Spheres, cones and pyramids Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Spheres, cones and pyramids questions. See exactly how to solve problems on sphere, volume and surface area formulas, exact answers in terms of pi, diameter to radius.

spherevolume and surface area formulasexact answers in terms of pidiameter to radiushemispherehalf a sphere
GCSE Foundation70 questionsStep-by-step solutions
Question 1
2 markseasy
A sphere has radius 33 cm. Work out the volume of the sphere. Give your answer in terms of π\pi.

Worked solution

  1. Write down the formula for the volume of a sphere.

    V=43πr3V = \frac{4}{3}\pi r^3

    The volume of a sphere depends only on its radius rr, and the radius is cubed.

  2. Substitute the radius into the formula.

    V=43π×33V = \frac{4}{3}\pi \times 3^3

    The radius is 33 cm, so put r=3r = 3 into the formula.

  3. State the volume of the sphere.

    V=43π×27=36πV = \frac{4}{3}\pi \times 27 = 36\pi

    The volume of the sphere is 36π36\pi cm3\text{cm}^3.

Answer
V=36π cm3V = 36\pi \text{ cm}^3
Question 2
2 markseasy
A sphere has radius 66 cm. Work out the volume of the sphere. Give your answer in terms of π\pi.

Worked solution

  1. Write down the formula for the volume of a sphere.

    V=43πr3V = \frac{4}{3}\pi r^3

    The volume of a sphere depends only on its radius rr, and the radius is cubed.

  2. Substitute the radius into the formula.

    V=43π×63V = \frac{4}{3}\pi \times 6^3

    The radius is 66 cm, so put r=6r = 6 into the formula.

  3. State the volume of the sphere.

    V=43π×216=288πV = \frac{4}{3}\pi \times 216 = 288\pi

    The volume of the sphere is 288π288\pi cm3\text{cm}^3.

Answer
V=288π cm3V = 288\pi \text{ cm}^3
Question 3
2 markseasy
A sphere has diameter 1212 cm. Work out the volume of the sphere. Give your answer in terms of π\pi.

Worked solution

  1. Halve the diameter to get the radius.

    r=122=6r = \frac{12}{2} = 6

    The formula needs the radius, so halve the diameter: the radius is 66 cm.

  2. Write down the formula for the volume of a sphere and substitute.

    V=43πr3=43π×63V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi \times 6^3

    Put r=6r = 6 into the volume formula for a sphere.

  3. State the volume of the sphere.

    V=43π×216=288πV = \frac{4}{3}\pi \times 216 = 288\pi

    The volume of the sphere is 288π288\pi cm3\text{cm}^3.

Answer
V=288π cm3V = 288\pi \text{ cm}^3
Question 4
1 markeasy
A sphere has radius 55 cm. Work out the surface area of the sphere. Give your answer in terms of π\pi.

Worked solution

  1. Write down the formula for the surface area of a sphere.

    A=4πr2A = 4\pi r^2

    The surface of a sphere is entirely curved, and its area is four times the area of a great circle through the centre.

  2. Substitute the radius into the formula.

    A=4π×52A = 4\pi \times 5^2

    The radius is 55 cm, so put r=5r = 5 into the formula.

  3. State the surface area of the sphere.

    A=4π×25=100πA = 4\pi \times 25 = 100\pi

    The surface area of the sphere is 100π100\pi cm2\text{cm}^2.

Answer
A=100π cm2A = 100\pi \text{ cm}^2
Question 5
2 markseasy
A sphere has radius 77 cm. Work out the surface area of the sphere. Give your answer in terms of π\pi.

Worked solution

  1. Write down the formula for the surface area of a sphere.

    A=4πr2A = 4\pi r^2

    The surface of a sphere is entirely curved, and its area is four times the area of a great circle through the centre.

  2. Substitute the radius into the formula.

    A=4π×72A = 4\pi \times 7^2

    The radius is 77 cm, so put r=7r = 7 into the formula.

  3. State the surface area of the sphere.

    A=4π×49=196πA = 4\pi \times 49 = 196\pi

    The surface area of the sphere is 196π196\pi cm2\text{cm}^2.

Answer
A=196π cm2A = 196\pi \text{ cm}^2

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