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.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Identify spherical sides as great-circle arcs.
- Explain three right angles.
- Use spherical excess in radians.
Before you start
Degrees, radians, the area 4πR² of a sphere, and the plane triangle angle sum.
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
On a radius-2 sphere, join the North Pole to two equatorial points 90° apart in longitude using shorter great-circle arcs. Close the triangle along the equator. Find its angle sum and area.
Why this math matters
A sphere has different intrinsic geometry from a flat map. The example shows exactly which assumption changes when a familiar triangle rule fails.
Set up the model
A useful answer starts with clear assumptions:
- The surface is an ideal sphere, not a flattened Earth model.
- The two equatorial longitudes differ by exactly 90°.
- The triangle uses the indicated shorter arcs and its smaller enclosed region.
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.
Build a triangle with three right angles on a sphere
PausedQuestion: Start with the question. Paused.
Question
Start with the question
On a radius-2 sphere, join the North Pole to two equatorial points 90° apart in longitude using shorter great-circle arcs. Close the triangle along the equator. Find its angle sum and 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.
Measure the corner angles
90° + 90° + 90° = 270°
Each meridian meets the equator at a right angle. At the pole, the meridians meet at their 90° longitude difference.
Calculate excess
E = 270° − 180° = 90° = π/2 radians
Spherical excess measures how much the angle sum exceeds the plane total. Convert it before using the area formula.
Find and check area
A = R²E = 4(π/2) = 2π square units
The whole sphere has area 16π. This triangle is one of eight congruent regions made by three perpendicular great circles, confirming the fraction 1/8.
The result
The triangle has angle sum 270° and area 2π square units.
The 180° rule remains correct for Euclidean plane triangles. Here the sides are curved in space but locally straight within the sphere's own surface geometry.
Common mistakes to catch
- Arbitrary latitude arcs are not great-circle sides.
- Multiplying R² by an excess measured in degrees gives an incorrect area.
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
What area does the same angular triangle have on a radius-3 sphere?
Show a hint
Keep E = π/2.
Reveal answer and explanation
9π/2 square units
Area scales as R², so A = 3²π/2.
Practice 2
A shorter-arc spherical triangle has angles 80°, 70°, and 60°. Find its excess.
Show a hint
Subtract 180° from their sum.
Reveal answer and explanation
30°, or π/6 radians
The sum is 210°; the excess is 30°, which equals π/6.
Take the idea with you
Name the geometric surface before applying a theorem learned for a flat plane.
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.
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