0580 · 4.2Geometrical constructions

Measuring lengths and angles

0:00A flat piece of card can become a box, but only if its faces fit together correctly. Similarly, three given lengths can determine a triangle without any angle measurements. In this video, you will understand accurate measurement, ruler and compass constructions, and nets. You will also use dimensions from nets to calculate surface area and volume. You will need a ruler, a protractor and a pair of compasses. A line segment is a straight part of a line between two endpoints.

0:27An angle measures the turn between two rays meeting at a vertex. Use a ruler for every straight edge, including the outlines of nets. To measure a segment, normally place the ruler’s zero mark at one endpoint. Read the scale directly above the other endpoint, looking straight down. Here, the endpoints lie at one point two and seven point eight centimetres. Subtract one point two from seven point eight to obtain six point six centimetres.

0:53The ruler’s physical end is not necessarily its zero mark. Starting at the wrong place adds an unwanted length to your measurement. To draw a segment measuring six centimetres, mark points at zero and six centimetres. Join those points with your ruler rather than estimating the length by eye. For an angle, place the protractor’s centre exactly on the vertex. Align its baseline with one ray, then read where the other ray crosses the scale.

1:18Choose the scale that starts at zero along your starting ray. Before accepting the reading, check whether the diagram shows an acute or obtuse angle. To draw an angle of sixty-five degrees, first draw the ray from P to Q. Centre the protractor at P and align its baseline with that ray. Mark sixty-five degrees on the correct scale. Remove the protractor and use a ruler to draw the ray from P through your mark.

1:42The two protractor scales can give sixty-five degrees and one hundred and fifteen degrees at the same position. Only one measures the turn from your chosen starting ray. An acute angle must be less than ninety degrees, so this provides a useful check. Good alignment matters just as much as choosing the correct scale.

Compass constructions

1:59Constructing a triangle from three side lengths uses a different idea. A compass keeps its pencil a fixed distance from its sharp point. Every point on the circle it draws therefore has the same distance from the centre. An arc is part of that circle, and can locate a point satisfying a length requirement. Consider an exam-style instruction to construct triangle PQR with sides six centimetres, five centimetres and four centimetres.

2:24Use a ruler and pair of compasses only. First, draw the line from P to Q measuring six centimetres. This fixes two vertices and gives us a base for locating the third. Set the compass opening to five centimetres using your ruler. With its point at P, draw an arc above the base. Change the opening to four centimetres and draw another arc centred at Q. Label their intersection R, then join P to R and Q to R using your ruler.

2:49The first arc guarantees that R is five centimetres from P. The second guarantees that R is four centimetres from Q. Their intersection satisfies both conditions at once, so no angle measurement is needed. Leave the construction arcs visible because they show how you located the vertex. There is also a matching intersection below the base, producing a reflected triangle with the same side lengths. Usually, either position works unless the question specifies one.

3:17Some sets of lengths cannot form a triangle because the arcs do not intersect. The two shorter sides must add to more than the longest side. The same method can construct a rhombus by drawing two triangles. Begin with a six centimetre diagonal from P to Q. Draw arcs of radius five centimetres from both endpoints, locating intersections R above and S below. Join P to R, R to Q, Q to S and S to P.

3:42Each boundary side measures five centimetres because each was fixed by a compass radius. The resulting quadrilateral is a rhombus because all four sides are equal. Its diagonal divides it into two constructed triangles. Guessing the vertices or measuring angles instead would not demonstrate the required ruler and compass method.

Nets, surface area and volume

4:01A net is a flat arrangement of faces that can fold to form a solid. The faces remain the same size when folded. Edges that meet must have matching lengths, and faces must not overlap in the completed solid. Imagine folding each face around its shared edge rather than judging only the flat outline. For a cube net, draw four equal squares in a row. Add one square above the second square and another below it.

4:25The row folds around the sides, while the extra squares close the top and bottom. Six squares alone are not enough to guarantee a valid net because their arrangement also matters. This exam-style net represents a cuboid measuring five centimetres by three centimetres by two centimetres. Its four side faces form a strip, with a common height of two centimetres. The remaining rectangles close the top and bottom. Opposite faces have matching dimensions, giving three pairs of equal rectangles.

4:54First, calculate the base area as 5 × 3, giving fifteen square centimetres. Multiply fifteen by the height of two to obtain a volume of thirty cubic centimetres. Volume measures the space inside the folded solid. The flat net itself has area, not volume. For surface area, the three different face areas are fifteen, ten and six square centimetres. There are two faces of each size. Doubling those areas gives thirty, twenty and twelve square centimetres.

5:23Adding thirty, twenty and twelve gives a total surface area of sixty-two square centimetres. A prism has the same cross-section throughout its length. This triangular prism has two matching triangular ends and three rectangular side faces. To draw its net, make rectangles six centimetres long with widths three, four and five centimetres. Attach the triangular ends along opposite three centimetre edges of the first rectangle. Put each right angle at the corner next to the four centimetre rectangle, so matching edges meet.

5:55The triangular ends are right-angled, with perpendicular sides of three and four centimetres. Half multiplied by 3 × 4 gives an area of six square centimetres. Multiply that cross-sectional area by the prism length of six centimetres. The volume is therefore thirty-six cubic centimetres. The three rectangles have areas eighteen, twenty-four and thirty square centimetres. Adding these gives seventy-two square centimetres. The two triangular ends contribute twice six, giving twelve square centimetres.

6:26Adding seventy-two and twelve gives a total surface area of eighty-four square centimetres. A square-based pyramid net has a square base with four triangular faces attached around it. Draw the square first, then place one triangle along each edge. In this example, each triangle has base four centimetres and perpendicular height three centimetres. When folded, the four outer vertices meet at the pyramid’s apex. The triangle height shown on this net lies along a sloping face.

6:55It is not the perpendicular height of the completed pyramid. For surface area, the square base gives sixteen square centimetres and each triangle gives six. The total is 16 + 24, which is forty square centimetres. Do not use a sloping face height as the solid’s vertical height when calculating volume.

Exam checks and recap

7:13In an examination, follow the command word carefully: measure, draw, construct or calculate. Construction arcs provide evidence of your method, so do not erase them. Check nets for missing faces, mismatched edges and faces that would overlap. For surface area, count every face exactly once and use square units. These drawing and construction methods must work without a calculator. Paper two is non-calculator, while Paper four allows a calculator. Our example answers are exact, so rounding is unnecessary.

7:44When calculations produce longer decimals, keep intermediate values and give three significant figures unless the question instructs otherwise. Accurate measurement depends on aligning your tools, not estimating by eye. Triangle constructions work because intersecting arcs enforce two distances simultaneously. Nets connect flat faces to solids, revealing both matching edges and surface areas. Volume comes from the folded shape, so identify its dimensions before calculating.