Variables are empty sockets. Constants and coefficients are the rules that come with them. Today you plug numbers into ten different shape formulas and watch them light up.
Part 1 · The Big Idea
A variable is an empty socket
A variable is a letter holding a spot for a number we don't know yet. Substitution means swapping that letter for a real number, then doing the arithmetic. The letter leaves; the number takes its seat.
The one rule: replace every copy of the variable with your number, then compute. Order of operations still applies.
Part 2 · Know Your Parts
Constants, coefficients, and variables
Before we start plugging in, we need to know what each piece of a formula actually is. Take — three different kinds of pieces working together.
Variable
A letter that changes. You pick the value. In , the variable is — you decide the radius.
Coefficient
A number that multiplies a variable. In , the is a coefficient — it scales whatever is.
Constant
A number that never changes. is always ... It's the same in every circle formula, everywhere.
In geometry, constants are everywhere. The in circle formulas, the in triangle area, the in cone and pyramid volume — these never change, no matter what values you plug in for the variables.
Part 3 · Warm Up
The Substitution Machine
Pick a formula below to see what it represents, set the value with the slider, and watch every substitution step in proper math. Constants stay put while variables get replaced.
Substitution Machine Beginner → Intermediate
Violet = variable · Amber = plugged-in value · Green = result · Teal = constant
Part 4 · Formula Browser
Explore all twelve formulas
Tap any formula to see the shape it describes, what each letter means, and a live substitution you can drive with the sliders. Variables in violet, constants in teal, coefficients in amber.
Interactive Formula Browser Interactive
Click a formula card, then move the sliders to plug in values.
Notice the pattern: every formula is just constants and coefficients combined with variables. Plug in the variables — the rest is arithmetic.
Part 5 · Area Explorer
Plug into area formulas
Slide the dimensions, watch the shape redraw, and follow the substitution steps live. Five 2-D shapes, one idea.
Area Explorer Interactive
Part 6 · Volume Explorer
Plug into volume formulas
Now we go 3-D. Volume counts unit cubes (or cone-slices) inside a solid. Five 3-D shapes — all sharing the same substitution move.
Volume Explorer Intermediate
Part 7 · Show What You Know
The Plug-it-In Challenge
A shape, a formula, and some measurements. Substitute and solve. Decimals are rounded to two places — use ≈ 3.14.
Plug-it-In Challenge All Shapes
Difficulty:
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Walk Away With This
Swap the letter. Do the math. Every time.
Every formula — from the simplest square to a sphere or cone — is just variables waiting for values. Constants stay. Coefficients scale. Variables get replaced. That one move unlocks all of geometry.