Volumes with Cross Sections: Triangles and Semicircles
Use the specified triangle, semicircle, or circular geometry, distinguishing a supplied radius from a diameter.
College Board Topic 8.8 · CED p. 154 (PDF p. 159) · CED effective Fall 2020; project course outline 2026–27.
Keep V=∫A; change only the geometry of a slice. A triangle has area ½bh, so the problem must tell you the relationship between its base and height. An equilateral triangle with side s has area √3·s²/4. A semicircle with diameter s has radius s/2 and area ½π(s/2)²=πs²/8.
Other specified shapes work the same way: a circle of radius r has A=πr², or the problem may supply A(x) directly. A funnel with circular sections is a cross-section problem even when it is not described as a rotation. Distinguish a base segment described as a diameter from one described as a radius; that factor of 2 changes area by a factor of 4. State the slice direction and keep matching bounds.
Let's work through it
- A base segment has length s=2−x on 0≤x≤2, and semicircular slices use it as diameter. Reason: The geometry fixes radius s/2.
- Set A(x)=π(2−x)²/8. Reason: A semicircle is half a circle, with the radius squared.
- V=(π/8)∫₀²(2−x)²dx=π/3. Reason: Integrate the slice area over its extent.
Try it from memory
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Question 1
If a semicircle’s diameter is 4, what is its area?
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Retrieval 1
If a semicircle’s diameter is 4, what is its area?
Worked answer and reason
2π.
Radius is 2, so half-circle area is ½π·2².
Retrieval 2
A triangle has base s and height 2s. What is its area?
Worked answer and reason
s².
Use ½·s·2s.
Catch the tempting mistake
- All triangular cross sections have the same area formula in s.
- The relation of height to the stated base matters. Equilateral, isosceles, and right triangles require their specified geometry.
Calculator workspace and limits
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Put it to work
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Explain and apply the mathematics
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Build a triangular solid from a curved base
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A diameter makes a semicircular volume
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For the teacher
- Pair triangle and semicircle problems with the same base region. Ask students to identify the geometric coefficient before integration. Include one directly supplied A(x) to emphasize the general slice principle.
Glossary and deeper reading · 3 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Cross section
A cross section is the planar slice of a solid made by a specified cutting plane. For volume by integration, its area must be expressed as a function of the position along the slicing direction. For example, a square slice with side s(x) has area s(x)². Integrating those areas over the allowed positions yields volume under the method’s assumptions.
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Integral
Integration can mean accumulating a function over an interval or finding an antiderivative. A definite integral gives a number, while an indefinite integral represents a family of antiderivatives. For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4. The Fundamental Theorem of Calculus connects these two operations under its hypotheses.
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Definite integral
A definite integral accumulates a function over a specified interval as a limit of Riemann sums when that limit exists. Its sign includes the function’s sign and the orientation of the limits. For example, the integral of the constant −2 from 0 to 3 is −6, although the geometric area is 6. In a rate model, the result has rate units multiplied by input units.