Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 10 chapters and video availability01 · Read and understand
What you will learn
- Express a triangle's height using sine of the included angle.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Algebraic equations, ratios, angle and area facts, and basic coordinate geometry.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A triangle has sides 8 cm and 10 cm enclosing a 30° angle. Find its area.
Why this math matters
Calculate the area of a triangular survey model from two sides and their enclosed angle. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use Euclidean geometry and the angle, parallelism, similarity, or congruence conditions stated; a sketch alone does not establish them.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Find area without a stated altitude
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A triangle has sides 8 cm and 10 cm enclosing a 30° angle. Find its area.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
Using base 10, height = 8 sin 30° = 4 cm
Project the other side onto the direction perpendicular to the base.
Develop the calculation
A=1/2 × base × height
The familiar triangle formula still applies.
Check and interpret
A=1/2 × 10 × 4 = 20 cm²
Equivalently A=1/2 ab sin C when C is the included angle.
The result
A=1/2 × 10 × 4 = 20 cm²
Equivalently A=1/2 ab sin C when C is the included angle.
Common mistakes to catch
- Use the angle between the two given sides.
- Lengths must share a unit before an area formula is applied.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find area for sides 6 and 9 with included angle 90°.
Show a hint
The sine factor is one.
Reveal answer and explanation
27 square units
(1/2)(6)(9)=27.
Practice 2
For fixed side lengths, which included angle maximizes area?
Show a hint
Sine's maximum for a triangle angle is one.
Reveal answer and explanation
90°
A right angle makes the perpendicular height as large as the fixed side allows.
Take the idea with you
Calculate the area of a triangular survey model from two sides and their enclosed angle.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Separate length, area, and volume scale factors
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