Lines, conics, triangles and areas show up all over the FE, inside statics, dynamics and fluids problems as often as on their own. This chapter makes them automatic.
Choose any section below to get a free practice problem. Work it with the FE Reference Handbook open, tap your answer, then check the solution.
Every straight line on the exam reduces to two numbers: a slope and a point.
What is the slope of the line 3x − 6y = 12?
Solve for y: 6y = 3x − 12, so y = (1/2)x − 2. The slope is 1/2.
Common trap: Choice (D) comes from forgetting that −6y flips the sign when you divide.
Straight Lineslopepoint-slopeThat was 1 of 4 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
For ax2 + bx + c = 0, the quadratic formula always works: x = [−b ± √(b2 − 4ac)]/2a.
What are the roots of x2 + 4x + 13 = 0?
Discriminant: 16 − 52 = −36. x = [−4 ± √(−36)]/2 = (−4 ± 6i)/2 = −2 ± 3i.
Common trap: (C) divides only the ± term by 2. Both −b and the root go over 2a.
Quadratic EquationrootsThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
Circles, parabolas, ellipses and hyperbolas all come from the general second-degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.
Where is the focus of the parabola y2 = 12x?
Match to (y − k)2 = 2p(x − h) with vertex (0, 0): 2p = 12, so p = 6. The focus sits p/2 from the vertex along the axis: (3, 0).
Common trap: (D) uses p itself instead of p/2. Check which form your source uses: 4px and 2px define p differently.
Conic SectionsCircleParabolaEllipseHyperbolaThat was 1 of 4 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
Distance is the Pythagorean theorem in disguise, and it extends to three dimensions by adding a z term: d = √[(Δx)2 + (Δy)2 + (Δz)2].
What is the distance between (−2, −3) and (4, 5)?
d = √[(4 − (−2))2 + (5 − (−3))2] = √(36 + 64) = √100 = 10.
Common trap: (A) adds the legs (6 + 8) instead of combining them with the Pythagorean theorem.
distance between two pointsSphereThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
SOH-CAH-TOA covers most right-triangle problems: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
From a point on level ground 30 m from the base of a 40 m tower, what is the angle of elevation to the top?
tan θ = opposite/adjacent = 40/30. θ = arctan(1.333) = 53.1°.
Common trap: (A) is arctan(30/40), the angle measured at the top of the tower.
Trigonometrysin θradianThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
You do not need to memorize identities for the FE, but you do need to recognize which one a problem wants and find it fast.
Using an angle-sum identity, sin 75° equals:
sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4 ≈ 0.966.
Common trap: (A) is sin 15°, from the difference formula.
Identitiesdouble anglesum of two anglesThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
When a triangle has no right angle, reach for the Law of Sines (a side and its opposite angle are known) or the Law of Cosines (two sides and the angle between them, or all three sides).
In triangle ABC, A = 40°, B = 60° and side a = 10. What is side b?
Law of Sines: b/sin B = a/sin A, so b = 10 sin 60°/sin 40° = 10(0.8660)/0.6428 = 13.47.
Common trap: (C) inverts the ratio. The larger angle (60°) must face the longer side.
Law of SinesLaw of CosinesThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
The Handbook lists area formulas for common shapes, including sectors, segments, trapezoids and ellipses.
What is the arc length of a 6 cm radius circle subtended by a 120° central angle?
Convert to radians: 120° = 2π/3 = 2.094 rad. s = rφ = 6(2.094) = 12.57 cm.
Common trap: (B) multiplies by degrees. Arc length needs radians.
MensurationCircular SectorCircular SegmentTrapezoidEllipseThat was 1 of 4 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
Newton’s Notes covers all 14 FE Mechanical exam areas: lessons, worked examples, hundreds of practice problems with full solutions, a 110-question practice exam, calculator keystrokes and Ask Newton, the AI tutor. Try it free for 3 days.
Start your 3-day free trialSee plans →