Checkpoint: areas without heights

A circle of radius 7 has an area of 154. What does a 180° sector hold?

• 77 square units, since half of 154 is 77
• 22 square units, since half of 44 is 22
• 44 square units, since that is the arc's length

A triangular plot has sides 13, 14 and 15. No height is given. What is its area?

• 84 square units
• It cannot be found without a height
• 91 square units, from 13 × 14 ÷ 2

A cyclic quadrilateral has sides 3, 4, 5, 6. What does the rule give for sides 3, 4, 5, 0?

• 6, which is Heron's answer for the 3, 4, 5 triangle
• 18.97, the same as before
• 0, a side has vanished

A rectangle 9 by 4 becomes a square by the drawing. What is true of the result?

• The square's side is 6, but its area is smaller
• The square's side is 6, and both shapes hold 36 square units
• The square's side is 13, the whole baseline