GED Mathematical Reasoning
Volume & Surface Area
Volume is how much fits inside a solid object — water in a tank. Surface area is how much wrapping paper it would take to cover the outside. Volume is in cubic units; surface area is in square units.
The words first
- volume
- the space inside a 3-D object, counted in cubes
- surface area
- the total area of every outside face added together
- prism
- a box shape — same flat shape all the way through
- cylinder
- a can — a circle all the way through
The method
THE BIG IDEA FOR ANYTHING BOX-LIKE OR CAN-LIKE
volume = area of the flat end × how long it is
BOX (rectangular prism)
volume = length × width × height
surface area = 2(lw) + 2(lh) + 2(wh) — six faces, in matching pairs
CYLINDER
volume = π × radius² × height
( the circle's area )
CONE and PYRAMID — exactly one third of the full shape
volume = ⅓ × (area of the base) × height
SPHERE
volume = 4/3 × π × radius³
Worked all the way through
The problem
A cylindrical water tank has a radius of 3 feet and a height of 10 feet. What is its volume? (Use π ≈ 3.14)
- Step 1 — Find the area of the flat end, which is a circle.area = π × radius² = 3.14 × 3²
- Step 2 — Do the exponent before the multiplication.3² = 3 × 3 = 9. So area = 3.14 × 9 = 28.26 square feet.
- Step 3 — Multiply by how tall it is.28.26 × 10 = 282.6
- Step 4 — Label the units.Volume is three dimensions, so cubic feet.
On the board
V = π × r² × h
the formula
V = 3.14 × 3² × 10
put the numbers in
V = 3.14 × 9 × 10
3² = 3 × 3 = 9, exponent first
V = 28.26 × 10
the circle's area
V = 282.6 cu ft
Answer
282.6 cubic feet
A second one, in a different form
A box is 4 ft long, 3 ft wide, and 2 ft tall. What is its volume?
V = l × w × h
the formula
V = 4 × 3 × 2
V = 24 cu ft
Answer
24 cubic feet
The usual mistake
Confusing volume with surface area, or forgetting to square the radius.
Check yourself
The most common error is using 3 × 2 instead of 3 × 3 for the radius squared. Squared means times itself, never times two.
Practice it
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