Grade 10 — Mathematics 10C › C10U02 Surface Area & Volume

Grade 10 › C10U02 Surface Area & Volume › L7
GRADE 10 · ALBERTAOUTCOME AB-MAT10C-M2

L7 · Volume of Right Cylinders, Cones and Spheres

Learning goal

“I can find the volumes of the round solids - cylinder V = πr²h, cone V = (1/3)πr²h, sphere V = (4/3)πr³ - using the VERTICAL height for the cone, and I can run a formula backwards to find a missing dimension.”

Cylinder volume — a prism whose base is a circle: V = (area of circular base) x height = πr²h. Lesson L6's formula, with B = πr².
Cone volume — the pyramid of the round family: exactly 1/3 of the cylinder with the same base and height, V = (1/3)πr²h (indicator 3.6).
Sphere volume — V = (4/3)πr³. Note the r³: the radius is used three times, matching the three dimensions of space the sphere fills.
Worked example

Sketch and label first. A right cylinder has radius 3 cm and height 7 cm. Find its volume - then the volume of a cone with the same base and height. Step 1: Cylinder: B = π x 3² = 9π; V = B x h = 9π x 7 = 63π ≈ 197.92 cm3. Step 2: Cone (same base, same height): V = 1/3 x 63π = 21π ≈ 65.97 cm3. Say it plainly: the cone is not a new idea - it is Lesson L6's 1/3 wearing a round base, and it uses the VERTICAL height, never the slant height.

Check yourself — explain it

A cone problem lists radius 5, vertical height 8, slant height 9.2 (approx.). For the volume you use...

Paired problem — example still visible

Cylinder, r = 2 cm, h = 9 cm -> V = π x 4 x 9 = 36π ≈ 113.10 cm3

Faded completion — last step is yours

"Cone, r = 6 cm, vertical height 5 cm: V = 1/3 x π x 36 x 5 = 1/3 x ___π = ___π ≈ ___ cm3" -> 180π; 60π ≈ 188.50 cm3

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