grade 7 Surface Area & Volume of Solids

Surface Area & Volume

Volume and surface area of solids become clearer once coins are stacked. One coin, a circle of area πr², becomes a cylinder of volume πr²h once you stack it to height h; a cone is a third of that and a sphere two thirds when the height equals the diameter. Unrolling the curved side into a rectangle also makes the surface-area formulas easy to see. Change the radius and height and switch among cylinder, cone, and sphere, and the volumes compare as 3 : 1 : 2.

Intuition — start by stacking coins

Radius r3

Height h6

🪙 Stack the Coins

①Area of one coin (circle) = πr²

②Stack h coins → cylinder → V = πr²h

③A cone is a stack of progressively smaller coins → 1/3 of the cylinder

④A sphere fits inside a cylinder, taking 2/3 of it (when h = 2r)

Volume and Surface Area of a Prism/Cylinder

Prism Volume

V = Sh = base area × height

Holds for any base shape — base area × height

Cylinder Volume

V = πr²h

A prism whose base is a circle of radius r

Cylinder Surface Area

S = 2πr² + 2πrh = 2πr(r + h)

Two end circles + side (unrolls to a rectangle: width 2πr, height h)

🔍 Unroll the Side

①Cut the cylinder's side and flatten it — you get a rectangle

②Width = circumference = 2πr

③Height = h

④Lateral area = 2πrh

Cone Volume — why 1/3?

Cone Volume

V = 13Sh

Exactly 1/3 of a prism with the same base and height

Cone Volume (formula)

V = 13πr²h

1/3 of πr²h

Cone Lateral Area

S = πrl

l is the slant length. Unrolling gives a sector. l = √(r²+h²)

💡 Verify by Experiment

①Use a cone-shaped cup with the same base and height as a cylinder cup

②Fill the cone with water and pour into the cylinder — exactly 3 cone-fills

③That confirms the 1/3 intuition without calculus!

Sphere Volume and Surface Area

Sphere Volume

V = 43πr³

Sphere of radius r — 2/3 of a cylinder of height 2r

Sphere Surface Area

S = 4πr²

Equal to four circles of the same radius

🏀 Sphere = 2/3 of the Cylinder

①Inscribe a sphere in a cylinder of height = diameter

②Cylinder V = πr² × 2r = 2πr³

③Sphere V = (4/3)πr³ = 2/3 of the cylinder

④A beautiful result Archimedes discovered!

Work It Out

Example 1

Find the volume of a cylinder with base radius 3 and height 5. (Leave π as is.)

Volume of a prism/cylinder = (base area) × (height) = πr² × h.

volume = π × 3² × 5

Compute.

= 9π × 5 = 45π

▸ 45π

A cylinder volume multiplies base area by height.

Example 2

Find the volume of a square pyramid with base side 6 and height 10.

Volume of a pyramid = (1/3) × (base area) × (height). Base area = 6² = 36.

volume = (1/3) × 36 × 10

Compute.

= (1/3) × 360 = 120

▸ 120

A pyramid is one-third of the prism with the same base and height.

Exam Key Points

Volume Summary

Sh, 13Sh, 43πr³

Cylinder : Cone : Sphere = 3 : 1 : 2 (when h = 2r)

Grade-7 school exam type

What is the volume of a sphere of radius 3? (Leave π as is.)

①12π

②27π

③36π

④72π

⑤108π

▸ ③ 36π

Volume of a sphere = (4/3)πr³.

volume = (4/3) × π × 3³

Compute.

= (4/3) × 27π = 36π

🎯 Exam Key Points

①Prism V = Sh; Cone V = Sh/3 — same regardless of base shape

②Cylinder surface 2πr(r+h) is common

③Cone lateral area πrl (l = slant length)

④Sphere surface 4πr² is 4× a circle's area — explain why in essay questions

⑤Hemisphere surface = 2πr² + πr² = 3πr² (curved + flat)

⑥Prism and pyramid surface area = sum of the faces on the net

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Polygon & Circle

Data Organization & Analysis

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