0580 · 5.2Area and perimeter

Area, perimeter and rectangles

0:00A rectangular garden needs a fence around its edge and grass across its surface. These jobs involve different measurements, even though they concern the same shape. The fence depends on perimeter, while the grass depends on area. By the end of this video, you will understand both measurements for rectangles, triangles, parallelograms and trapezia. Perimeter is the total distance around the boundary of a shape. Area measures the amount of surface inside that boundary.

0:24Perimeter uses length units, such as centimetres or metres. Area uses square units, such as square centimetres or square metres, because we are counting how many unit squares cover the surface. Before calculating, make sure all the lengths use the same unit. Also recognise a right angle, because perpendicular height means a distance measured at ninety degrees to the chosen base. A base is a side we choose as the starting point for an area calculation.

0:49It does not have to be the side at the bottom of the page. A rectangle has four right angles, and its opposite sides are equal. Imagine a rectangle eight centimetres long and five centimetres wide, covered with one centimetre squares. Each row contains eight squares, and there are five rows. 8 × 5 counts all forty squares without counting them individually. This explains why rectangle area equals length multiplied by width.

1:13For the perimeter, we count the four boundary lengths instead. There are two lengths and two widths, so rectangle perimeter equals two multiplied by the sum of length and width. Multiplication counts the surface, while addition measures the journey around its edge. For this rectangle, the area calculation is 8 × 5. That gives forty square centimetres. For perimeter, first add eight and five to obtain thirteen centimetres. Doubling thirteen counts both pairs of opposite sides.

1:41The perimeter is twenty-six centimetres. Area and perimeter are not interchangeable, and one does not determine the other. A rectangle ten centimetres long and four centimetres wide also has area forty square centimetres. Its perimeter is twenty-eight centimetres, rather than twenty-six. The same amount of surface can therefore have a different boundary length.

Parallelograms and triangles

2:01A parallelogram has two pairs of parallel opposite sides. Its sloping side and its perpendicular height are usually different lengths. In this diagram, the base is nine centimetres, the sloping side is five centimetres, and the height is four centimetres. The height measures straight across the gap between the parallel bases. Imagine cutting off the triangular piece at the left and moving it to the right. It fills the gap there, making a rectangle without adding or removing any surface.

2:27That rectangle has the same base and perpendicular height as the parallelogram. This is why parallelogram area equals base multiplied by perpendicular height. To calculate the area, multiply the base of nine by the perpendicular height of four. The result is thirty-six square centimetres. For perimeter, add nine and five to obtain fourteen centimetres. Double fourteen to get twenty-eight centimetres, since opposite sides of a parallelogram are equal. Notice that the height was used for area but not perimeter.

2:56The dashed height lies inside the shape, so it is not part of the boundary. Using the sloping side for area would measure the wrong distance between the parallel bases. Using the internal height for perimeter would count a line that we never travel along. A triangle has half the area of a parallelogram with the same base and perpendicular height. Imagine making an identical copy and rotating it to join the original along one side.

3:18The two triangles form a parallelogram, and each contributes half its area. Therefore, triangle area equals one half multiplied by base multiplied by perpendicular height. For the right-angled triangle shown, the base is eight centimetres and the height is six centimetres. The ten centimetre side slopes, so it is not perpendicular to our chosen base. It is still part of the boundary and must be included in the perimeter. The right-angle mark identifies which two lengths can serve as base and height together.

3:45Start the area calculation with ½ × 8 × 6. Half of eight is four, so this becomes 4 × 6. The area is twenty-four square centimetres. For perimeter, 8 + 6 gives fourteen, and adding ten gives twenty-four centimetres. The numerical answers happen to match, but they describe different measurements. Twenty-four centimetres measures the boundary, while twenty-four square centimetres measures the surface. For a triangle that is not right-angled, the height may need to be drawn separately.

4:16It can even meet an extension of the base outside the triangle.

Trapezium area and perimeter

4:19For our trapezium calculations, identify the pair of parallel sides first. The perpendicular height measures the distance between them. Here, the parallel sides are ten centimetres and four centimetres, and the height is four centimetres. Each sloping boundary side is five centimetres, but those lengths are not needed for area. Make an identical copy of the trapezium, rotate it, and join the shapes along a sloping side. Together, they form a parallelogram whose base is the sum of the original parallel sides.

4:47Its height is unchanged, and its area is twice the original area. Therefore, trapezium area equals one half multiplied by the sum of the parallel sides multiplied by the perpendicular height. First add the parallel sides, ten and four, to obtain fourteen centimetres. Half of fourteen is seven centimetres. Multiplying seven by the height of four gives twenty-eight square centimetres. For perimeter, the parallel sides total fourteen and the sloping sides total ten, giving twenty-four centimetres altogether.

5:17Another way to understand the formula is average width multiplied by height. Half the sum of the parallel sides gives their average length. The straight sloping edges mean the width changes steadily between those two lengths. However, the formula uses only the parallel pair, not any two sides you happen to choose.

Exam methods and common errors

5:35Cambridge questions can ask you to calculate an area or perimeter, or work backwards to find a missing length. Show the formula, substitute the measurements, and keep the units attached to the final answer. Except for triangle area, these formulas are not given, so you need to know them. Their explanations make them easier to reconstruct and check. Consider an exam-style question asking for the perpendicular height of this trapezium.

5:57Its area is forty-five square centimetres, and its parallel sides measure twelve and six centimetres. We use the trapezium formula because it connects those three known measurements with the missing height. The unknown height will be called h. Substitute the values to give 45 = ½ multiplied by the sum of twelve and 6 × h. Adding the parallel lengths gives eighteen, so the right-hand side becomes ½ × 18 × h.

6:23Half of eighteen is nine, leaving 45 = 9 × h. Divide forty-five by nine to obtain a height of five centimetres. These methods must work without a calculator on Paper Two, and also apply on calculator Paper Four. Simplifying before multiplying often makes the arithmetic easier, particularly when a formula contains one half. Our answers here are exact, so no rounding is needed. When rounding is required, normally give three significant figures unless instructed otherwise, and avoid rounding intermediate results.

6:55Common errors include forgetting the half in triangle area and using a sloping side instead of perpendicular height. Another mistake is counting an internal height as part of the perimeter. Check whether each length measures the boundary or the gap perpendicular to the base. Also check that the answer uses length units for perimeter and square units for area. Perimeter always comes from adding the boundary lengths. Area formulas count unit squares, sometimes after rearranging or duplicating a shape.

7:19Rectangles and parallelograms use base multiplied by perpendicular height, while triangles use half that amount. Trapezia use the average of the parallel side lengths multiplied by perpendicular height.