Geometric feature and trigonometry | FE Civil Mathematics

This chapter covers the following topics:

This chapter covers formulas for common geometric shapes and trigonometric relationships from the FE reference handbook.

Parabola
Ellipse
Circular segment
Circular sector
Sphere
Parallelogram
Regular polygon (n equal sides)
Prismoid
Right circular cone
Right circular cylinder
Paraboloid of revolution
Trigonometric functions
Law of sines
Law of cosines
Trigonometric identities

All of the formulas below come from the FE Reference Handbook’s mensuration of areas and volumes and trigonometry pages. The exam gives you the handbook, so the skill worth building here is locating and applying the right formula quickly, not memorizing it.

Parabola

A parabolic segment is the region bounded by a parabola and a straight line (the chord).

Area under a parabolic segment:

$A = \frac{2}{3} bh$

where $b$ is the base and $h$ is the height.

Perimeter: The arc length of a parabola is found using an integral and does not have a simple closed-form formula.

Ellipse

Area:

$A = πab$

where $a$ and $b$ are the semi-major and semi-minor axes.

Perimeter (approximate):

$P = π (3 (a + b) - (3 a + b) (a + 3 b))$

Circular segment

A circular segment is the region cut off from a circle by a chord.

Area:

$A = \frac{r ^{2} ( θ - sin θ )}{2}$

where $r$ is the radius and $θ$ is the central angle in radians.

Perimeter:

$P = 2 r sin (\frac{θ}{2}) + r θ$

Circular sector

A circular sector is the region bounded by two radii and the included arc.

Area:

$A = \frac{1}{2} r^{2} θ$

Perimeter:

$P = 2 r + r θ$

Example: circular sector area and perimeter

A circular sector has a radius of $8$ cm and a central angle of $50°$ . Find its area and perimeter.

Both formulas require $θ$ in radians, so convert first: $θ = 50° \times \frac{π}{180°} = 0.8727$ rad.

Area: $A = \frac{1}{2} r^{2} θ = \frac{1}{2} (8)^{2} (0.8727) = 27.9 cm^{2}$

Perimeter: $P = 2 r + r θ = 2 (8) + 8 (0.8727) = 16 + 6.98 = 23.0$ cm

Answer: $A \approx 27.9 cm^{2}$ , $P \approx 23.0$ cm

Watch out: the circular segment and sector formulas only work with $θ$ in radians - set your calculator to radian mode (or convert degrees to radians first), since plugging in degrees gives a plausible but wrong answer. It’s also worth carrying full precision through multi-step area and volume problems and rounding only at the final answer; FE answer choices are often close enough together that early rounding lands you on the wrong one.

Sphere

Surface area:

$A = 4 π r^{2}$

Volume:

$V = \frac{4}{3} π r^{3}$

Parallelogram

Area:

$A = bh$

where $b$ is the base and $h$ is the height.

Perimeter:

$P = 2 (a + b)$

where $a$ and $b$ are the lengths of the sides.

Regular polygon (n sides)

Area:

$A = \frac{1}{4} n s^{2} cot (\frac{π}{n})$

where $s$ is the side length.

Perimeter:

$P = n s$

Prismoid

Volume:

$V = \frac{h}{6} (A_{1} + 4 A_{m} + A_{2})$

where $A_{1}$ and $A_{2}$ are the areas of the parallel bases, $A_{m}$ is the midsection area, and $h$ is the height.

Right circular cone

Lateral surface area:

$A_{L} = π r l$

where $l$ is the slant height.

Total surface area:

$A = π r (r + l)$

Volume:

$V = \frac{1}{3} π r^{2} h$

Example: cone total surface area

A right circular cone has a base radius of $5$ in and a slant height of $13$ in. Find its total surface area.

$A = π r (r + l) = π (5) (5 + 13) = π (5) (18) = 90 π \approx 282.7 in^{2}$

Answer: $\approx 282.7 in^{2}$

Right circular cylinder

Lateral surface area:

$A_{L} = 2 π r h$

Total surface area:

$A = 2 π r (r + h)$

Volume:

$V = π r^{2} h$

Paraboloid of revolution

A paraboloid of revolution is formed by rotating a parabola around its axis.

For a paraboloid:

$z = \frac{x ^{2} + y ^{2}}{4 f}$

Volume:

$V = \frac{1}{2} π r^{2} h$

where $r$ is the base radius and $h$ is the height. The FE Reference Handbook lists only the volume for this shape - there’s no simple closed-form surface area formula for a paraboloid of revolution.

Trigonometric functions

Trigonometric functions can be defined using a right triangle.

$sin θ = \frac{y}{r}$

$cos θ = \frac{x}{r}$

$tan θ = \frac{y}{x}$

$cot θ = \frac{x}{y}$

$csc θ = \frac{r}{y}$

$sec θ = \frac{r}{x}$

Here:

$θ$ is one of the non-right angles
$r$ is the hypotenuse (the longest side, opposite the right angle)
$y$ is the side opposite $θ$
$x$ is the side adjacent to $θ$

Example: angle of elevation

From a point $120$ ft from the base of a tower, the angle of elevation to the top of the tower is $35°$ . How tall is the tower?

The horizontal distance is the side adjacent to $θ$ , and the tower height is the side opposite $θ$ , so use $tan θ = \frac{y}{x}$ with $x = 120$ ft and $θ = 35°$ :

$y = x tan θ = 120 tan (35°) = 120 (0.7002) = 84.0$ ft

Answer: $\approx 84.0$ ft

Law of sines

$\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$

where $a, b, c$ are the sides of the triangle, and $A, B, C$ are the angles opposite to sides $a, b, c$ respectively.

Law of cosines

The law of cosines relates a triangle’s three side lengths to the cosine of one of its angles. There are three equivalent versions of the formula, one for each angle.

$a^{2} = b^{2} + c^{2} - 2 b c cos A$

$b^{2} = a^{2} + c^{2} - 2 a c cos B$

$c^{2} = a^{2} + b^{2} - 2 ab cos C$

where $a, b, c$ are the sides of the triangle, and $A, B, C$ are the angles opposite to sides $a, b, c$ respectively.

Trigonometric identities

Basic identities

These identities relate sine and cosine to each other through a quarter-turn shift. They also define cosecant, secant, tangent, and cotangent as reciprocals or ratios of sine and cosine.

$cos θ = sin (θ + \frac{π}{2}) = - sin (θ - \frac{π}{2})$

$sin θ = cos (θ - \frac{π}{2}) = - cos (θ + \frac{π}{2})$

$csc θ = \frac{1}{sin θ}$

$sec θ = \frac{1}{cos θ}$

$tan θ = \frac{sin θ}{cos θ}$

$cot θ = \frac{1}{tan θ}$

Pythagorean identities

$sin^{2} θ + cos^{2} θ = 1$

$tan^{2} θ + 1 = sec^{2} θ$

$cot^{2} θ + 1 = csc^{2} θ$

Angle sum and difference identities

These identities expand sine and cosine of a sum of two angles. They also give the double-angle formulas for sine, cosine, tangent, and cotangent.

$sin (α + β) = sin α cos β + cos α sin β$

$cos (α + β) = cos α cos β - sin α sin β$

$sin 2 α = 2 sin α cos α$

$cos 2 α = cos^{2} α - sin^{2} α = 1 - 2 sin^{2} α = 2 cos^{2} α - 1$

$tan 2 α = \frac{2 tan α}{1 - tan ^{2} α}$

$cot 2 α = \frac{cot ^{2} α - 1}{2 cot α}$

Sum and difference of tangent and cotangent

These identities give the tangent of a sum of two angles. They also give the sine, cosine, and tangent of a difference of two angles.

$tan (α + β) = \frac{tan α + tan β}{1 - tan α tan β}$

$sin (α - β) = sin α cos β - cos α sin β$

$cos (α - β) = cos α cos β + sin α sin β$

$tan (α - β) = \frac{tan α - tan β}{1 + tan α tan β}$

The cotangent sum and difference identities, $cot (α \pm β)$ , follow the same pattern and are listed in the FE Reference Handbook’s trigonometry table.

Half-angle identities

These identities express sine, cosine, tangent, and cotangent of half an angle. Each is written in terms of the cosine of the full angle.

$sin (\frac{α}{2}) = \pm \frac{1 - cos α}{2}$

$cos (\frac{α}{2}) = \pm \frac{1 + cos α}{2}$

$tan (\frac{α}{2}) = \pm \frac{1 - cos α}{1 + cos α}$

$cot (\frac{α}{2}) = \pm \frac{1 + cos α}{1 - cos α}$

Product-to-sum formulas

These convert a product of sines and/or cosines into a sum - for example, $sin α sin β = \frac{1}{2} [cos (α - β) - cos (α + β)]$ . The full set is listed in the FE Reference Handbook’s trigonometry table.

Sum-to-product formulas

These convert a sum of sines and/or cosines into a product - for example, $sin α + sin β = 2 sin (\frac{1}{2} (α + β)) cos (\frac{1}{2} (α - β))$ . The full set is listed in the FE Reference Handbook’s trigonometry table.

Parabola

Parabolic segment area: $A = \frac{2}{3} bh$
Perimeter (arc length): requires integral, no simple formula

Ellipse

Area: $A = πab$
Approximate perimeter: $P = π [3 (a + b) - (3 a + b) (a + 3 b)]$

Circular segment

Area: $A = \frac{r ^{2}}{2} (θ - sin θ)$
Perimeter: $P = 2 r sin (θ /2) + r θ$

Circular sector

Area: $A = \frac{1}{2} r^{2} θ$
Perimeter: $P = 2 r + r θ$

Sphere

Surface area: $A = 4 π r^{2}$
Volume: $V = \frac{4}{3} π r^{3}$

Parallelogram

Area: $A = bh$
Perimeter: $P = 2 (a + b)$

Regular polygon (n sides)

Area: $A = \frac{1}{4} n s^{2} cot (π / n)$
Perimeter: $P = n s$

Prismoid

Volume: $V = \frac{h}{6} (A_{1} + 4 A_{m} + A_{2})$

Right circular cone

Lateral surface area: $A_{L} = π r l$
Total surface area: $A = π r (r + l)$
Volume: $V = \frac{1}{3} π r^{2} h$

Right circular cylinder

Lateral surface area: $A_{L} = 2 π r h$
Total surface area: $A = 2 π r (r + h)$
Volume: $V = π r^{2} h$

Paraboloid of revolution

Surface area: $A = π r r^{2} + 4 h^{2}$
Volume: $V = \frac{1}{2} π r^{2} h$

Trigonometric functions

Defined using right triangle ratios:
- $sin θ = \frac{y}{r}$ , $cos θ = \frac{x}{r}$ , $tan θ = \frac{y}{x}$
- $cot θ = \frac{x}{y}$ , $csc θ = \frac{r}{y}$ , $sec θ = \frac{r}{x}$

Law of sines

$\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$
Relates sides and opposite angles in a triangle

Law of cosines

$a^{2} = b^{2} + c^{2} - 2 b c cos A$ (and cyclic permutations)
Generalizes Pythagorean theorem for any triangle

Trigonometric identities

Basic identities:
- $cos θ = sin (θ + π /2)$ , $sin θ = cos (θ - π /2)$
- $csc θ = 1/ sin θ$ , $sec θ = 1/ cos θ$
- $tan θ = sin θ / cos θ$ , $cot θ = 1/ tan θ$
Pythagorean identities:
- $sin^{2} θ + cos^{2} θ = 1$
- $tan^{2} θ + 1 = sec^{2} θ$
- $cot^{2} θ + 1 = csc^{2} θ$
Angle sum/difference identities:
- $sin (α \pm β) = sin α cos β \pm cos α sin β$
- $cos (α \pm β) = cos α cos β \mp sin α sin β$
- $tan (α \pm β) = \frac{t a n α \pm t a n β}{1 \mp t a n α t a n β}$
Double and half-angle identities:
- $sin 2 α = 2 sin α cos α$
- $cos 2 α = cos^{2} α - sin^{2} α$
- $sin (α /2) = \pm \frac{1 - c o s α}{2}$
- $cos (α /2) = \pm \frac{1 + c o s α}{2}$
Product-to-sum and sum-to-product formulas:
- $sin α sin β = \frac{1}{2} [cos (α - β) - cos (α + β)]$
- $sin α + sin β = 2 sin \frac{α + β}{2} cos \frac{α - β}{2}$

This chapter covers the following topics:

This chapter covers formulas for common geometric shapes and trigonometric relationships from the FE reference handbook.

Parabola
Ellipse
Circular segment
Circular sector
Sphere
Parallelogram
Regular polygon (n equal sides)
Prismoid
Right circular cone
Right circular cylinder
Paraboloid of revolution
Trigonometric functions
Law of sines
Law of cosines
Trigonometric identities

Parabola

A parabolic segment is the region bounded by a parabola and a straight line (the chord).

Area under a parabolic segment:

$A = \frac{2}{3} bh$

where $b$ is the base and $h$ is the height.

Perimeter: The arc length of a parabola is found using an integral and does not have a simple closed-form formula.

Ellipse

Area:

$A = πab$

where $a$ and $b$ are the semi-major and semi-minor axes.

Perimeter (approximate):

$P = π (3 (a + b) - (3 a + b) (a + 3 b))$

Circular segment

A circular segment is the region cut off from a circle by a chord.

Area:

$A = \frac{r ^{2} ( θ - sin θ )}{2}$

where $r$ is the radius and $θ$ is the central angle in radians.

Perimeter:

$P = 2 r sin (\frac{θ}{2}) + r θ$

Circular sector

A circular sector is the region bounded by two radii and the included arc.

Area:

$A = \frac{1}{2} r^{2} θ$

Perimeter:

$P = 2 r + r θ$

Example: circular sector area and perimeter

A circular sector has a radius of $8$ cm and a central angle of $50°$ . Find its area and perimeter.

Both formulas require $θ$ in radians, so convert first: $θ = 50° \times \frac{π}{180°} = 0.8727$ rad.

Area: $A = \frac{1}{2} r^{2} θ = \frac{1}{2} (8)^{2} (0.8727) = 27.9 cm^{2}$

Perimeter: $P = 2 r + r θ = 2 (8) + 8 (0.8727) = 16 + 6.98 = 23.0$ cm

Answer: $A \approx 27.9 cm^{2}$ , $P \approx 23.0$ cm

Sphere

Surface area:

$A = 4 π r^{2}$

Volume:

$V = \frac{4}{3} π r^{3}$

Parallelogram

Area:

$A = bh$

where $b$ is the base and $h$ is the height.

Perimeter:

$P = 2 (a + b)$

where $a$ and $b$ are the lengths of the sides.

Regular polygon (n sides)

Area:

$A = \frac{1}{4} n s^{2} cot (\frac{π}{n})$

where $s$ is the side length.

Perimeter:

$P = n s$

Prismoid

Volume:

$V = \frac{h}{6} (A_{1} + 4 A_{m} + A_{2})$

where $A_{1}$ and $A_{2}$ are the areas of the parallel bases, $A_{m}$ is the midsection area, and $h$ is the height.

Right circular cone

Lateral surface area:

$A_{L} = π r l$

where $l$ is the slant height.

Total surface area:

$A = π r (r + l)$

Volume:

$V = \frac{1}{3} π r^{2} h$

Example: cone total surface area

A right circular cone has a base radius of $5$ in and a slant height of $13$ in. Find its total surface area.

$A = π r (r + l) = π (5) (5 + 13) = π (5) (18) = 90 π \approx 282.7 in^{2}$

Answer: $\approx 282.7 in^{2}$

Right circular cylinder

Lateral surface area:

$A_{L} = 2 π r h$

Total surface area:

$A = 2 π r (r + h)$

Volume:

$V = π r^{2} h$

Paraboloid of revolution

A paraboloid of revolution is formed by rotating a parabola around its axis.

For a paraboloid:

$z = \frac{x ^{2} + y ^{2}}{4 f}$

Volume:

$V = \frac{1}{2} π r^{2} h$

Trigonometric functions

Trigonometric functions can be defined using a right triangle.

$sin θ = \frac{y}{r}$

$cos θ = \frac{x}{r}$

$tan θ = \frac{y}{x}$

$cot θ = \frac{x}{y}$

$csc θ = \frac{r}{y}$

$sec θ = \frac{r}{x}$

Here:

$θ$ is one of the non-right angles
$r$ is the hypotenuse (the longest side, opposite the right angle)
$y$ is the side opposite $θ$
$x$ is the side adjacent to $θ$

Example: angle of elevation

From a point $120$ ft from the base of a tower, the angle of elevation to the top of the tower is $35°$ . How tall is the tower?

The horizontal distance is the side adjacent to $θ$ , and the tower height is the side opposite $θ$ , so use $tan θ = \frac{y}{x}$ with $x = 120$ ft and $θ = 35°$ :

$y = x tan θ = 120 tan (35°) = 120 (0.7002) = 84.0$ ft

Answer: $\approx 84.0$ ft

Law of sines

$\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$

where $a, b, c$ are the sides of the triangle, and $A, B, C$ are the angles opposite to sides $a, b, c$ respectively.

Law of cosines

The law of cosines relates a triangle’s three side lengths to the cosine of one of its angles. There are three equivalent versions of the formula, one for each angle.

$a^{2} = b^{2} + c^{2} - 2 b c cos A$

$b^{2} = a^{2} + c^{2} - 2 a c cos B$

$c^{2} = a^{2} + b^{2} - 2 ab cos C$

where $a, b, c$ are the sides of the triangle, and $A, B, C$ are the angles opposite to sides $a, b, c$ respectively.

Trigonometric identities

Basic identities

These identities relate sine and cosine to each other through a quarter-turn shift. They also define cosecant, secant, tangent, and cotangent as reciprocals or ratios of sine and cosine.

$cos θ = sin (θ + \frac{π}{2}) = - sin (θ - \frac{π}{2})$

$sin θ = cos (θ - \frac{π}{2}) = - cos (θ + \frac{π}{2})$

$csc θ = \frac{1}{sin θ}$

$sec θ = \frac{1}{cos θ}$

$tan θ = \frac{sin θ}{cos θ}$

$cot θ = \frac{1}{tan θ}$

Pythagorean identities

$sin^{2} θ + cos^{2} θ = 1$

$tan^{2} θ + 1 = sec^{2} θ$

$cot^{2} θ + 1 = csc^{2} θ$

Angle sum and difference identities

These identities expand sine and cosine of a sum of two angles. They also give the double-angle formulas for sine, cosine, tangent, and cotangent.

$sin (α + β) = sin α cos β + cos α sin β$

$cos (α + β) = cos α cos β - sin α sin β$

$sin 2 α = 2 sin α cos α$

$cos 2 α = cos^{2} α - sin^{2} α = 1 - 2 sin^{2} α = 2 cos^{2} α - 1$

$tan 2 α = \frac{2 tan α}{1 - tan ^{2} α}$

$cot 2 α = \frac{cot ^{2} α - 1}{2 cot α}$

Sum and difference of tangent and cotangent

These identities give the tangent of a sum of two angles. They also give the sine, cosine, and tangent of a difference of two angles.

$tan (α + β) = \frac{tan α + tan β}{1 - tan α tan β}$

$sin (α - β) = sin α cos β - cos α sin β$

$cos (α - β) = cos α cos β + sin α sin β$

$tan (α - β) = \frac{tan α - tan β}{1 + tan α tan β}$

The cotangent sum and difference identities, $cot (α \pm β)$ , follow the same pattern and are listed in the FE Reference Handbook’s trigonometry table.

Half-angle identities

These identities express sine, cosine, tangent, and cotangent of half an angle. Each is written in terms of the cosine of the full angle.

$sin (\frac{α}{2}) = \pm \frac{1 - cos α}{2}$

$cos (\frac{α}{2}) = \pm \frac{1 + cos α}{2}$

$tan (\frac{α}{2}) = \pm \frac{1 - cos α}{1 + cos α}$

$cot (\frac{α}{2}) = \pm \frac{1 + cos α}{1 - cos α}$

Product-to-sum formulas

Sum-to-product formulas

Parabola

Parabolic segment area: $A = \frac{2}{3} bh$
Perimeter (arc length): requires integral, no simple formula

Ellipse

Area: $A = πab$
Approximate perimeter: $P = π [3 (a + b) - (3 a + b) (a + 3 b)]$

Circular segment

Area: $A = \frac{r ^{2}}{2} (θ - sin θ)$
Perimeter: $P = 2 r sin (θ /2) + r θ$

Circular sector

Area: $A = \frac{1}{2} r^{2} θ$
Perimeter: $P = 2 r + r θ$

Sphere

Surface area: $A = 4 π r^{2}$
Volume: $V = \frac{4}{3} π r^{3}$

Parallelogram

Area: $A = bh$
Perimeter: $P = 2 (a + b)$

Regular polygon (n sides)

Area: $A = \frac{1}{4} n s^{2} cot (π / n)$
Perimeter: $P = n s$

Prismoid

Volume: $V = \frac{h}{6} (A_{1} + 4 A_{m} + A_{2})$

Right circular cone

Lateral surface area: $A_{L} = π r l$
Total surface area: $A = π r (r + l)$
Volume: $V = \frac{1}{3} π r^{2} h$

Right circular cylinder

Lateral surface area: $A_{L} = 2 π r h$
Total surface area: $A = 2 π r (r + h)$
Volume: $V = π r^{2} h$

Paraboloid of revolution

Surface area: $A = π r r^{2} + 4 h^{2}$
Volume: $V = \frac{1}{2} π r^{2} h$

Trigonometric functions

Defined using right triangle ratios:
- $sin θ = \frac{y}{r}$ , $cos θ = \frac{x}{r}$ , $tan θ = \frac{y}{x}$
- $cot θ = \frac{x}{y}$ , $csc θ = \frac{r}{y}$ , $sec θ = \frac{r}{x}$

Law of sines

$\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$
Relates sides and opposite angles in a triangle

Law of cosines

$a^{2} = b^{2} + c^{2} - 2 b c cos A$ (and cyclic permutations)
Generalizes Pythagorean theorem for any triangle

Trigonometric identities

Basic identities:
- $cos θ = sin (θ + π /2)$ , $sin θ = cos (θ - π /2)$
- $csc θ = 1/ sin θ$ , $sec θ = 1/ cos θ$
- $tan θ = sin θ / cos θ$ , $cot θ = 1/ tan θ$
Pythagorean identities:
- $sin^{2} θ + cos^{2} θ = 1$
- $tan^{2} θ + 1 = sec^{2} θ$
- $cot^{2} θ + 1 = csc^{2} θ$
Angle sum/difference identities:
- $sin (α \pm β) = sin α cos β \pm cos α sin β$
- $cos (α \pm β) = cos α cos β \mp sin α sin β$
- $tan (α \pm β) = \frac{t a n α \pm t a n β}{1 \mp t a n α t a n β}$
Double and half-angle identities:
- $sin 2 α = 2 sin α cos α$
- $cos 2 α = cos^{2} α - sin^{2} α$
- $sin (α /2) = \pm \frac{1 - c o s α}{2}$
- $cos (α /2) = \pm \frac{1 + c o s α}{2}$
Product-to-sum and sum-to-product formulas:
- $sin α sin β = \frac{1}{2} [cos (α - β) - cos (α + β)]$
- $sin α + sin β = 2 sin \frac{α + β}{2} cos \frac{α - β}{2}$