Mathematics · Geometry
Cone Radius–Height Volume Calculator
Calculate cone volume from base radius and perpendicular height.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Use c=πa²b/3 with base radius=5 and perpendicular height=12.
- cone volume=314.1592653589793.
Understand Cone Radius–Height Volume
One idea, three depths
Choose how deeply to explain Cone Radius–Height Volume
Cone Radius–Height Volume: Calculate cone volume from base radius and perpendicular height.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Cone Radius–Height Volume to answer this question: calculate cone volume from base radius and perpendicular height? Enter base radius and perpendicular height; the calculator shows cone volume. For example: base radius=5 and perpendicular height=12 produce cone volume=314.1592653589793. The answer tells you cone volume.
Age 15Explain it to a 15-year-oldConnect it to the formula
A cone has one third of the matching cylinder's volume. This page evaluates the relationship directly. The rule is c=πa²b/3. Its input values are base radius, perpendicular height, and the main result is cone volume. For example: base radius=5 and perpendicular height=12 produce cone volume=314.1592653589793.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated cone radius–height volume relation over the valid real-number domain stated below. The implemented relation is c=πa²b/3, evaluated from base radius, perpendicular height to produce cone volume. A cone has one third of the matching cylinder's volume. This page evaluates the relationship directly. Use perpendicular height and the base radius, not diameter or slant height.
Inputs and valid domain
- base radius must be a finite real number.
- perpendicular height must be a finite real number.
Important boundary: Use perpendicular height and the base radius, not diameter or slant height.
The formula
c=πa²b/3
How the calculator works through it
It substitutes base radius, perpendicular height into the formula and exposes every numerical step above. The main output is cone volume.
Read the result correctly
The cone volume is the direct answer to “calculate cone volume from base radius and perpendicular height.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
base radius=5 and perpendicular height=12 produce cone volume=314.1592653589793.
Where this model stops being reliable
Use perpendicular height and the base radius, not diameter or slant height.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Cone Radius–Height Volume works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Cone Radius–Height Volume uses c=πa²b/3. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Ratios between measured quantities
Ratios help you check the scale, units and proportional meaning of Cone Radius–Height Volume.
Review this foundation about 4 min
Optional enrichment
- Angles and geometric relationships
Angle language provides useful geometric context for extending Cone Radius–Height Volume to related shapes and constructions.
Review this foundation about 4 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read base radius, perpendicular height.
- Evaluate the principal relationship: c=πa²b/3.
- Return cone volume and check the domain conditions described above.
Python
from math import *
def cone_radius_height_volume_calculator(a, b) -> float:
return ((pi * ((a * a) * b)) / 3.0)
assert abs(cone_radius_height_volume_calculator(5, 12) - 314.1592653589793) < 1e-6 * max(1.0, abs(314.1592653589793))
C
#include <assert.h>
#include <math.h>
double cone_radius_height_volume_calculator(double a, double b) {
return ((3.141592653589793 * ((a * a) * b)) / 3.0);
}
int main(void) {
const double expected = 314.1592653589793;
const double actual = cone_radius_height_volume_calculator(5, 12);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double cone_radius_height_volume_calculator(double a, double b) {
return ((std::numbers::pi * ((a * a) * b)) / 3.0);
}
int main() {
constexpr double expected = 314.1592653589793;
const double actual = cone_radius_height_volume_calculator(5, 12);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double cone_radius_height_volume_calculator(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global cone_radius_height_volume_calculator
section .text
cone_radius_height_volume_calculator:
push rbp
mov rbp, rsp
sub rsp, 64
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-40], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-8]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-56]
mulsd xmm0, [rbp-16]
movsd [rbp-48], xmm0
movsd xmm0, [rbp-40]
mulsd xmm0, [rbp-48]
movsd [rbp-32], xmm0
mov rax, 0x4008000000000000
movq xmm0, rax
movsd [rbp-64], xmm0
movsd xmm0, [rbp-32]
divsd xmm0, [rbp-64]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = cone_radius_height_volume_calculator(a, b)
result = ((pi * ((a * a) * b)) / 3.0);
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_] := ((Pi * ((a * a) * b)) / 3.0);
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Cone Radius–Height Volume Calculator. MW SysArc Tools. https://math.mwsysarc.com/geometry/cone-radius-height-volume-calculator
MLA 9
MW SysArc. “Cone Radius–Height Volume Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/geometry/cone-radius-height-volume-calculator. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Cone Radius–Height Volume Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/geometry/cone-radius-height-volume-calculator.
Harvard
MW SysArc (2026) ‘Cone Radius–Height Volume Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/geometry/cone-radius-height-volume-calculator (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_cone_radius_height_volume_calculator_2026,
author = {{MW SysArc}},
title = {Cone Radius–Height Volume Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/geometry/cone-radius-height-volume-calculator},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Cone Radius–Height Volume Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/geometry/cone-radius-height-volume-calculator
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Cone Radius–Height Volume do?
Calculate cone volume from base radius and perpendicular height.
How does the Cone Radius–Height Volume work?
The calculator applies c=πa²b/3. A cone has one third of the matching cylinder's volume. This page evaluates the relationship directly.
What can I learn from the Cone Radius–Height Volume?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .