Geometry and measures, Year 10: plans, area and volume formulae

The plan of a solid

HintArchitects use the same word for a drawing of a building's floor layout.

The whyThe front elevation is the view from the front and the side elevation is the view from one side. Together the three views describe a 3D shape on flat paper.

An elevation, which is a rectangle for the cylinder

HintWalk round to look at each solid from the front, at eye level.

The whyFrom the front or the side, the upright cylinder looks like a rectangle, while the sphere still looks like a circle. One view alone rarely fixes a solid.

A rectangle 5 cm wide and 2 cm high

HintFrom the front you cannot see how far back the solid goes.

The whyAn elevation shows width and height only. Looking at a 5 cm side, the width seen is 5 cm and the height is 2 cm; the 3 cm depth points away from you and does not appear.

3

HintFrom overhead, one cube hides another.

The whyFrom above, the stacked cube sits exactly over the middle cube, so the plan shows one row of three squares. The front elevation, looking at the row side-on, shows all four: three in a row with one above the middle.

A cone

HintIt stands on a round base and narrows to a point.

The whyThe round base gives the circular plan. From any side the outline is two slanting edges meeting at the top, which is a triangle.

A = 1/2 × b × h

HintA triangle is half of the parallelogram with the same base and height.

The whyTwo copies of any triangle fit together to make a parallelogram of area b × h, so one triangle is half of that. AQA expects this formula to be known; it is not given in the exam.

32 cm²

HintOne of the three measurements is not needed: choose the one at right angles to the base.

The whyArea of a parallelogram = base × perpendicular height = 8 × 4 = 32 cm². Cutting a triangle off one end and moving it to the other turns the parallelogram into an 8 cm by 4 cm rectangle.

8; 40

HintAverage the two parallel sides first, then multiply by the distance between them.

The whyArea of a trapezium = 1/2 × (a + b) × h. Here 1/2 × (6 + 10) = 8, and 8 × 5 = 40 cm². The formula treats the trapezium as a rectangle whose width is the average of the parallel sides. AQA expects it to be known or derived; it is not given.

Volume = area of cross-section × length

HintThink of the solid as one flat shape pushed through a distance.

The whyA prism has the same cross-section all the way along, so its volume is that area multiplied by how far it extends. A cuboid and a cylinder are both special cases. AQA expects this formula to be known or derived; it is not given.

60

HintFind the area of the triangular end first, remembering the half.

The whyCross-section area = 1/2 × 3 × 4 = 6 cm². Volume = 6 × 10 = 60 cm³.

C = 2πr

HintThe diameter version is πd, and a diameter is two radii.

The whySince d = 2r, πd and 2πr are the same formula. AQA expects it to be known; it is not given in the exam.

36π cm²

HintSquare the radius and leave the symbol alone, as you would a letter in algebra.

The whyArea = πr² = π × 6² = 36π cm². Leaving π in the answer keeps it exact; 113.1 cm² is only a rounded value.

πr² — an area multiplies two lengths, so r must be squared

HintThink about the units of each expression when r is in centimetres.

The why2πr contains one length, so it is measured in cm and must be the circumference. πr² contains r × r, which gives cm², so it is the area.

7

HintSet the formula for the distance round equal to the given value and cancel what matches.

The whyC = 2πr, so 2πr = 14π. Dividing both sides by 2π gives r = 7 cm.