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

The plan of a solid

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

为什么The 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

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

为什么From 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

提示From the front you cannot see how far back the solid goes.

为什么An 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

提示From overhead, one cube hides another.

为什么From 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

提示It stands on a round base and narrows to a point.

为什么The 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

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

为什么Two 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

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

为什么Area 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

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

为什么Area 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

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

为什么A 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

提示Find the area of the triangular end first, remembering the half.

为什么Cross-section area = 1/2 × 3 × 4 = 6 cm². Volume = 6 × 10 = 60 cm³.

C = 2πr

提示The diameter version is πd, and a diameter is two radii.

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

36π cm²

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

为什么Area = 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

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

为什么2πr 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

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

为什么C = 2πr, so 2πr = 14π. Dividing both sides by 2π gives r = 7 cm.