Cone, Sphere & Hemisphere — Forming Equations

Secondary Mathematics — Mensuration & Algebra
3 questions (5 marks each)
Prepared by Miss Clarissa Ng
www.clartutors.com
Formulae you need: curved surface area of a cone = πrl  ·  curved surface area of a hemisphere = 2πr2  ·  surface area of a sphere = 4πr2
r = radius, l = slant height of the cone. Leave π in your working until the last step.
Surface Area Relationships — Form the Equation, Then Solve

Q1. A cone and a hemisphere have the same radius r. The curved surface area of the cone is equal to the curved surface area of the hemisphere.

(a) Form an equation and simplify it into the form l = kr, where l is the slant height of the cone and r is the radius of both solids.  [2]

(b) Given that the sum of the radius and twice the slant height is 40 cm, find the radius and the slant height of the cone.  [3]

h r l cone r hemisphere

Q2. A cone and a sphere have the same radius r. The curved surface area of the cone is equal to the surface area of the sphere.

(a) Form an equation and simplify it into the form l = kr, where l is the slant height of the cone.  [2]

(b) Given that the slant height exceeds the radius by 27 cm, find the radius and the slant height of the cone.  [3]

Q3. The curved surface area of a cone is three times the curved surface area of a hemisphere of the same radius r.

(a) Form an equation and simplify it into the form l = kr, where l is the slant height of the cone.  [2]

(b) Given that the sum of the radius and the slant height is 35 cm, find the radius and the slant height of the cone.  [3]

Answer Key

Surface Area Relationships

Q1(a)πrl = 2πr2  →  l = 2r
Q1(b)r + 2l = 40, with l = 2r:  r + 4r = 40  →  5r = 40  →  r = 8 cm, l = 16 cm
Q2(a)πrl = 4πr2  →  l = 4r
Q2(b)l − r = 27, with l = 4r:  4r − r = 27  →  3r = 27  →  r = 9 cm, l = 36 cm
Q3(a)πrl = 3(2πr2) = 6πr2  →  l = 6r
Q3(b)r + l = 35, with l = 6r:  7r = 35  →  r = 5 cm, l = 30 cm
Miss Clarissa Ng | www.clartutors.com