Quadric Surface (SPHERE)

Mathematics · FE Reference Handbook section

Mathematics

3 formulas

10 exam-style examples

~51 min

All Mathematics lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Quadric Surface (SPHERE) within Mathematics. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what quadric surface (sphere) describes physically and when it applies.
State every one of the 3 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: radians vs degrees — set the calculator before the first trig entry.

Lecture

Why this section exists. Quadric Surface (SPHERE) is the part of Mathematics that lets you connect an algebraic or calculus expression that must be evaluated exactly to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as short symbolic manipulations with one numeric evaluation at the end. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. radians vs degrees — set the calculator before the first trig entry. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Three engineers in hard hats and safety vests reviewing drawings on a truck tailgate. — Photo 1. Where this shows up in practice: quadric surface (sphere).
Capstone Studio instructional photograph

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes an algebraic or calculus expression that must be evaluated exactly. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 3 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

The standard form of the equation is
with center at (h, k, m).
In a three-dimensional space, the distance between two points is
2 2 2

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Volume and surface area of a cylindrical tank — Quadric Surface (SPHERE)

A cylindrical tank is 13 ft in diameter and 13 ft tall. Most nearly, what is its volume in gallons and its total surface area?

Given

d = 13 ft
h = 13 ft
1 ft³ = 7.48 gal

Find

Volume (gal) and total surface area (ft²)

Start with the thinking

Volume uses the circular area times height.
Total surface adds both end caps to the lateral shell.

Step-by-step solution

Volume
$V = (π /4) d^{2} h$
Substituting
$V = (π /4) (13)^{2} (13) = 1, 726 f t^{3}$
Convert
$V = 1, 726 \times 7.48 = 12, 907 g a l$
Surface — S = πdh + 2(π/4)d²
Substituting
$S = π (13) (13) + 2 (π /4) (13)^{2} = 796.4 f t^{2}$

Answer: V ≈ 12,907 gal; S ≈ 796.4 ft²

Why the other options are there

6,902 ft³ (radius and diameter confused)
530.9 ft² (end caps omitted)

Reference: FE Reference Handbook — Mathematics → Quadric Surface (SPHERE)

Example 2

Ellipse: eccentricity, focal distance and enclosed area — Quadric Surface (SPHERE)

An elliptical culvert opening has a semi-major axis a = 5 ft and semi-minor axis b = 2 ft. Determine the eccentricity of this conic section, the distance from the centre to each focus, and the enclosed area.

Given

a = 5 ft
b = 2 ft

Find

Eccentricity e, focal distance c, and area A of the ellipse

Start with the thinking

Every conic section is characterised by its eccentricity: e = 0 circle, 0 < e < 1 ellipse, e = 1 parabola.
For an ellipse c² = a² − b², so the foci sit on the major axis.

Step-by-step solution

Formula
$e = (1 - b^{2} / a^{2})$
Substituting
$e = (1 - 2^{2} / 5^{2}) = (1 - 0.1600) = 0.9165$
Formula
$c = a e$
Substituting
$c = 5 \times 0.9165 = 4.583 f t$
Formula
$A = π ab$
Substituting
$A = π (5) (2) = 31.42 f t^{2}$

Answer: e = 0.917, c = 4.58 ft, A = 31.4 ft²

Why the other options are there

A = 78.5 ft² (used a twice)
e = 0.400 (ratio of axes, not eccentricity)

Reference: FE Reference Handbook — Mathematics → Quadric Surface (SPHERE)

Example 3

Volume and surface area of a cylindrical tank — Quadric Surface (SPHERE) (2)

A cylindrical tank is 5 ft in diameter and 26 ft tall. Most nearly, what is its volume in gallons and its total surface area?

Given

d = 5 ft
h = 26 ft
1 ft³ = 7.48 gal

Find

Volume (gal) and total surface area (ft²)

Start with the thinking

Volume uses the circular area times height.
Total surface adds both end caps to the lateral shell.

Step-by-step solution

Volume
$V = (π /4) d^{2} h$
Substituting
$V = (π /4) (5)^{2} (26) = 510.5 f t^{3}$
Convert
$V = 510.5 \times 7.48 = 3, 819 g a l$
Surface — S = πdh + 2(π/4)d²
Substituting
$S = π (5) (26) + 2 (π /4) (5)^{2} = 447.7 f t^{2}$

Answer: V ≈ 3,819 gal; S ≈ 447.7 ft²

Why the other options are there

2,042 ft³ (radius and diameter confused)
408.4 ft² (end caps omitted)

Reference: FE Reference Handbook — Mathematics → Quadric Surface (SPHERE)

Example 4

Ellipse: eccentricity, focal distance and enclosed area — Quadric Surface (SPHERE) (2)

An elliptical culvert opening has a semi-major axis a = 11 ft and semi-minor axis b = 3 ft. Determine the eccentricity of this conic section, the distance from the centre to each focus, and the enclosed area.

Given

a = 11 ft
b = 3 ft

Find

Eccentricity e, focal distance c, and area A of the ellipse

Start with the thinking

Every conic section is characterised by its eccentricity: e = 0 circle, 0 < e < 1 ellipse, e = 1 parabola.
For an ellipse c² = a² − b², so the foci sit on the major axis.

Step-by-step solution

Formula
$e = (1 - b^{2} / a^{2})$
Substituting
$e = (1 - 3^{2} /1 1^{2}) = (1 - 0.0744) = 0.9621$
Formula
$c = a e$
Substituting
$c = 11 \times 0.9621 = 10.583 f t$
Formula
$A = π ab$
Substituting
$A = π (11) (3) = 103.7 f t^{2}$

Answer: e = 0.962, c = 10.58 ft, A = 103.7 ft²

Why the other options are there

A = 380.1 ft² (used a twice)
e = 0.273 (ratio of axes, not eccentricity)

Reference: FE Reference Handbook — Mathematics → Quadric Surface (SPHERE)

Example 5

Volume and surface area of a cylindrical tank — Quadric Surface (SPHERE) (3)

A cylindrical tank is 19 ft in diameter and 30 ft tall. Most nearly, what is its volume in gallons and its total surface area?

Given

d = 19 ft
h = 30 ft
1 ft³ = 7.48 gal

Find

Volume (gal) and total surface area (ft²)

Start with the thinking

Volume uses the circular area times height.
Total surface adds both end caps to the lateral shell.

Step-by-step solution

Volume
$V = (π /4) d^{2} h$
Substituting
$V = (π /4) (19)^{2} (30) = 8, 506 f t^{3}$
Convert
$V = 8, 506 \times 7.48 = 63, 624 g a l$
Surface — S = πdh + 2(π/4)d²
Substituting
$S = π (19) (30) + 2 (π /4) (19)^{2} = 2, 358 f t^{2}$

Answer: V ≈ 63,624 gal; S ≈ 2,358 ft²

Why the other options are there

34,023 ft³ (radius and diameter confused)
1,791 ft² (end caps omitted)

Reference: FE Reference Handbook — Mathematics → Quadric Surface (SPHERE)

Example 6

Ellipse: eccentricity, focal distance and enclosed area — Quadric Surface (SPHERE) (3)

An elliptical culvert opening has a semi-major axis a = 5 ft and semi-minor axis b = 3 ft. Determine the eccentricity of this conic section, the distance from the centre to each focus, and the enclosed area.

Given

a = 5 ft
b = 3 ft

Find

Eccentricity e, focal distance c, and area A of the ellipse

Start with the thinking

Every conic section is characterised by its eccentricity: e = 0 circle, 0 < e < 1 ellipse, e = 1 parabola.
For an ellipse c² = a² − b², so the foci sit on the major axis.

Step-by-step solution

Formula
$e = (1 - b^{2} / a^{2})$
Substituting
$e = (1 - 3^{2} / 5^{2}) = (1 - 0.3600) = 0.8000$
Formula
$c = a e$
Substituting
$c = 5 \times 0.8000 = 4.000 f t$
Formula
$A = π ab$
Substituting
$A = π (5) (3) = 47.12 f t^{2}$

Answer: e = 0.800, c = 4.00 ft, A = 47.1 ft²

Why the other options are there

A = 78.5 ft² (used a twice)
e = 0.600 (ratio of axes, not eccentricity)

Reference: FE Reference Handbook — Mathematics → Quadric Surface (SPHERE)

Example 7

Volume and surface area of a cylindrical tank — Quadric Surface (SPHERE) (4)

A cylindrical tank is 8 ft in diameter and 26 ft tall. Most nearly, what is its volume in gallons and its total surface area?

Given

d = 8 ft
h = 26 ft
1 ft³ = 7.48 gal

Find

Volume (gal) and total surface area (ft²)

Start with the thinking

Volume uses the circular area times height.
Total surface adds both end caps to the lateral shell.

Step-by-step solution

Volume
$V = (π /4) d^{2} h$
Substituting
$V = (π /4) (8)^{2} (26) = 1, 307 f t^{3}$
Convert
$V = 1, 307 \times 7.48 = 9, 776 g a l$
Surface — S = πdh + 2(π/4)d²
Substituting
$S = π (8) (26) + 2 (π /4) (8)^{2} = 754.0 f t^{2}$

Answer: V ≈ 9,776 gal; S ≈ 754.0 ft²

Why the other options are there

5,228 ft³ (radius and diameter confused)
653.5 ft² (end caps omitted)

Reference: FE Reference Handbook — Mathematics → Quadric Surface (SPHERE)

Example 8

Ellipse: eccentricity, focal distance and enclosed area — Quadric Surface (SPHERE) (4)

An elliptical culvert opening has a semi-major axis a = 6 ft and semi-minor axis b = 3 ft. Determine the eccentricity of this conic section, the distance from the centre to each focus, and the enclosed area.

Given

a = 6 ft
b = 3 ft

Find

Eccentricity e, focal distance c, and area A of the ellipse

Start with the thinking

Every conic section is characterised by its eccentricity: e = 0 circle, 0 < e < 1 ellipse, e = 1 parabola.
For an ellipse c² = a² − b², so the foci sit on the major axis.

Step-by-step solution

Formula
$e = (1 - b^{2} / a^{2})$
Substituting
$e = (1 - 3^{2} / 6^{2}) = (1 - 0.2500) = 0.8660$
Formula
$c = a e$
Substituting
$c = 6 \times 0.8660 = 5.196 f t$
Formula
$A = π ab$
Substituting
$A = π (6) (3) = 56.55 f t^{2}$

Answer: e = 0.866, c = 5.20 ft, A = 56.5 ft²

Why the other options are there

A = 113.1 ft² (used a twice)
e = 0.500 (ratio of axes, not eccentricity)

Reference: FE Reference Handbook — Mathematics → Quadric Surface (SPHERE)

Example 9

Volume and surface area of a cylindrical tank — Quadric Surface (SPHERE) (5)

A cylindrical tank is 14 ft in diameter and 14 ft tall. Most nearly, what is its volume in gallons and its total surface area?

Given

d = 14 ft
h = 14 ft
1 ft³ = 7.48 gal

Find

Volume (gal) and total surface area (ft²)

Start with the thinking

Volume uses the circular area times height.
Total surface adds both end caps to the lateral shell.

Step-by-step solution

Volume
$V = (π /4) d^{2} h$
Substituting
$V = (π /4) (14)^{2} (14) = 2, 155 f t^{3}$
Convert
$V = 2, 155 \times 7.48 = 16, 120 g a l$
Surface — S = πdh + 2(π/4)d²
Substituting
$S = π (14) (14) + 2 (π /4) (14)^{2} = 923.6 f t^{2}$

Answer: V ≈ 16,120 gal; S ≈ 923.6 ft²

Why the other options are there

8,621 ft³ (radius and diameter confused)
615.8 ft² (end caps omitted)

Reference: FE Reference Handbook — Mathematics → Quadric Surface (SPHERE)

Example 10

Ellipse: eccentricity, focal distance and enclosed area — Quadric Surface (SPHERE) (5)

An elliptical culvert opening has a semi-major axis a = 7 ft and semi-minor axis b = 2 ft. Determine the eccentricity of this conic section, the distance from the centre to each focus, and the enclosed area.

Given

a = 7 ft
b = 2 ft

Find

Eccentricity e, focal distance c, and area A of the ellipse

Start with the thinking

Every conic section is characterised by its eccentricity: e = 0 circle, 0 < e < 1 ellipse, e = 1 parabola.
For an ellipse c² = a² − b², so the foci sit on the major axis.

Step-by-step solution

Formula
$e = (1 - b^{2} / a^{2})$
Substituting
$e = (1 - 2^{2} / 7^{2}) = (1 - 0.0816) = 0.9583$
Formula
$c = a e$
Substituting
$c = 7 \times 0.9583 = 6.708 f t$
Formula
$A = π ab$
Substituting
$A = π (7) (2) = 43.98 f t^{2}$

Answer: e = 0.958, c = 6.71 ft, A = 44.0 ft²

Why the other options are there

A = 153.9 ft² (used a twice)
e = 0.286 (ratio of axes, not eccentricity)

Reference: FE Reference Handbook — Mathematics → Quadric Surface (SPHERE)

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given an algebraic or calculus expression that must be evaluated exactly, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Quadric Surface (SPHERE) contains 3 relations; you must be able to find this page in under 15 seconds.
Exam style: short symbolic manipulations with one numeric evaluation at the end.
Unit rule: radians vs degrees — set the calculator before the first trig entry.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

radians vs degrees — set the calculator before the first trig entry
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.