IB AA · Unit 2 · 5A Volumes and Surface Area
Find the volume and surface area of a sphere or a solid hemisphere, then work backwards from a given volume or area.
Sphere
Radius r.
V = 43 π r³
A = 4 π r²
A diameter is twice the radius. Halve it before you substitute.
Solid hemisphere
Volume is half a sphere: V = 23 π r³
Curved surface = 2 π r²
The flat base is a circle: π r²
Total surface of a solid hemisphere = 3 π r²
Exact, then a decimal
Keep π in the exact answer when the radius is a whole number.
When a decimal is needed, round to 3 significant figures.
Given the volume or the area
Write the formula, rearrange for r, then substitute.
From A = 4 π r², r² = A ÷ (4 π).
Whole number radii stay in terms of π. A decimal radius is rounded to 3 significant figures.
V = 43 π r³
V = 43 π (5)³
V = 500π3 cm³
As a decimal, V is about 524 cm³
Answer: 500π3 cm³, about 524 cm³
V = 43 π r³
V = 43 π (7)³
V = 1372π3 cm³
As a decimal, V is about 1440 cm³
Answer: 1372π3 cm³, about 1440 cm³
A = 4 π r²
A = 4 π (4)²
A = 64π cm²
As a decimal, A is about 201 cm²
Answer: 64π cm², about 201 cm²
V = 43 π r³
V = 43 π (9)³
V = 972π m³
As a decimal, V is about 3050 m³
Answer: 972π m³, about 3050 m³
A = 4 π r²
A = 4 π (8)²
A = 256π cm²
As a decimal, A is about 804 cm²
Answer: 256π cm², about 804 cm²
V = 43 π r³
V = 43 π (2.40)³
V = 57.9 m³
Answer: 57.9 m³
A solid hemisphere includes the flat circular base in the total surface area.
Curved surface = 2 π r²
Base = π r²
Total surface = 3 π r²
Total surface = 3 π (6)²
Total surface = 108π cm²
As a decimal, the total surface is about 339 cm²
Answer: 108π cm², about 339 cm²
V = 23 π r³
V = 23 π (10)³
V = 2000π3 cm³
As a decimal, V is about 2090 cm³
Answer: 2000π3 cm³, about 2090 cm³
Curved surface = 2 π r²
Curved surface = 2 π (8)²
Curved surface = 128π m²
As a decimal, the curved surface is about 402 m²
Answer: 128π m², about 402 m²
Curved surface = 2 π r²
Base = π r²
Total surface = 3 π r²
Total surface = 3 π (9)²
Total surface = 243π cm²
As a decimal, the total surface is about 763 cm²
Answer: 243π cm², about 763 cm²
V = 43 π r³
V = 43 π (5.50)³
V = 697 cm³
Answer: 697 cm³
V = 23 π r³
V = 23 π (3.20)³
V = 68.6 cm³
Answer: 68.6 cm³
Rearrange the formula before you substitute.
A = 4 π r²
256 π = 4 π r²
r² = 256 π4 π
r² = 64
r = 8 cm
Answer: 8 cm
V = 43 π r³
288 π = 43 π r³
r³ = 288 × 34
r³ = 216
r = 6 cm
Answer: 6 cm
V = 23 π r³
144 π = 23 π r³
r³ = 144 × 32
r³ = 216
r = 6 m
Answer: 6 m
V = 43 π r³
500 = 43 π r³
r³ = 500 × 34 π
r = 4.92 cm
Answer: 4.92 cm
V = 43 π r³
900 = 43 π r³
r³ = 900 × 34 π
r = 5.99 cm
Diameter = 2 × 5.99
Diameter = 12.0 cm
Answer: 12.0 cm
Curved surface = 2 π r²
98 π = 2 π r²
r² = 98 π2 π
r² = 49
r = 7 cm
Answer: 7 cm
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