Geometry

Area with sine

Open a triangle. Follow its area.

Predict, test and explain

Step 1 of 3

Predict

With a = 4 and b = 6, do included angles 30° and 150° give the same area?

K=12absin⁡C≈15.16K = \frac12ab\sin C \approx 15.16
A≈43.9∘A \approx 43.9^\circ
B≈76.1∘B \approx 76.1^\circ
C=60∘C = 60^\circ
a=5a = 5
b=7b = 7
c≈6.24c \approx 6.24

Scroll the diagram sideways to see every label.

area≈15.16\text{area} \approx 15.16h=asin⁡C≈4.33h=a\sin C \approx 4.33

InvestigateWhich two angles give the same area?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

A triangle has a base of 8 cm and a perpendicular height of 5 cm. Find its area.

Hint

A triangle uses half the area of a rectangle with the same base and height.

Reveal answer

The area is 20 square centimetres.

A=12×8×5=20 cm2A=\frac12\times8\times5=20\,\text{cm}^2

Build an explanation

Hint 1

Use the angle between the two given sides.

Hint 2

Reveal the perpendicular height and connect it to sine.

Hint 3

The area is half × first side × second side × sine of the included angle.

Worked example

This example uses fixed values, separate from the diagram controls.

Two sides are 5 cm and 7 cm, with included angle 60°. Find the area.

  1. Use area = ½ × 5 × 7 × sin 60°.
  2. Calculate 17.5 × sin 60°.

Answer: Approximately 15.16 cm².

Connect this idea

Watch out: Use the included angle and square units; a sloping side is not the perpendicular height.