Comprehensive Geometry and Measurement Formula Sheet - Seventh Grade Mathematics
This comprehensive reference guide consolidates all essential formulas, geometric relationships, proportional equations, and statistical metrics across seventh grade mathematics.
Two-Dimensional Perimeter, Circumference, and Area Formulas
Two-dimensional measurements quantify linear boundaries and planar regions, where perimeter is measured in linear units and area is measured in square units.
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| Geometric Shape | Perimeter / Boundary | Area (Square Units) |
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| Square (side s) | P = 4s | A = s^2 |
| Rectangle (l, w) | P = 2l + 2w = 2(l + w)| A = l * w |
| Triangle (base b, h) | P = s1 + s2 + s3 | A = (1/2) * b * h |
| Parallelogram (b, h) | P = 2a + 2b | A = b * h |
| Trapezoid (b1, b2, h) | P = s1 + s2 + s3 + s4 | A = (1/2) * (b1 + b2) * h |
| Circle (radius r) | C = 2 * pi * r = pi*d | A = pi * r^2 |
| Semicircle (radius r) | P = (1/2)*pi*d + d | A = (1/2) * pi * r^2 |
| Sector (angle theta) | Arc = (theta/360)*pi*d| A = (theta / 360) * pi * r^2|
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Circle Reference:
d = 2r r = d / 2
pi approx 3.14 or 22/7
Three-Dimensional Surface Area and Volume Formulas
Three-dimensional metrics measure boundary coverage in square units and internal cubic capacity in cubic units.
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| 3D Solid Type | Total Surface Area (SA) | Volume (V) |
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| General Right Prism | SA = 2B + P * h | V = B * h |
| Rectangular Prism | SA = 2lw + 2lh + 2wh | V = l * w * h |
| Cube (edge s) | SA = 6s^2 | V = s^3 |
| Triangular Prism | SA = 2B + (s1+s2+s3)*H | V = (1/2)*b*h_tri * H |
| Regular Pyramid | SA = B + (1/2) * P * l | V = (1/3) * B * h |
| Square Pyramid (s, l) | SA = s^2 + 2 * s * l | V = (1/3) * s^2 * h |
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In the pyramid formulas, l represents the slant height along the triangular face, while h represents the true perpendicular altitude from the apex to the center of the base.
Proportional Relationships, Percents, and Financial Formulas
Algebraic models govern proportional scaling, financial interest, and commercial transactions.
PROPORTIONAL EQUATIONS & RATES
Constant of Proportionality: k = y / x
Proportional Equation: y = k * x
Cross-Product of Proportions: If a/b = c/d, then a * d = b * c
PERCENT EQUATIONS & FINANCE
Core Percent Equation: Part = (Percent / 100) * Whole
Percent Change (Inc/Dec): Percent Change = (|New - Orig| / Orig) * 100%
Percent Error: Percent Error = (|Exp - Actual| / Actual) * 100%
Retail Markup Price: Retail = Wholesale * (1 + Markup Rate)
Discounted Sale Price: Sale = Original * (1 - Discount Rate)
Simple Interest Formula: I = P * r * t
Total Accrued Balance: A = P + I = P * (1 + r * t)
In the simple interest formula, P represents the principal sum, r represents the annual interest rate as a decimal, and t represents the time strictly measured in years.
Statistics and Probability Formulas
Statistical calculations describe distributional centers and variations, while probability rules quantify chance occurrences.
STATISTICAL METRICS
Arithmetic Mean: x_bar = ( Sum of all values ) / n
Median: Middle value of numerically ordered data
Interquartile Range (IQR): IQR = Q3 - Q1
Mean Absolute Deviation (MAD): MAD = ( Sum of |x_i - x_bar| ) / n
PROBABILITY RULES
Theoretical Probability: P(E) = n(E) / n(S)
Experimental Relative Frequency: P_exp(E) = Frequency / Total Trials
Complement Rule: P(not A) = 1 - P(A)
Independent Events (A and B): P(A and B) = P(A) * P(B)
Dependent Events (A and B): P(A and B) = P(A) * P(B given A)
Fundamental Counting Principle: Total Outcomes = n_1 * n_2 * ... * n_k
Measurement and Unit Conversion Benchmarks
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| Domain | Customary Conversions | Metric Conversions |
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| Length | 1 foot = 12 inches | 1 meter = 100 cm |
| | 1 yard = 3 feet = 36 inches | 1 km = 1,000 meters |
| | 1 mile = 5,280 feet | 1 cm = 10 millimeters |
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| Volume & Capacity | 1 cup = 8 fluid ounces | 1 liter = 1,000 mL |
| | 1 pint = 2 cups | 1 cubic meter = 1,000 L|
| | 1 quart = 2 pints = 4 cups | |
| | 1 gallon = 4 quarts | |
| | 1 cubic yard = 27 cu ft | |
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| Mass & Weight | 1 pound = 16 ounces | 1 kilogram = 1,000 g |
| | 1 ton = 2,000 pounds | 1 gram = 1,000 mg |
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Exercises
Work through each problem carefully in your notebook before reading the provided solutions.
Exercise 1. A trapezoidal drainage canal cross-section has parallel bases measuring eight meters and twelve meters, with a perpendicular height of four meters. Using the trapezoid area formula from the formula sheet, calculate the cross-sectional area.
Exercise 2. A triangular prism has right-triangular bases with legs of six centimeters and eight centimeters. The height of the prism connecting the bases is ten centimeters. Using the formula sheet, calculate the volume of the prism.
Exercise 3. A consumer deposits eight hundred dollars into a savings account earning a simple annual interest rate of three point five percent for eighteen months. Use the simple interest formulas from the formula sheet to calculate the interest earned and the final total balance.
Exercise 4. A dataset consists of the numbers ten, twelve, fourteen, and twenty. Use the statistical formulas from the formula sheet to calculate the mean and the Mean Absolute Deviation (MAD).
Solutions
Solution to Exercise 1. Locate the trapezoid area formula on the sheet: A equals one-half times the quantity b1 plus b2 times height h. Substitute base one equals eight, base two equals twelve, and height equals four: A equals one-half times the quantity eight plus twelve times four. Sum the bases: eight plus twelve equals twenty. Multiply: one-half times twenty is ten; ten times four equals forty square meters. The cross-sectional area of the canal is forty square meters.
Solution to Exercise 2. Locate the triangular prism volume formula: V equals one-half times base b times triangle height h_tri times prism height H. The legs of the right triangle are six centimeters and eight centimeters, serving as base and height: one-half times six times eight equals twenty-four square centimeters for base area B. Multiply by the prism height H of ten centimeters: twenty-four times ten equals two hundred forty cubic centimeters. The volume of the prism is two hundred forty cubic centimeters.
Solution to Exercise 3. Locate the simple interest formula: I equals P times r times t. The principal P is eight hundred dollars. The annual rate r is three point five percent, which converts to zero point zero three five (0.035). Convert eighteen months into years: eighteen divided by twelve equals one point five years (t = 1.5). Multiply: eight hundred times zero point zero three five times one point five. Eight hundred times zero point zero three five equals twenty-eight; twenty-eight times one point five equals forty-two dollars in interest. To find the total balance, add principal and interest: eight hundred plus forty-two equals eight hundred forty-two dollars ($842.00).
Solution to Exercise 4. First find the mean x-bar: sum ten, twelve, fourteen, and twenty to get fifty-six. Divide fifty-six by four: fifty-six divided by four equals fourteen. The mean is fourteen. Next compute absolute deviations from fourteen: vertical bar ten minus fourteen vertical bar equals four; vertical bar twelve minus fourteen vertical bar equals two; vertical bar fourteen minus fourteen vertical bar equals zero; vertical bar twenty minus fourteen vertical bar equals six. Sum the absolute deviations: four plus two plus zero plus six equals twelve. Divide by count four: twelve divided by four equals three. The Mean Absolute Deviation (MAD) is three.