SAT geometry: angles, area, similarity, circles, and trig

01Start with the relationships the diagram guarantees

A drawing can suggest a right angle, equal sides, or parallel lines without establishing them. Use the labels and stated conditions. If the problem says two lines are parallel, you can apply parallel-line angle relationships; if it doesn't, appearance alone isn't enough.

Vertical angles are equal. Angles along a straight line sum to $180^{\circ }$. A triangle's interior angles sum to $180^{\circ }$. If two triangle angles are $48^{\circ }$ and $67^{\circ }$, the third is $180^{\circ }-48^{\circ }-67^{\circ }=65^{\circ }$.

An exterior angle equals the sum of the two nonadjacent interior angles. For that triangle, the exterior angle beside the $65^{\circ }$ angle is $115^{\circ }$, matching $48^{\circ }+67^{\circ }$. Write the equation directly onto your scratch diagram.

02Match corresponding parts of similar figures

Similar triangles have equal corresponding angles and proportional corresponding sides. If triangle ABC is similar to triangle DEF, the order identifies A with D, B with E, and C with F.

Suppose AB is 6, DE is 9, and BC is 8. The scale factor from ABC to DEF is $9/6=1.5$, so EF is $8(1.5)=12$. Using a different side pairing would solve a different proportion.

The area multiplier is the square of the side multiplier. Here the larger triangle has $1.5^2=2.25$ times the smaller triangle's area. For similar solids, the volume multiplier is cubed. Keep those separate from perimeter, which scales like a length.

03Area needs the perpendicular height

A triangle with base 12 and perpendicular height 5 has area $\frac{1}{2}(12)(5)=30$ square units. A slanted side of length 7 is not a substitute for the height unless it is perpendicular to the chosen base.

For a composite shape, split it into pieces whose dimensions you know. A rectangle 10 units wide and 8 units tall with a 3-by-4 rectangular corner removed has area $10(8)-3(4)=68$ square units.

Volume extends area through a third dimension. A cylinder with radius 3 and height 10 has volume $\pi (3^2)(10)=90\pi$ cubic units. Radius and diameter are different: a diameter of 6 gives radius 3. Write that conversion before substituting.

04Use the right-triangle facts in the right place

For a right triangle with legs 5 and 12, the hypotenuse is $\sqrt{5^2+12^2}=13$. The Pythagorean theorem applies because the triangle has a right angle. It does not justify using those squared lengths in any triangle.

A $45^{\circ }$–$45^{\circ }$–$90^{\circ }$ triangle has side ratio $1:1:\sqrt{2}$. A $30^{\circ }$–$60^{\circ }$–$90^{\circ }$ triangle has ratio $1:\sqrt{3}:2$, with the shortest side opposite $30^{\circ }$.

For trig, label opposite and adjacent relative to the requested angle. If opposite is 5 and hypotenuse is 13, the angle's sine is $5/13$. Changing the reference angle changes which leg is opposite. The right-triangle guide works through that distinction.

05Connect circle formulas to the question asked

For radius 4, a circle's circumference is $8\pi$ and its area is $16\pi$. A $90^{\circ }$ central angle is one quarter of a turn, so the corresponding arc length is $2\pi$ and the sector area is $4\pi$.

In the coordinate plane, $(x-2{)}^2+(y+3{)}^2=25$ has center $(2,-3)$ and radius 5. The right side is the squared radius. Reading 25 as the radius changes every later calculation.

A point on the circle is 5 units from the center. Use that fact or substitute coordinates to test an answer. The circles guide includes completing the square and radians.

06Choose a practice order from your errors

College Board groups geometry and trigonometry into a domain covering area and volume, lines and triangles, right triangles and trig, and circles. Use those categories to find the specific relationship you need to learn rather than reviewing every formula at once. College Board's Math specifications describe the scope.

Start with untimed questions in the Math bank. Explain why the formula applies, then use a timed module to check whether you can identify it while other topics are mixed in. If you miss a question, record whether the problem was the geometry relationship, a unit conversion, or the arithmetic. Those need different practice.