C6.4, C6.5; E6.4, E6.5 - Spheres and parts of solids

C6.4, C6.5; E6.4, E6.5 - Spheres and parts of solids

Cutting a solid in half halves its volume, but does it halve its total surface area? C6.4 and E6.4 supply the sphere formulas; C6.5 and E6.5 use them for parts and combinations of solids. You will separate volume from exposed area, including the extra flat face made by a cut and the hidden faces at a join.

Sphere formulas

A spherical ball is curved all the way round, so it has no circular base to add. Its radius runs from its centre to any point on its surface; a diameter passes through the centre and is twice the radius.

Surface area and volume of a sphere

For a sphere of radius rr,

S=4πr2,V=43πr3S=4\pi r^2,\qquad V=\frac43\pi r^3

Surface area is measured in square units; volume is measured in cubic units. Both formulas are on the supplied lists for Papers 1–4; required formulas are supplied within questions on Papers 5 and 6.

The powers distinguish the quantities. Surface area covers the outside; volume fills the interior. Squaring or cubing a given diameter instead of the radius would magnify the error differently in the two formulas.

Worked example 1: radius given

Question. A sphere has radius 6 cm. Calculate its surface area and volume in terms of π\pi.

Plan. Use the same radius in both formulas, with the appropriate power.

S=4π×62=144π cm2surface uses a squareV=43π×63=288π cm3volume uses a cube\begin{aligned} S&=4\pi\times6^2=144\pi\text{ cm}^2 &&\text{surface uses a square}\\ V&=\frac43\pi\times6^3=288\pi\text{ cm}^3 &&\text{volume uses a cube} \end{aligned}

Answer. Surface area 144π cm2144\pi\text{ cm}^2; volume 288π cm3288\pi\text{ cm}^3.

Check. The surface area is four times the area of a radius-6 circle: 4×36π=144π4\times36\pi=144\pi. Also, V/S=r/3=2V/S=r/3=2 cm, and 288π/144π=2288\pi/144\pi=2 cm, consistent with the two formulas.

A diameter is supplied next. One division is needed before either formula can be used.

Worked example 2: diameter given

Question. A sphere has diameter 10 cm. Calculate its surface area and volume in terms of π\pi.

Plan. Halve the diameter, then square or cube that radius.

r=10/2=5 cmradiusS=4π×52=100π cm2surface areaV=43π×53=500π3 cm3volume\begin{aligned} r&=10/2=5\text{ cm} &&\text{radius}\\ S&=4\pi\times5^2=100\pi\text{ cm}^2 &&\text{surface area}\\ V&=\frac43\pi\times5^3=\frac{500\pi}{3}\text{ cm}^3 &&\text{volume} \end{aligned}

Answer. Surface area 100π cm2100\pi\text{ cm}^2; volume 500π3 cm3\frac{500\pi}{3}\text{ cm}^3.

Check. A bounding cube of side 10 cm has volume 1000 cm31000\text{ cm}^3; the sphere's volume is about 524 cm3524\text{ cm}^3, smaller as it must be. The surface area is 4×25π=100π cm24\times25\pi=100\pi\text{ cm}^2.

Keep exact answers as multiples of π\pi. If a decimal is requested, retain the full calculator value until the final rounding.

Watch out

A sphere's surface area is already its complete outside area. Adding another πr2\pi r^2 invents a flat base that a sphere does not have.

Choose the radius before cubing.

Half a sphere

A plane cut through a sphere's centre makes two equal halves. Each half contains half the volume and half the original curved surface, but the cut exposes a new flat circle as well.

[DIAGRAM: asset_name: Curved and flat surfaces of half a sphere; asset_slug: half_sphere_surfaces; caption: A half-sphere solid has a curved surface and a new circular face.; description: A half-sphere's curved part is shown in pale blue, with its flat circular cut face separated below in pale green to show both surfaces clearly. A radius r is marked across the flat face from centre to rim. Separate labels identify curved area 2 pi r squared and flat area pi r squared; a second outline shows a dome whose base is excluded from the required covering area.]
A half-sphere solid has a curved surface and a new circular face.

Half-sphere solid

A sphere cut through its centre gives two half-sphere solids. Each is also called a hemisphere. Its curved area is half the sphere's area; its flat face is a circle of the same radius.

You can work with “half a sphere” directly: first halve the appropriate sphere quantity, then decide whether the flat cut face is included. These derived half-sphere formulas are not separate entries on the supplied lists.

Vhalf=12×43πr3=23πr3Scurved=12×4πr2=2πr2Stotal=2πr2+πr2=3πr2\begin{aligned} V_{\text{half}}&=\frac12\times\frac43\pi r^3=\frac23\pi r^3\\ S_{\text{curved}}&=\frac12\times4\pi r^2=2\pi r^2\\ S_{\text{total}}&=2\pi r^2+\pi r^2=3\pi r^2 \end{aligned}

Worked example 3: a solid cut through its centre

Question. A solid sphere of radius 4 cm is cut into two equal halves. Calculate the volume and total surface area of one half, in terms of π\pi.

Plan. Halve the sphere volume, then add the flat circle to half its surface area.

V=12×43π×43=128π3 cm3half the filled sphereS=12×4π×42+π×42curved part plus cut face=32π+16π=48π cm2\begin{aligned} V&=\frac12\times\frac43\pi\times4^3=\frac{128\pi}{3}\text{ cm}^3 &&\text{half the filled sphere}\\ S&=\frac12\times4\pi\times4^2+\pi\times4^2 &&\text{curved part plus cut face}\\ &=32\pi+16\pi=48\pi\text{ cm}^2 \end{aligned}

Answer. Volume 128π3 cm3\frac{128\pi}{3}\text{ cm}^3; total surface area 48π cm248\pi\text{ cm}^2.

Check. Rejoining two halves gives the whole sphere volume 256π/3256\pi/3. Their separate areas total 96π96\pi; hiding the two 16π16\pi cut faces leaves 64π64\pi, the whole sphere's area.

Only the exposed surfaces change next. The flat base lies against a support and needs no covering.

Worked example 4: covering a dome

Question. A dome is exactly half a sphere of radius 4 m. Only its curved outside is to be covered. Calculate the covering area in terms of π\pi.

Plan. Halve the sphere's surface area without adding a base.

A=12×4π×42curved half only=32π m2\begin{aligned} A&=\frac12\times4\pi\times4^2 &&\text{curved half only}\\ &=32\pi\text{ m}^2 \end{aligned}

Answer. 32π m232\pi\text{ m}^2.

Check. The excluded base has area 16π m216\pi\text{ m}^2. Adding it would give 48π m248\pi\text{ m}^2, the total area of a separate solid half-sphere of this radius.

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