Missing Measurements of 3D Shapes

📖 The Lesson

In the Missing Sides lesson, you learned to work backwards from a formula to find a missing dimension. This page extends that exact skill to 3D shapes — cylinders, cones, spheres, and prisms. The approach is identical: plug in what you know, simplify, then use the opposite operation to isolate the unknown.

The GED gives you the formula sheet. Your job is knowing which formula to use and plugging in the values correctly.

🎯 The GED Strategy

⭐ The 4-Step Method

Write the formula for the shape.
Plug in every value you know.
Do as much calculator math as possible on the known side — multiply and simplify first.
Use the opposite operation to isolate the missing value — usually divide, or take the square root if the variable is squared.

⭐ Key idea

Simplify the known side completely first — then use the opposite operation to find the missing piece.

📐 GED Formula Sheet — 3D Shapes

These are the exact formulas from the official GED formula sheet. You don't need to memorize them — the test provides them. What matters is knowing how to use them to find a missing value.

Shape	Surface Area (SA)	Volume (V)
Rectangular Prism	`SA = 2lw + 2lh + 2wh`	`V = lwh`
Right Prism	`SA = ph + 2B`	`V = Bh`
Cylinder	`SA = 2πrh + 2πr²`	`V = πr²h`
Pyramid	`SA = ½ps + B`	`V = ⅓Bh`
Cone	`SA = πrs + πr²`	`V = ⅓πr²h`
Sphere	`SA = 4πr²`	`V = ⁴⁄₃πr³`

⭐ Note on Cubes — derive, don't memorize

Cubes are not listed on the GED formula sheet, and you should not memorize a separate cube formula. If you see a cube problem, use the rectangular prism formula with all three dimensions set equal: l = w = h = s. For example, SA = 2lw + 2lh + 2wh becomes 2(s·s) + 2(s·s) + 2(s·s) — three pairs of s² faces. That derivation is the move the test expects.

p = perimeter of base with area B · s = slant height (for pyramid and cone) · π ≈ 3.14

🔄 Opposite Operations

To isolate the unknown, undo whatever operation is applied to it. These are the three you'll need most:

Undo multiplication

Divide both sides

314 = 78.5 × h → h = 314 ÷ 78.5 = 4

Undo a square (r²)

Take the square root

r² = 49 → r = √49 = 7

Undo a cube (r³ or s³)

Take the cube root

r³ = 27 → r = ∛27 = 3

Watch for ⅓ in cone and pyramid

Multiply all known values first

Compute ⅓ × π × r² first, then divide that result into V to find h.

🔢 Worked Examples

Example 1 — Find the height of a cylinder (circle-based shape)

A cylindrical storage tank has a volume of 628 cm³. The radius of the tank is 5 cm. What is the height of the tank? (Use π ≈ 3.14)

Calculator tip: This shape uses π, so the calculator will do the heavy lifting. Focus on setting up each step correctly — let the calculator handle the multiplication.

The formula for the volume of a cylinder is: V = π × r² × h

Plug in the values you know — V = 628 and r = 5:
628 = 3.14 × 5² × h

Simplify the right side first. Start with the exponent: 5² = 25. Then multiply: 3.14 × 25 = 78.5.
Now the equation reads: 628 = 78.5 × h

h is being multiplied by 78.5. The opposite of multiplication is division — divide both sides by 78.5:
h = 628 ÷ 78.5 = 8 cm

Example 2 — Find the radius from sphere surface area (circle-based shape)

A manufacturer is wrapping a spherical ornament. The total surface area of the ornament is 200.96 cm². What is the radius? (Use π ≈ 3.14)

Calculator tip: Sphere formulas involve π and exponents — use your calculator at every step. Don't try to do this in your head.

The formula for surface area of a sphere is: SA = 4 × π × r²

Plug in the value you know — SA = 200.96:
200.96 = 4 × 3.14 × r²

Simplify the right side first. Multiply 4 × 3.14 = 12.56.
Now the equation reads: 200.96 = 12.56 × r²

r² is being multiplied by 12.56. Divide both sides by 12.56:
r² = 200.96 ÷ 12.56 = 16

r² = 16 means r is being squared. The opposite of squaring is the square root:
r = √16 = 4 cm

Example 3 — Find the missing height of a rectangular prism

A moving company is designing a cardboard shipping box. The box needs to hold exactly 360 cubic inches of product. The base of the box is 12 inches long and 5 inches wide. How tall does the box need to be?

The formula for the volume of a rectangular prism is: V = l × w × h

Plug in the values you know — V = 360, l = 12, w = 5:
360 = 12 × 5 × h

Simplify the right side first. Multiply the two known dimensions together: 12 × 5 = 60.
Now the equation reads: 360 = 60 × h

h is being multiplied by 60. Divide both sides by 60:
h = 360 ÷ 60 = 6 inches

The box needs to be 6 inches tall to hold exactly 360 cubic inches.

Example 4 — Find the height of a cone (circle-based shape)

A road crew places a cone-shaped traffic marker on the highway. The marker has a volume of 75.36 cubic inches and a base radius of 3 inches. How tall is the marker? (Use π ≈ 3.14)

Calculator tip: The cone formula includes π and a fraction (⅓) — multiply all the known values together first before dividing. Use your calculator at every step.

The formula for the volume of a cone is: V = ⅓ × π × r² × h

Plug in the values you know — V = 75.36 and r = 3:
75.36 = ⅓ × 3.14 × 3² × h

Simplify the right side first. Start with the exponent: 3² = 9. Then multiply all the known values: ⅓ × 3.14 × 9 = 9.42.
Now the equation reads: 75.36 = 9.42 × h

h is being multiplied by 9.42. The opposite of multiplication is division — divide both sides by 9.42:
h = 75.36 ÷ 9.42 = 8 inches

The traffic cone is 8 inches tall.