Geometry-Based Algebra

Shape & Setup

Mode

L = 2W + 3, 2L + 2W = 54

Length = (m)·Width + (b), plus perimeter. Solves for Width and Length.

Multiplier m

Offset b

Perimeter P

Examples

Given area A and one side, find the other side. A = L × W.

Area A

Known side

Examples

Given P and A, solve x² − (P/2)x + A = 0 for both sides.

Perimeter P

Area A

Examples

Given perimeter and one side, find the other. P = 2L + 2W.

Perimeter P

Known side

Examples

Mode

x + 2x + (x+20) = 180

Angles: (a₁x + b₁)°, (a₂x + b₂)°, (a₃x + b₃)°. Sum = 180°.

a₁

b₁

a₂

b₂

a₃

b₃

Examples

Given area and base, solve for height. A = ½ · b · h.

Area A

Base b

Examples

Mode

A = 50π → r = √50 = 5√2

A = πr² → r = √(A/π). Enter area in terms of π multiples or plain units.

A/π coefficient k (A = kπ)

Leave blank to enter plain area below instead.

— OR — Plain Area (sq units)

Examples

C = 2πr → r = C / (2π).

Circumference C

Examples

V = πr²h. Give V and h, find r. Or give V and r, find h.

Volume V

Height h

Volume V

Radius r

Examples

Solution

Select a shape and mode, then press Solve to see the unknown dimension, all labeled values, and the formula used.

Answer

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All Dimensions Labeled

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Formula Used & Verification

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Key Formulas

Rectangle: P = 2l + 2w | A = l × w Triangle: A = ½bh | angles sum = 180° Circle: A = πr² | C = 2πr Cylinder: V = πr²h

Rectangle perimeter: Substitute a dimension expression like L = 2W + 3 into 2L + 2W = P, then solve the resulting one-variable linear equation.

Quadratic case (P+A): If you know both perimeter and area, set s = l + w = P/2, then the sides are roots of x² − sx + A = 0.

Triangle angles: Set the sum of angle expressions equal to 180, collect x terms, solve.

Circles/cylinders: Isolate r or h algebraically, then simplify the square root.

Strategy

1. Draw and label the shape 2. Pick the right formula 3. Substitute given info 4. Solve for the unknown

Geometry problems always give you a formula and a relationship. Write everything in terms of one variable, substitute into the formula, and solve.

Tip: After solving, always plug your answer back in to verify the original formula holds.

All dimensions must be positive — reject negative roots.
For quadratic cases, both roots are valid sides (just swap L and W).
Area is in square units; volume in cubic units.
Angle sums always equal exactly 180° — check your arithmetic.

Geometry algebra problems still tricky?