenthesis zero: $(x+4)^2+(y-3)^2=49$ has center $(-4,3)$ and radius $7$. From a center and a point, $r^2$ is the change in $x$ squared plus the change in $y$ squared, with no square root needed. A point is inside when the left side gives less than $r^2$, on the circle when it gives exactly $r^2$, and outside when it gives more. See SAT circle equations.

Completing the square

Halve the coefficient and square it: for $x^2+6x$, add $9$ to both sides, and $x^2+6x+9$ becomes $(x+3)^2$. Do the same for the $y$-terms. When $x^2$ and $y^2$ share a coefficient, divide every term by it first. See completing the square circle.

Circles from coordinates

The ends of a diameter give the center, their midpoint, and the squared diameter: divide it by $4$ for $r^2$. A circle tangent to an axis has a radius equal to its center's distance from that axis. With messy coordinates, Desmos gives midpoint(A,B) and (distance(A,B)/2)^2 exactly, as Desmos distance midpoint SAT shows. See SAT coordinate circle geometry.

Mistakes to avoid with circle equations

The center's signs are opposite: $(-4,3)$. The right side is $r^2$, so $4^2=16$. And whatever you add on the left goes on the right too: $12+9+4=25$.