1.2 Geometric feature and trigonometry

Achievable FE Civil

1. Mathematics

Our FE Civil course is currently in development and is a work-in-progress.

Geometric feature and trigonometry

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This chapter covers the following topics:

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:

$P = 2 r + r θ$

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:

$V = \frac{1}{3} π r^{2} h$

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:

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

Surface area:

$A = π r r^{2} + 4 h^{2}$

where $r$ is the base radius and $h$ is the height.

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 $θ$

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

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

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

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

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

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

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

$csc θ = \frac{1}{sin θ}$

$sec θ = \frac{1}{cos θ}$

$tan θ = \frac{sin θ}{cos θ}$

$cot θ = \frac{1}{tan θ}$

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

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

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

$sin 2 α = 2 sin α cos α$

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

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

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

Sum and difference of tangent and cotangent

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

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

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

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

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

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

Half-angle identities

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

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

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

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

Product-to-sum formulas

$sin α sin β = \frac{1}{2} [cos (α - β) - cos (α + β)]$

$cos α cos β = \frac{1}{2} [cos (α - β) + cos (α + β)]$

$sin α cos β = \frac{1}{2} [sin (α + β) + sin (α - β)]$

Sum-to-product formulas

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

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

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

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

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:
- $sin α sin β = \frac{1}{2} [cos (α - β) - cos (α + β)]$
- $sin α + sin β = 2 sin \frac{α + β}{2} cos \frac{α - β}{2}$

Next | 1.3 Algebra

Geometric feature and trigonometry

This chapter covers the following topics:

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:

$P = 2 r + r θ$

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:

$V = \frac{1}{3} π r^{2} h$

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:

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

Surface area:

$A = π r r^{2} + 4 h^{2}$

where $r$ is the base radius and $h$ is the height.

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 $θ$

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

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

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

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

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

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

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

$csc θ = \frac{1}{sin θ}$

$sec θ = \frac{1}{cos θ}$

$tan θ = \frac{sin θ}{cos θ}$

$cot θ = \frac{1}{tan θ}$

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

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

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

$sin 2 α = 2 sin α cos α$

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

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

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

Sum and difference of tangent and cotangent

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

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

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

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

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

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

Half-angle identities

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

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

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

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

Product-to-sum formulas

$sin α sin β = \frac{1}{2} [cos (α - β) - cos (α + β)]$

$cos α cos β = \frac{1}{2} [cos (α - β) + cos (α + β)]$

$sin α cos β = \frac{1}{2} [sin (α + β) + sin (α - β)]$

Sum-to-product formulas

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

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

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

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

Achievable FE Civil

1. Mathematics

Our FE Civil course is currently in development and is a work-in-progress.