Year 10 Standard · Unit 5 Solids, similarity and transformations · Lesson 1 of 6

Which height does the surface of a cone or pyramid use?

A cone and a pyramid each hide a right-angled triangle. This lesson finds it, and asks which of its sides the surface area needs.

Geometry and TrigonometryGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: a right-angled triangle has short sides 7 cm and 24 cm. How long is the hypotenuse?
    Answer25 cm, because 49 + 576 = 625 and √625 = 25.
  2. From Year 9: find the area of a circle of radius 4 cm, leaving π in the answer.
    Answer16π cm², because π × 4² = 16π.
  3. From Year 7: find the area of a triangle with base 10 cm and height 7 cm.
    Answer35 cm². The area is half of base × height: ½ × 10 × 7 = 35.
Words used in this lesson
slant height
the length of the sloping side from the apex to the edge of the base. For example: the sloping edge of a roof, from the ridge to the gutter.
vertical height
the distance straight up from the base to the apex. For example: h in the cone.
curved surface
the part of a cone's surface that is not its base. For example: πrs.
total surface area
the area of every face, including the base. For example: curved surface plus base.
apex
the point at the top of a cone or pyramid. For example: the tip of the hat.
hypotenuse
the longest side of a right-angled triangle, opposite the right angle. For example: the side of length 5 in a 3, 4, 5 triangle.

A hat from Hanoi

A craftsperson in Hanoi weaves a conical hat with base radius 12 cm and vertical height 16 cm. She wants to know how long a strip of straw must be to run from the tip straight down to the rim, along the outside of the hat.

Predict first: is that length 28 cm (12 + 16), 16 cm (the height), or something else? Write down a reason, then use the first figure to see which length belongs to the right-angled triangle.

2
5
√29 ≈ 5.39slant height s
46.40total surface area, base included

Radius 2, height 5: s² = 2² + 5² = 29, so s = √29 ≈ 5.39. Curved surface πrs ≈ 33.84, base πr² ≈ 12.57, total ≈ 46.40.

Show me the steps
  1. Radius 2, vertical height 5: the right-angled triangle gives s² = 2² + 5² = 29, so s = √29 ≈ 5.39. The slant height is longer than the vertical height 5.
  2. Make the radius 1 and the height 1: s = √2 ≈ 1.41. The slant side is the longest side of the triangle, whatever the radius.
  3. Radius 3 and height 4: s² = 25 and s = 5 exactly. The curved surface is π × 3 × 5 = 15π.
  4. Radius 5 and height 7: s = √74 ≈ 8.60. The total surface area is the base 25π plus the curved surface 5π√74.

The slant height s is the hypotenuse of a triangle inside the solid. For a cone, s = √(r² + h²), where h is the vertical height. The curved surface is πrs, the base is πr², and a closed cone has total surface area πr² + πrs.

A square-based pyramid has four triangular faces. Each face has height l = √(h² + (b/2)²), because the foot of l is the middle of a base edge. Total surface area = b² + 4 × (½bl) = b² + 2bl.

4
3
√13 ≈ 3.61slant height l of a face
44.84total surface area, base included

Base edge 4, height 3: l² = 3² + 2² = 13, so l = √13 ≈ 3.61. One face ≈ 7.21, four faces ≈ 28.84, base 16, total ≈ 44.84.

Show me the steps
  1. Base edge 4, vertical height 3: l² = 3² + 2² = 13, so l = √13 ≈ 3.61. One face is ½ × 4 × 3.61 ≈ 7.21; four faces ≈ 28.84; with the base 16 the total is ≈ 44.84.
  2. Make the base edge 8 and keep the height 3: the half edge is 4 and l = √(9 + 16) = 5 exactly. Four faces are 80 and the total is 64 + 80 = 144.
  3. Make the base edge 2 and the height 7: l² = 49 + 1 = 50, so l ≈ 7.07, only a little longer than the height 7.
  4. The slant height of a face is always longer than the vertical height, so using h in the face area always gives too small an answer.

Watch the two heights. The vertical height h is the one you measure straight up from the base. The surface needs the slant height. If you write πrh for the curved surface of a cone, you have used the shorter length and the area comes out too small.

Worked examples

1. A cone has base radius 15 cm and vertical height 20 cm. Find its total surface area in terms of π.

  1. s² = 15² + 20² = 225 + 400 = 625, so s = 25 cm.
  2. Curved surface = π × 15 × 25 = 375π. Base = π × 15² = 225π.
  3. Total = 375π + 225π = 600π cm².

2. A square-based pyramid has base edge 24 cm and vertical height 5 cm. Find its total surface area.

  1. Half the base edge is 12, so l² = 5² + 12² = 169 and l = 13 cm.
  2. Each face is ½ × 24 × 13 = 156, so four faces are 624. The base is 24² = 576.
  3. Total = 624 + 576 = 1200 cm².

3. A cone has radius 3 cm and vertical height 5 cm. Find its curved surface area to 1 decimal place.

  1. s² = 9 + 25 = 34, so s = √34. Keep it as a root.
  2. Curved surface = π × 3 × √34 = 55.0 cm² (to 1 d.p.).

Your go. The first step is given. A square-based pyramid has base edge 14 cm and vertical height 24 cm. Find its total surface area.

  1. Half the base edge is 7, so l² = 24² + 7². Work out l.
  2. Four faces: 2bl. Then add the base b².
Check the answer

l² = 576 + 49 = 625, so l = 25 cm. Four faces = 2 × 14 × 25 = 700, base = 196, total = 896 cm². Using h = 24 for the faces would give 2 × 14 × 24 + 196 = 868, which is too small.

With a GDC (optional)

A GDC is useful for the last step. Keep the slant height as a root, such as √29, until the end, then enter the whole expression once, with brackets, so that nothing is rounded in the middle. Many calculators have a π key; use it instead of 3.14.

Check the size by hand first, with π ≈ 3: for r = 2 and s ≈ 5.4, the curved surface is about 3 × 2 × 5 = 30. Everything here can be done by hand with π left in the answer.

Check before you go on

1. A cone has base radius 8 cm and vertical height 15 cm. What is its slant height?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A cone has base radius 6 cm and vertical height 8 cm. Its curved surface area is ?π cm². What number goes in front of π?

3. A square-based pyramid has base edge 10 cm and vertical height 12 cm. How long is the slant height of one triangular face, in cm?

Level 2 · Building

4. For the same pyramid (base edge 10 cm, slant height 13 cm), what is the total surface area, in cm²?

5. A conical tent has base radius 4 m and slant height 9 m. The canvas covers the curved surface only. How much canvas, in m², to 1 decimal place?

6. A square-based pyramid has base edge 16 m and the slant height of each face is 17 m. What is its vertical height, in m?

Level 3 · Stretch

7. A cone has base radius 7 cm and total surface area 105π cm². What is its slant height, in cm?

8. A paper party hat is a cone with base radius 10 cm and vertical height 24 cm, open at the bottom. How much paper makes its curved surface, in cm², to the nearest whole number?

Something to think about

A cone has radius 4 and vertical height 3, so its slant height is 5. If the height doubles to 6 with the same radius, does the slant height double too? Test it with the first figure.

One way to think about it

No. With radius 4 and height 6, s = √52 ≈ 7.21, which is not 10. Doubling the height adds 3 to the vertical leg but only about 2.2 to the slant height, because the slant height is a hypotenuse. The taller the cone, the closer s gets to h.

Exit ticket

9. A closed cone has base radius 9 cm and vertical height 12 cm. Which is its total surface area?