Pyramid and Cones – GCSE Maths

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Introduction

  • Pyramid and Cones are three-dimensional (3D) geometric shapes that play essential role in mathematics, architecture, engineering, and everyday life.
  • Both shapes have a base and an apex, but they differ in structure and properties.

Real life Examples:

Examples of cones and pyramids using everyday objects like an ice cream cone, traffic cone, birthday hat, tent, Egyptian pyramids, and house roof

What is a Cone?

  • A cone is a solid figure with a circular base that curves upward to meet at a single vertex, forming a pointed tip.

Key Features of a Cone:

  • Base – A circular flat surface.
  • Apex – The pointed top where all the sides meet.
  • Height (h) – The perpendicular distance from the base to the apex.
  • Slant Height (l) – The distance from the apex to any point on the edge of the base.
  • Radius (r) – The distance from the center of the base to its edge.

A labeled diagram of a cone showing the apex, slant height, radius, height, and base

What is a Pyramid?

  • A Pyramid is a three-dimensional geometric shape with a polygonal base and triangular faces that meet at a common point called the apex.

Key Features of a Pyramid:

  • Base – A polygon (e.g., triangle, square, pentagon, etc.).
  • Apex – The topmost point where all triangular faces meet.
  • Faces – Triangular sides connecting the base to the apex.
  • Edges – The line segments where two faces meet.
  • Height (h) – The perpendicular distance from the base to the apex.
  • Slant Height (l) – The height of a triangular face from the base to the apex.

A labeled diagram of a pyramid showing the apex, slant height, radius, and height with clear annotations

How to Find the Volume of Cones?

  • Volume is the amount of space occupied by an object.

Volume of a Cone:

  • A cone has a circular base and tapers to apex.
  • The formula for its volume is:

Volume formulas for pyramids and cones showing one-third multiplied by base area and height, and one-third πr²h for cones

Where,

    • r = Radius of the base
    • h = Height (perpendicular distance from the base to the apex)
    • π ≈ 3.1416 (pi)

Steps to Calculate Volume:

  • Step#1: Measure the radius (r) of the circular base.
  • Step#2: Measure the height (h) of the cone.
  • Step#3: Plug the values into the formula:

Formula for the volume of a cone: V equals one-third times pi times radius squared times height

  • Step#4: Compute the result.

How to Find the Volume of Pyramids?

Volume of a Pyramid:

  • A Pyramid has a polygonal base (e.g., square, triangle) and triangular faces that meet at an apex.
  • The formula for its volume is:

General volume formulas for pyramids and cones: one-third times base area times vertical height

Where,

    • B = Area of the base
    • h = Height (perpendicular distance from the base to the apex)

Steps to Calculate Volume:

  • Step#1: Find the area (B) of the base (depends on the base shape):
      • Square base: B = side2
      • Rectangular base: B = length × width
      • Triangular base: B = 1/2 × base × height
  • Step#2: Measure the height (h) of the pyramid.
  • Step#3: Plug the values into the formula:

Volume formula showing V equals one-third multiplied by B and h, used for pyramids and cones

  • Step#4: Compute the result.

certified Physics and Maths tutorSolved Example

Problem: Find the volume of a cone with radius 6 cm and height 10 cm. (Use π ≈ 3.14)

Solution: 

Step #1: Given

    • r = 6 cm
    • h = 10 cm

Step #2: Plug the Values into the formula:

Volume of a cone formula with example using π = 3.14, radius 6, and height 10

Step #3: Compute the result:

Step-by-step volume calculation of a cone using the formula ⅓ × π × r² × h with final answer 376.8 cm³

The Volume is 376.8 cm³

Final Answer: 376.8 cm³

certified Physics and Maths tutorSolved Example

Problem: Find the volume of a pyramid with a square base of side length 9 meters and a height of 12 meters.

Solution: 

Step #1: Find the Area of the base:

    • Since the base is a square:
    • Base Area= 9 × 9 = 81

Step #2: Plug the Values into the formula:

olume formula for a pyramid with a worked example using base area 81 and height 12

Step #3: Compute the result:

Step-by-step volume calculation of a pyramid using ⅓ × 81 × 12 with final answer 324 m³

The volume is 324 m³

Final Answer: 324 m³

How to Find the Surface Area of Cones

  • Pyramid and Cones Surface area is the Total Area of all the surfaces (faces, bases, and curved sides) that cover a 3D object.

Surface area of a Cone:

It includes:

  • Base area (a circle)
  • Lateral surface area (the curved side)
  • Formula:

Surface area formula showing Surface Area equals Base Area plus Lateral Area

Where,

    • Base Area = πr2 (where r=radius)
    • Lateral Area = πrℓ (where ℓ=slant height)

Steps to Calculate Surface Area:

  • Step#1: Identify Given Values.
  • Step#2: Find the Base Area
  • Step#3: Find the Lateral (Curved) Surface Area.
  • Step#4: Calculate Total Surface Area.

How to Find the Surface Area of Pyramids?

  • Pyramid and Cones Surface area is the Total Area of all the surfaces (faces, bases, and curved sides) that cover a 3D object.

Surface Area of a Pyramid:

It includes:

  • A base (which can be a square, triangle, rectangle, etc.)
  • Triangular lateral faces (number depends on the base shape)
  • Formula:

Surface area formula showing Surface Area equals Base Area plus Lateral Area in red

Where,

    • Lateral Area = ​1/2 × Perimeter of Base × ℓ
    • ℓ = Slant height

Steps to Calculate Surface Area:

  • Step#1: Identify Given Values.
  • Step#2: Find the Base Area
      • Square base: Area = s2 (side length s)
      • Triangular base: Area = 1/2 bh (base b, height h)
      • Rectangular base: Area= lw (length l, width w)
  • Step#3: Find the lateral area.
  • Step#4: Calculate Total Surface Area.

certified Physics and Maths tutorSolved Example

Problem: A cone has a radius of 5 cm and a slant height of 13 cm. Calculate its total surface area. Using π ≈ 3.14

Solution: 

Step #1: Identify Given Values:

    • Radius (r) = 5 cm
    • Slant height (ℓ) = 13 cm

Step #2: Find the Base Area:

The base is a circle, so its area is:

Base area formula for a cone using πr² with r = 5, showing the result as 25π cm²

Step #3: Find the Lateral (Curved) Surface Area:

The lateral area of a cone is given by:

Lateral area formula for a cone using πrl with r = 5 and slant height = 13, giving 65π cm²

Step #4: Calculate Total Surface Area:

Full surface area calculation of a cone using base area 25π and lateral area 65π, resulting in 282.6 cm²

The Surface Area is 282.6 cm2

Final Answer: 282.6 cm

certified Physics and Maths tutorSolved Example

Problem: A square pyramid has a base side length of 6 m and a slant height of 5 m. Find its total surface area.

Solution: 

Step #1: Identify Given Values:

    • Base side length (s) = 6 m
    • Slant height (ℓ) = 5 m

Step #2: Find the Base Area:

The base is a square, so its area is:

Base area formula for a square-based pyramid using s² with side length 6 m, result is 36 m²

Step #3: Find the Lateral (Curved) Surface Area:

A square pyramid has 4 triangular faces.

Lateral area formula for a pyramid showing ½ × perimeter of base × slant height (ℓ)

Where,

Perimeter of a square base calculated using 4s = 4 × 6 = 24 meters

Now,

Lateral area of a pyramid calculated using ½ × 24 × 5 = 60 square meters

Step #4: Calculate Total Surface Area:

Final surface area of a pyramid calculated by adding base area 36 and lateral area 60 to get 96 m²

The Surface Area is 96 m2

Final Answer: 96 m2

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