Lines, conics, triangles and areas: the geometry and trigonometry inside the FE Civil’s Mathematics and Statistics questions, and the same tools you use again in surveying, statics and highway geometry.
NCEES lists these Mathematics and Statistics topics for the FE Civil: analytic geometry; single-variable calculus; vector operations; statistics: distributions, mean, mode, standard deviation, confidence intervals, regression and curve fitting; probability: estimation and expected value. On Newton’s Notes this area spans 4 chapters: this one and Algebra and Linear Algebra, Calculus and Probability and Statistics.
Choose any section below to get a free FE Civil 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.
Which line passes through (1, 4) and is perpendicular to y = −(1/3)x + 2?
The perpendicular slope is the negative reciprocal of −1/3, which is 3. Point-slope: y − 4 = 3(x − 1), so y = 3x + 1.
Common trap: (B) has the right slope but misses the point: 3(1) − 1 = 2, not 4.
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.
For what positive value of k does x2 + kx + 9 = 0 have exactly one (repeated) real root?
One repeated root means the discriminant is zero: k2 − 4(1)(9) = 0, so k2 = 36 and k = 6.
Common trap: (B) is the size of the repeated root (x = −3 when k = 6), not the value of k.
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.
What is the eccentricity of the ellipse x2/169 + y2/144 = 1?
a = 13, b = 12. e = √(1 − b2/a2) = √(1 − 144/169) = √(25/169) = 5/13 = 0.385.
Common trap: An ellipse always has e < 1, which rules out (A) on sight.
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 midpoint of the segment from (−3, 5) to (7, −1)?
Average the coordinates: ((−3 + 7)/2, (5 + (−1))/2) = (2, 2).
Common trap: (D) subtracts instead of averaging.
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.
What is 150° in radians?
Multiply by π/180: 150(π/180) = 5π/6 = 2.618.
Common trap: (A) is 120° and (C) is 300°.
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.
If sin θ = 0.6 and θ is acute, what is sin 2θ?
cos θ = √(1 − 0.36) = 0.8. sin 2θ = 2 sin θ cos θ = 2(0.6)(0.8) = 0.96.
Common trap: (B) doubles sin θ. Sine of a double angle is never just double the sine.
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).
A triangle has sides 5, 7 and 9. What is its largest angle?
The largest angle is opposite the longest side (9): cos C = (52 + 72 − 92)/(2·5·7) = −7/70 = −0.1. C = arccos(−0.1) = 95.7°.
Common trap: A negative cosine means the angle is obtuse, which rules out (D) and (C).
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.
A circle of radius 10 m is cut by a chord that subtends a 90° central angle. What is the area of the smaller circular segment?
Segment: A = r2(φ − sin φ)/2 with φ = π/2. A = 100(1.5708 − 1)/2 = 28.5 m2.
Common trap: (A) is the whole sector. The segment subtracts the triangle (50 m2) from it.
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 FE Civil has 18 chapters (11 shared with Mechanical, 7 Civil-only) with lessons, worked examples and 420+ FE-style practice problems with full solutions, 171 of them Civil-only. Pro adds a timed full-length FE Civil practice exam (110 problems), a custom test builder and Ask Newton, the AI tutor. Try it free for 3 days, no card to start.
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