Mensuration and trigonometry Revision Notes

Core method

Perimeter is the total distance around the outside boundary of a shape. For a composite shape, trace the boundary once, include every outside edge, calculate any missing lengths from aligned totals, and exclude internal joining lines. All lengths must use the same unit. The essential approach is to identify the mathematical structure before calculating, retain exact values where possible, and show enough working for each conclusion to be checked independently.

Worked reasoning and checks

On an L-shape, a missing horizontal step can be found by subtracting known horizontal parts from the total width; the same idea applies vertically. Then add only the exterior segments. A reliable check is that total rightward length equals total leftward length, and total upward length equals total downward length. Check the final result against the original information, including signs, units, scale and whether the requested form or degree of accuracy has been supplied.

The Formula for Area

The area of a triangle is calculated using the formula: Area = (1/2) × base × height. This formula is fundamental for finding the space a triangle occupies. It comes from the idea that any triangle can be seen as half of a parallelogram or rectangle. 'Base' refers to any side of the triangle, and 'height' refers to the perpendicular distance from that base to the opposite vertex. Always ensure the units of base and height are consistent before calculation.

Identifying Base and Perpendicular Height

To use the formula correctly, you must identify the base and its corresponding perpendicular height. The 'base' can be any side of the triangle. The 'height' is the straight line distance from the chosen base to the opposite vertex, measured at a 90-degree angle to the base. It's crucial that these two measurements are perpendicular. For right-angled triangles, two of the sides (the legs) can serve as base and height. For other triangles, you may need to drop a perpendicular line from a vertex to the opposite side (or an extension of it) to find the height.

Perpendicularity is Key

The most common mistake when calculating the area of a triangle is using a side length instead of the perpendicular height, or using a height that isn't perpendicular to the chosen base. Always look for the right angle symbol or be prepared to identify the line that is exactly perpendicular to the base. If the height is drawn outside the triangle, you still use the original length of the base side in your calculation, not the extended length.

Units for Area

When calculating area, the units will always be square units. For example, if the base is in centimetres (cm) and the height is in centimetres (cm), the area will be in square centimetres (cm²). If the base is in metres (m) and the height is in metres (m), the area will be in square metres (m²). It is important to include the correct units in your final answer.

The Sine Rule for Area of a Triangle

The area of any triangle can be calculated using the formula: Area = ½ab sin C. In this formula, 'a' and 'b' represent the lengths of two sides of the triangle, and 'C' represents the size of the angle included between those two sides. It is crucial that the angle used is the one between the two sides whose lengths are known. If you use a different angle, the formula will give an incorrect result. For example, if you have sides 'x' and 'y' and the angle 'Z' between them, the area would be ½xy sin Z.

When to Use This Formula

This formula is particularly useful when the perpendicular height of the triangle is not known or cannot be easily calculated. It is often applied in contexts involving non-right-angled triangles where side lengths and angles are given. You should primarily consider using this method when you are given two side lengths and the angle between them (SAS - Side-Angle-Side). If you are given different combinations (e.g., ASA or SSS), you may first need to use the Sine Rule or Cosine Rule to find the necessary side lengths and the included angle.

Steps for Calculation

1. Identify two known side lengths (e.g., 'a' and 'b').
2. Identify the angle included between these two sides (e.g., 'C').
3. Substitute these values into the formula: Area = ½ab sin C.
4. Calculate the sine of the angle 'C'.
5. Multiply all the values together. Ensure your calculator is in degree mode if the angle is given in degrees. The result will be in square units (e.g., cm², m²).

Example Application

Consider a triangle with sides measuring 7 cm and 10 cm, and the angle between these sides is 60°. Using the formula: Area = ½ × 7 × 10 × sin(60°). Area = ½ × 70 × (√3 / 2) Area = 35 × (√3 / 2) Area = 17.5√3 cm² or approximately 30.3 cm² (to 3 significant figures). This demonstrates how to directly apply the formula when the required information is provided.

What is a Parallelogram?

A parallelogram is a quadrilateral (a four-sided polygon) where opposite sides are parallel and equal in length. Key features include two pairs of parallel sides, and opposite angles are equal. Adjacent angles sum to 180 degrees. It looks like a 'slanted' rectangle.

Identifying Base and Perpendicular Height

To calculate the area, you need to identify the 'base' and the 'perpendicular height'. The base can be any side of the parallelogram. The perpendicular height is the shortest distance between that base and its opposite side, measured at a 90-degree angle. It is NOT the length of the slanted side.

Formula for Area of a Parallelogram

The area (A) of a parallelogram is calculated using the formula: Area = base × perpendicular height. This can be written as A = b × h, where 'b' is the base and 'h' is the perpendicular height. The units for area will be square units, such as cm² or m².

Visualising the Transformation

Think of a parallelogram as a rectangle. You can cut a right-angled triangle from one end of the parallelogram and move it to the other end. This forms a rectangle with the same base and the same perpendicular height as the original parallelogram, reinforcing why the area formula is base × height.