Spheres, Cones and Pyramids

Questions on these solids are short and do not come up often. You may be shown a solid, such as a cone, and asked to write down its name. You may be asked to count the faces or edges of a pyramid, or to draw the plan or front elevation of a pyramid. These are worth 1 or 2 marks each.

The longer questions, for about 4 marks, ask for a surface area: the total area of the faces of a pyramid, or the surface area of a sphere.

For a sphere, the formula you need is printed in the question: "Surface area of sphere = 4\pi r^2". It is not on the formulae sheet. Your job is to know what r stands for and to put the right number in. For a pyramid there is no formula: you add up the areas of the square and the triangles, so learn the area of a square and of a triangle.

On Paper 1, with no calculator, leave \pi in a sphere's surface area, such as 64\pi\text{ cm}^2. With a calculator, use the \pi key and round only at the end.

A pyramid has a flat base and triangular faces that slope up to a point at the top. A square-based pyramid has a square base and four triangles.

A cone has a circular base and a curved surface that rises to a point. A sphere is a ball shape: every point on its surface is the same distance from the centre. That distance is the radius, r.

Learn the three names. A question may show a solid, such as a cone, and ask you to write down its name, for 1 mark.

Give one name, the name of a solid. A cone is a "cone", not a "triangle cone" or a "cylinder cone". A sphere is a solid, so it is not a circle.

A face is a flat surface of a solid. An edge is a line where two faces meet. The corners, where edges meet, are called vertices.

A square-based pyramid has 5 faces: the square base and 4 triangles. It has 8 edges: 4 round the base and 4 that slope up to the top.

In a drawing, the edges you cannot see are drawn dashed. Count them too. Go round the base first, then count the edges that go up to the top.

The plan is the view of a solid from above. The front elevation is the view from the front. Each one is a flat 2-D shape: not a 3-D drawing, and not the faces laid out together.

Here are the plan and front elevation of a square-based pyramid.

From above, the four sloping edges run from the corners of the square to the top, which is over the middle. So the plan is a square with both diagonals drawn. A square on its own is only half right.

From the front you see a triangle. Its base is the side of the square. Its height is the vertical height of the pyramid: the height straight up from the middle of the base to the top, not the length up a sloping face.

The surface area of a solid is the total area of all its outside surfaces. It is measured in square units, such as \text{cm}^2

For a pyramid, add up the areas of its faces. A square-based pyramid has a square base and four identical triangles.

Each triangle has area \frac{1}{2} \times 8 \times 11 = 44\text{ cm}^2, so the four triangles make 4 \times 44 = 176\text{ cm}^2. Add the square base, 8 \times 8 = 64\text{ cm}^2. The total surface area is 176 + 64 = 240\text{ cm}^2

The height of each triangle is measured up the middle of its sloping face. It is longer than the vertical height of the pyramid. Use the height of the triangle for the surface area. The vertical height is for the front elevation.

The formula is printed in the question:

\text{Surface area of sphere} = 4\pi r^2

where r is the radius of the sphere.

4\pi r^2 means 4 \times \pi \times r \times r. Square the radius first, then multiply by 4. A sphere of radius 2 cm has surface area 4 \times \pi \times 2^2 = 4 \times 4 \times \pi = 16\pi\text{ cm}^2

The formula uses the radius. If you are given the diameter, halve it first.

If the question says "in terms of \pi", leave \pi in the answer, such as 16\pi. Do not put in 3.14. If you can use a calculator and the answer is to be rounded, use the \pi key and round only at the end.

• Giving two names, or the name of a flat shape. Write one name of a solid: "cone", not "triangle cone"; a sphere is not a circle.
• Not counting the hidden edges. Count the dashed edges as well as the ones you can see.
• Drawing a 3-D picture for a plan or elevation. Draw a flat shape. The plan of a square-based pyramid is a square with both diagonals.
• Using the vertical height for the triangular faces. Each triangle's area uses the height of the triangle, measured up the sloping face.
• Using the diameter instead of the radius. Halve the diameter before you put it into 4\pi r^2
• Putting in a number for \pi when the answer must be in terms of \pi. Keep \pi in the answer, such as 196\pi