UK K12 GCSE/A-Level: Year 11 GCSE Mathematics - Mensuration, Surface Area and Volume of Spheres and Cones

Learning objectives

By the end of this topic, students will be able to calculate the surface area and volume of spheres and cones, and apply these calculations to solve problems in a variety of contexts. Students will also be able to demonstrate an understanding of the key concepts and formulas involved, including the use of-and the application of mathematical techniques such as substitution and rearrangement.

Core concepts

Surface Area and Volume of Spheres

The surface area (A) of a sphere is given by the formula:

A = 4?r^2

where r is the radius of the sphere.

The volume (V) of a sphere is given by the formula:

V = (4/3)?r^3

where r is the radius of the sphere.

Surface Area and Volume of Cones

The surface area (A) of a cone is given by the formula:

A = ?r^2 + ?rl

where r is the radius of the base of the cone and l is the slant height.

The volume (V) of a cone is given by the formula:

V = (1/3)?r^2h

where r is the radius of the base of the cone and h is the height.

Key concepts and formulas

• The use of? (pi) as a mathematical constant
• The application of mathematical techniques such as substitution and rearrangement
• The use of formulas to calculate surface area and volume
• The ability to apply mathematical concepts to solve problems in a variety of contexts

Worked examples

Example 1: Surface Area of a Sphere

Find the surface area of a sphere with a radius of 5 cm.

Solution:

A = 4?r^2 = 4 ×-× (5)^2 = 4 ×-× 25 ? 314.16 cm^2

Example 2: Volume of a Cone

Find the volume of a cone with a radius of 4 cm and a height of 10 cm.

Solution:

V = (1/3)?r^2h = (1/3) ×-× (4)^2 × 10 = (1/3) ×-× 16 × 10 ? 418.88 cm^3

Example 3: Surface Area of a Cone

Find the surface area of a cone with a radius of 3 cm and a slant height of 10 cm.

Solution:

A = ?r^2 + ?rl =-× (3)^2 +-× 3 × 10 =-× 9 +-× 30 ? 88.19 + 94.25 ? 182.44 cm^2

Common misconceptions

• Students may confuse the formulas for surface area and volume of spheres and cones.
• Students may not understand the concept of? (pi) and its application in mathematical formulas.
• Students may not apply mathematical techniques such as substitution and rearrangement correctly.

Exam tips

• Make sure to read the question carefully and understand what is being asked.
• Use the formulas and mathematical techniques learned in class to solve problems.
• Check your answers for accuracy and make sure to show all working.
• Use? (pi) correctly and consistently throughout the exam.

MCQs with explanations

MCQ 1: Surface Area of a Sphere [F]

What is the surface area of a sphere with a radius of 4 cm?

A) 100 cm^2 B) 200 cm^2 C) 400 cm^2 D) 800 cm^2

Correct answer: C) 400 cm^2

Why the distractors fail: Students may have confused the formula for surface area of a sphere or not applied it correctly.

MCQ 2: Volume of a Cone [H]

What is the volume of a cone with a radius of 6 cm and a height of 12 cm?

A) 200 cm^3 B) 400 cm^3 C) 600 cm^3 D) 800 cm^3

Correct answer: C) 600 cm^3

Why the distractors fail: Students may have not applied the formula for volume of a cone correctly or not used the correct values.

MCQ 3: Surface Area of a Cone [F]

What is the surface area of a cone with a radius of 2 cm and a slant height of 8 cm?

A) 20 cm^2 B) 40 cm^2 C) 80 cm^2 D) 160 cm^2

Correct answer: C) 80 cm^2

Why the distractors fail: Students may have not applied the formula for surface area of a cone correctly or not used the correct values.

MCQ 4: Volume of a Sphere [H]

What is the volume of a sphere with a radius of 8 cm?

A) 1000 cm^3 B) 2000 cm^3 C) 4000 cm^3 D) 8000 cm^3

Correct answer: C) 4000 cm^3

Why the distractors fail: Students may have not applied the formula for volume of a sphere correctly or not used the correct values.

MCQ 5: Surface Area of a Sphere [H]

What is the surface area of a sphere with a radius of 10 cm?

A) 1000 cm^2 B) 2000 cm^2 C) 4000 cm^2 D) 8000 cm^2

Correct answer: C) 4000 cm^2

Why the distractors fail: Students may have not applied the formula for surface area of a sphere correctly or not used the correct values.

Short-answer questions

Question 1

Find the surface area of a sphere with a radius of 6 cm.

Question 2

Find the volume of a cone with a radius of 5 cm and a height of 15 cm.

Question 3

Find the surface area of a cone with a radius of 4 cm and a slant height of 12 cm.

Question 4

Find the volume of a sphere with a radius of 9 cm.

Question 5

Find the surface area of a sphere with a radius of 7 cm.

Note: Students should show all working and check their answers for accuracy.