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

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:

$$A=\pi ab$$

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

Perimeter (approximate):

$$P=\pi \left(3(a+b)-\sqrt{(3a+b)(a+3b)}\right)$$

Circular segment

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

Area:

$$A=\frac{r^2(\theta -\sin \theta )}{2}$$

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

Perimeter:

$$P=2r\sin \left(\frac{\theta }{2}\right)+r\theta$$

Circular sector

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

Area:

$$A=\frac{1}{2}{r}^2\theta$$

Perimeter:

$$P=2r+r\theta$$

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 $\theta$ in radians, so convert first: $\theta =50°\times \frac{\pi }{180°}=0.8727$ rad.

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

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

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

Watch out: the circular segment and sector formulas only work with $\theta$ 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:

$$A=4\pi r^2$$

Volume:

$$V=\frac{4}{3}\pi r^3$$

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:

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

where $s$ is the side length.

Perimeter:

$$P=ns$$

Prismoid

Volume:

$$V=\frac{h}{6}(A_1+4A_m+A_2)$$

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:

$$A_L=\pi rl$$

where $l$ is the slant height.

Total surface area:

$$A=\pi r(r+l)$$

Volume:

$$V=\frac{1}{3}\pi r^{2}h$$

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.

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

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

Right circular cylinder

Lateral surface area:

$$A_L=2\pi rh$$

Total surface area:

$$A=2\pi r(r+h)$$

Volume:

$$V=\pi r^{2}h$$

Paraboloid of revolution

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

For a paraboloid:

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

Volume:

$$V=\frac{1}{2}\pi r^{2}h$$

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.

$$\sin \theta =\frac{y}{r}$$

$$\cos \theta =\frac{x}{r}$$

$$\tan \theta =\frac{y}{x}$$

$$\cot \theta =\frac{x}{y}$$

$$\csc \theta =\frac{r}{y}$$

$$\sec \theta =\frac{r}{x}$$

Here:

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

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 $\theta$, and the tower height is the side opposite $\theta$, so use $\tan \theta =\frac{y}{x}$ with $x=120$ ft and $\theta =35°$:

$y=x\tan \theta =120\tan (35°)=120(0.7002)=84.0$ 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

$$a^2=b^2+c^2-2bc\cos A$$

$$b^2=a^2+c^2-2ac\cos B$$

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

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

$$\cos \theta =\sin \left(\theta +\frac{\pi }{2}\right)=-\sin \left(\theta -\frac{\pi }{2}\right)$$

$$\sin \theta =\cos \left(\theta -\frac{\pi }{2}\right)=-\cos \left(\theta +\frac{\pi }{2}\right)$$

$$\csc \theta =\frac{1}{\sin \theta }$$

$$\sec \theta =\frac{1}{\cos \theta }$$

$$\tan \theta =\frac{\sin \theta }{\cos \theta }$$

$$\cot \theta =\frac{1}{\tan \theta }$$

Pythagorean identities

$${\sin }^2\theta +{\cos }^2\theta =1$$

$${\tan }^2\theta +1={\sec }^2\theta$$

$${\cot }^2\theta +1={\csc }^2\theta$$

Angle sum and difference identities

$$\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$$

$$\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$$

$$\sin 2\alpha =2\sin \alpha \cos \alpha$$

$$\cos 2\alpha ={\cos }^2\alpha -{\sin }^2\alpha =1-2{\sin }^2\alpha =2{\cos }^2\alpha -1$$

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

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

Sum and difference of tangent and cotangent

$$\tan (\alpha +\beta )=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }$$

$$\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta$$

$$\cos (\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta$$

$$\tan (\alpha -\beta )=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }$$

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

Half-angle identities

$$\sin \left(\frac{\alpha }{2}\right)=\pm \sqrt{\frac{1-\cos \alpha }{2}}$$

$$\cos \left(\frac{\alpha }{2}\right)=\pm \sqrt{\frac{1+\cos \alpha }{2}}$$

$$\tan \left(\frac{\alpha }{2}\right)=\pm \frac{\sqrt{1-\cos \alpha }}{\sqrt{1+\cos \alpha }}$$

$$\cot \left(\frac{\alpha }{2}\right)=\pm \frac{\sqrt{1+\cos \alpha }}{\sqrt{1-\cos \alpha }}$$

Product-to-sum formulas

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