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SPI Mathematics · Mensuration 3D

Spheres and Hemispheres

A sphere's surface is four times the area of its great circle and its volume is four thirds of π times the radius cubed; a hemisphere is exactly half the solid.

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Why this matters

Nine questions in the bank, five EASY and four MODERATE, one of them from the 2022 paper. Each needs one of three formulas, or the rule that areas scale with the square of the radius and volumes with its cube. The usual slip is the flat face of a solid hemisphere.

Concept 1 of 3: Surface area and volume of a sphere

A sphere has one measurement, its radius. The surface needs the radius squared and the volume the radius cubed, so working back from a volume means taking a cube root.

Definition

  • Surface area =4πr2= 4\pi r^2.
  • Volume =43πr3= \frac{4}{3}\pi r^3.
  • If the diameter is given, halve it before using either formula.
  • Working back from a volume: get r3r^3 alone, then look for a perfect cube. 343=73343 = 7^3, 9261=2139261 = 21^3, and a value ending in .875 or .125 is often a half-number cubed, such as 3.53=42.8753.5^3 = 42.875.

Sphere

S=4πr2,V=43πr3S = 4\pi r^2, \qquad V = \frac{4}{3}\pi r^3

Worked example

A sphere has radius 7 cm. Find its surface area and its volume.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Mensuration 3D · Spheres and Hemispheres

The volume of a sphere is 4851 cm3. Its diameter is

Using the diameter as the radius

A ball of diameter 14 cm has radius 7 cm. Putting 14 into 43πr3\frac{4}{3}\pi r^3 makes the volume 8 times too big.

Answering with the radius when the diameter is asked

After the cube root gives rr, check the question. If it asks for the diameter, double it; the radius is usually printed as a wrong option.
Drill 2 more on surface area and volume of a sphere

Concept 2 of 3: Surface area and volume of a hemisphere

Cut a ball in half. Each half keeps half the curved skin, but the cut makes a brand new flat circle on each piece. A bowl has no flat face; a solid half-ball does.

Definition

  • Curved surface =2πr2= 2\pi r^2. This is all a hollow bowl has.
  • Total surface of a solid hemisphere =2πr2+πr2=3πr2= 2\pi r^2 + \pi r^2 = 3\pi r^2.
  • Volume =23πr3= \frac{2}{3}\pi r^3, half the sphere.
  • Cutting a sphere into two halves adds two new circles: the two pieces together show 6πr26\pi r^2, more than the sphere's 4πr24\pi r^2.

Hemisphere

CSA=2πr2,TSA=3πr2,V=23πr3\text{CSA} = 2\pi r^2, \quad \text{TSA} = 3\pi r^2, \quad V = \frac{2}{3}\pi r^3

Worked example

A solid hemisphere has radius 21 cm. Find its total surface area and its volume.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Mensuration 3D · Spheres and Hemispheres

A solid sphere with total surface area 48 cm2\text{cm}^2 is bisected into two hemispheres, then the total surface area of any one of the hemispheres is

Bowl or solid

A hollow bowl is 2πr22\pi r^2. A solid hemisphere also has its flat circle, so its total is 3πr23\pi r^2. Read whether the question says solid.

A hemisphere is not half the sphere's surface

Half of 4πr24\pi r^2 is 2πr22\pi r^2, the curved part only. A solid half also has its new flat face, so each half shows 3πr23\pi r^2, three quarters of the whole sphere's surface.
Drill 3 more on surface area and volume of a hemisphere

Concept 3 of 3: Ratios and percent changes: square for area, cube for volume

Double a ball's radius and its skin grows four times but the stuff inside grows eight times. Lengths, areas and volumes change by k, k squared and k cubed.

Definition

  • If two spheres have radii in the ratio kk, their surfaces are in k2k^2 and their volumes in k3k^3.
  • Going back from volumes, take the cube root: volumes 8:278 : 27 mean radii 2:32 : 3. From areas, take the square root.
  • Percent change: write the new radius as a multiple of the old. 20% less is 0.8r0.8r, so the area becomes 0.640.64 of the old (36% less).
  • Diameters are in the same ratio as radii, and a difference of diameters is twice the difference of radii.

Scaling rule

S1S2=(r1r2)2,V1V2=(r1r2)3\frac{S_1}{S_2} = \left(\frac{r_1}{r_2}\right)^2, \qquad \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3

Worked example

Two spheres have surface areas in the ratio 4 : 9. Find the ratio of their volumes.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q42Easy

Example 3 · Mensuration 3D · Spheres and Hemispheres

The ratio of the volumes of two spheres is 27:64. If sum of their radii is 28cm, then the difference in their diameters is ____cm.

Using the volume ratio as the radius ratio

Volumes 1 : 8 do not mean radii 1 : 8. Take the cube root: radii 1 : 2.

Copying the percent of the length to the area

A diameter 20% shorter gives an area 0.82=0.640.8^2 = 0.64 of the old, so 36% less, not 20% and not 40%.
Drill 1 more on ratios and percent changes: square for area, cube for volume

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Watch out for (6)

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