Cogito
Trigonometry · Chapter 7 · Lesson 3
Area of Any Triangle
When you cannot see the height.
12 problems · about 21 minutes · G-SRT.D.9, G-SRT.D.11
What this lesson teaches
The student computes triangle area using ½ab sin C and Heron’s formula.
- Area = ½ab sin C, where C is the angle between the two sides.
- With three sides and no angle, use Heron’s formula.
- A right angle makes sin C equal 1, recovering the familiar ½ × base × height.
Warm Up
Straightforward practice. Get the method working first.
5 problemsSides 10 and 4, included angle 30°. What is the area?
Answer 10
Why 10.
The angle in ½ab sin C must be which one?
Answer The angle between the two sides.
Why The included angle.
Sides 8 and 10, included angle 30°. Area?
Answer 20
Why ½ × 8 × 10 × 0.5.
Sides 6 and 6, included angle 90°. Area?
Answer 18
Why sin 90° = 1.
Sides 3, 4, 5 with s = 6. What is the Heron area?
Answer 6
Why √(6 × 3 × 2 × 1).
Build It Up
The same ideas with more to keep track of.
3 problemsSides 5, 5, 6. What is s?
Answer 8
Why Half of 16.
Sides 5, 5, 6 with s = 8. What is the area?
Answer 12
Why √(8 × 3 × 3 × 2).
Sides 20 and 14, included angle 90°. Area?
Answer 140
Why ½ × 20 × 14.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each situation by which formula fits.
Answer Use ½ab sin C: Two sides and the angle between them, Sides 7 and 9 with a 40° angle between · Use Heron: All three sides, no angles, Sides 6, 7 and 8
Why No angle given means no sine to use.
Sides 12 and 5, included angle 90°. Area?
Answer 30
Why ½ × 12 × 5.
The Garden: Sides 10 and 12 with an included angle of 30°. What is the area?
Answer 30
Why 30.
The Three Sides: A triangle has sides 3, 4 and 5. What is s in Heron’s formula?
Answer 6
Why 6.