Understanding the attributes of solids is fundamental in geometry, as it forms the basis for recognising and calculating the properties of three-dimensional shapes. This section delves into various solids including cubes, cuboids, prisms, cylinders, pyramids, cones, spheres, hemispheres, and frustums. Additionally, we will explore the concept of faces, surfaces, and edges, which are crucial for identifying and describing these solids.

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Cube

A cube is a three-dimensional shape with six square faces, all of the same size, 12 edges, and 8 vertices.

Image courtesy of CueMath

• Formulae:
  • Surface area: $6a^2$ (where $a$ is the length of a side)
  • Volume: $a^3$

Example: Calculate the surface area and volume of a cube with a side length of 5 cm.

• Surface area: $6 \times 5^2 = 150 \text{ cm}^2$
• Volume: $5^3 = 125 \text{ cm}^3$

Cuboid

A cuboid is a three-dimensional shape with 6 rectangular faces, 12 edges, and 8 vertices.

• Formulae:
  • Surface area: $2(lw + lh + wh)$ (where $l$, $w$, and $h$ are the length, width, and height respectively)
  • Volume: $lwh$

Example: Calculate the surface area and volume of a cuboid with dimensions 3 cm by 4 cm by 5 cm.

• Surface area: $2(3\cdot4 + 3\cdot5 + 4\cdot5 = 94 \text{ cm}^2$
• Volume: $(3\cdot4\cdot5 = 60) \text{ cm}^3$

Prism

A prism is a solid with two parallel faces called bases that are congruent polygons and all other faces are parallelograms.

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• Formulae:
  • Volume: $Base Area \times Height$

Example: A triangular prism has a base area of $30 \text { cm}^2$ and a height of 5 cm. Calculate its volume.

• Volume: $30 \times 5 = 150 \text{ cm}^3$

Cylinder

A cylinder has two parallel circular bases and a curved surface, connecting the bases.

• Formulae:
  • Surface area: $2\pi r(h + r)$ (where $r$ is the radius and $h$ is the height)
  • Volume: $\pi r^2h$

Example: Calculate the surface area and volume of a cylinder with a radius of 3 cm and a height of 7 cm.

• Surface area: $2\pi \times 3(7 + 3) = 188.4 \text{ cm}^2$ (approx.)
• Volume: $\pi \times 3^2 \times 7 = 198.9 \text{ cm}^3$ (approx.)

Pyramid

A pyramid has a polygon base and triangular faces that meet at a point (the apex).

• Formulae:
  • Volume: $\frac{1}{3} \times Base Area \times Height$

Example: A pyramid with a square base of side 4 cm and a height of 9 cm. Calculate its volume.

• Volume: $\frac{1}{3} \times 4^2 \times 9 = 48 \text{ cm}^3$

Cone

A cone has a circular base and a curved surface that tapers to a point.

Image courtesy of Vedantu

• Formulae:
  • Surface area: $\pi r(r + \sqrt{h^2 + r^2})$ (where $r$ is the radius and $h$ is the height)
  • Volume: $\frac{1}{3}\pi r^2h$

Example: Calculate the surface area and volume of a cone with a radius of 3 cm and a height of 4 cm.

• Surface area: $\pi \times 3(3 + \sqrt{4^2 + 3^2}) = 75.8 \text{ cm}^2$(approx.)
• Volume: $\frac{1}{3}\pi \times 3^2 \times 4 = 37.7 \text{ cm}^3$ (approx.)

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, like a round ball.

• Formulae:
  • Surface area: $4\pi r^2$
  • Volume: $\frac{4}{3}\pi r^3$

Example: Calculate the surface area and volume of a sphere with a radius of 5 cm.

• Surface area: $4\pi \times 5^2 = 314 \text{ cm}^2$
• Volume: $\frac{4}{3}\pi \times 5^3 = 523.6 \text{ cm}^3$ (approx.)

Hemisphere

A hemisphere is half of a sphere.

Image courtesy of Coding Hero

• Formulae:
  • Surface area: $3\pi r^2$
  • Volume: $\frac{2}{3}\pi r^3$

Example: Calculate the surface area and volume of a hemisphere with a radius of 3 cm.

• Surface area: $3\pi \times 3^2 = 84.8 \text{ cm}^2$ (approx.)
• Volume: $\frac{2}{3}\pi \times 3^3 = 56.5 \text{ cm}^3$ (approx.)

Frustum

A frustum is a portion of a solid (normally a cone or pyramid) that lies between two parallel planes cutting it.

Image courtesy of CueMath

• Formulae:
  • Volume: $\frac{1}{3}\pi h(r_1^2 + r_2^2 + r_1r_2)$ (where $h$ is the height, $r_1$ and $r_2$ are the radii of the two bases)

Example: A frustum of a cone with radii 3 cm and 5 cm and a height of 6 cm. Calculate its volume.

• Volume: $\frac{1}{3}\pi \times 6(3^2 + 5^2 + 3\times5) = 197.9 \text{ cm}^3$(approx.)