Mensuration and trigonometry Revision Notes

What is a Polyhedron?

A polyhedron is a 3D solid bounded by flat polygonal faces. It does not have any curved surfaces. Examples include prisms and pyramids. A sphere or a cylinder is not a polyhedron because they have curved surfaces. Understanding this definition is the first step in identifying and working with polyhedra.

Key Components: Faces, Edges, and Vertices

Every polyhedron has three main components: Faces (F), Edges (E), and Vertices (V). Faces are the flat surfaces, which are always polygons. Edges are the straight lines where two faces meet. Vertices are the points where three or more edges meet. Being able to correctly identify and count these components is fundamental to describing polyhedra.

Euler's Formula for Polyhedra (F - E + V = 2)

For any convex polyhedron, there is a consistent relationship between the number of faces (F), edges (E), and vertices (V), given by Euler's Formula: F - E + V = 2. This formula is incredibly useful for verifying your counts for a given polyhedron, or for finding one of the values if the other two are known. For example, a triangular prism has 5 faces, 9 edges, and 6 vertices, so 5 - 9 + 6 = 2.

Sphere Formulas

A sphere is a perfectly round 3D shape. Its properties are defined solely by its radius, r. You need to know these two key formulas for a sphere:

1. Volume of a sphere: $V = \frac{4}{3}\pi r^3$
2. Surface area of a sphere: $A = 4\pi r^2$

Remember to identify the radius correctly from the problem. If the diameter is given, divide it by 2 to find the radius before using the formulas. Pay attention to units and ensure your final answer has the correct cubic units for volume and square units for area, e.g., cm3 or m2.

Cone Formulas

A cone has a circular base and tapers to a single point (apex). For a cone, you need to be aware of its radius r, its perpendicular height h (from the centre of the base to the apex), and its slant height l (from the apex to any point on the edge of the base).

Here are the important formulas for a cone:

1. Volume of a cone: $V = \frac{1}{3}\pi r^2 h$
2. Curved surface area of a cone: $A_{curved} = \pi r l$
3. Total surface area of a cone: $A_{total} = \pi r l + \pi r^2$ (curved area + base area)

Note that the height h, radius r, and slant height l form a right-angled triangle, so Pythagoras' theorem applies: $r^2 + h^2 = l^2$. You might need to use this to find a missing dimension before calculating volume or area.

Applying the Formulas: Step-by-Step

When solving problems involving cones and spheres, follow these steps:

1. Read the question carefully: Identify what shape(s) are involved and what quantity you need to calculate (volume, surface area, curved surface area, total surface area).
2. Identify given values: Extract the radius, diameter, height, or slant height provided in the problem. If a diameter is given, find the radius.
3. Choose the correct formula: Select the appropriate formula based on the shape and the required calculation. For cones, distinguish between curved surface area and total surface area.
4. Check for missing values: If a cone problem requires slant height for area but only height and radius are given, use Pythagoras' theorem ($r^2 + h^2 = l^2$) to find l. For volume of a cone, if slant height is given instead of perpendicular height, use Pythagoras' theorem to find h.
5. Substitute values and calculate: Plug the numbers into the formula and perform the calculation. You can use a calculator for $\pi$ or use 3.142 or $\frac{22}{7}$ if specified.
6. State units and round: Ensure your answer has the correct units (e.g., cm3 for volume, cm2 for area) and round to the specified degree of accuracy if required.

What is a Pyramid?

A pyramid is a 3D shape with a polygonal base and triangular faces that meet at a single point called the apex. The type of pyramid is named after the shape of its base (e.g., square-based pyramid, triangular-based pyramid). Key parts include the base, apex, lateral faces (triangles), edges, and vertices. The height is the perpendicular distance from the apex to the base's center. The slant height is the height of a triangular face.

Calculating the Volume of a Pyramid

The formula for the volume of any pyramid is given by: Volume = (1/3) × Base Area × Perpendicular Height.

First, identify the shape of the base and calculate its area. For a square base with side length 's', Base Area = s². For a rectangular base with length 'l' and width 'w', Base Area = l × w. For a triangular base, Base Area = (1/2) × base × height of triangle.

Next, find the perpendicular height (h) of the pyramid, which is the vertical distance from the apex to the centre of the base. Substitute these values into the volume formula. Remember your units for volume will be cubic (e.g., cm³).

Calculating the Surface Area of a Pyramid

To find the total surface area of a pyramid, you need to sum the area of its base and the areas of all its triangular lateral faces.

Total Surface Area = Area of Base + Sum of Areas of Lateral Faces.

Calculate the area of the base first, depending on its shape. Then, calculate the area of each triangular lateral face. The area of a triangle is (1/2) × base × height. For the lateral faces, the 'base' is an edge of the pyramid's base, and the 'height' is the slant height of the triangular face. If the base is a regular polygon, all lateral faces will be identical, so you can calculate the area of one face and multiply by the number of faces. Units for surface area are square units (e.g., cm²).

What is Pythagoras' Theorem?

Pythagoras' Theorem applies only to right-angled triangles. It describes the relationship between the lengths of the three sides. The theorem states: the square of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the other two sides (legs). Mathematically, this is written as a² + b² = c², where 'c' is always the hypotenuse, and 'a' and 'b' are the other two sides of the triangle.

Finding the Hypotenuse

If you need to find the length of the hypotenuse (c) when you know the lengths of the two shorter sides (a and b), you use the formula directly. First, square 'a' and square 'b'. Then, add these squared values together. Finally, take the square root of the sum to find 'c'. So, if a² + b² = c², then c = √(a² + b²). Always ensure your final answer is a positive length.

Finding a Shorter Side

If you need to find the length of one of the shorter sides (say, 'a') when you know the hypotenuse (c) and the other shorter side (b), you need to rearrange the formula. Start with a² + b² = c². To find 'a', subtract b² from both sides, giving a² = c² - b². Then, take the square root of the result: a = √(c² - b²). Remember that 'c' (the hypotenuse) is always the longest side, so its square will always be larger than the square of any other side.

Key Steps for Problem Solving

1. Identify if the triangle is right-angled. Pythagoras' Theorem only works for right-angled triangles. 2. Label the sides: 'c' for the hypotenuse (opposite the right angle), and 'a' and 'b' for the other two legs. 3. Write down the formula: a² + b² = c². 4. Substitute the known values into the formula. 5. Rearrange and solve for the unknown side. Make sure to take the square root as the final step to get the side length, not its square.