Geometry reference

Area Formulas

Q: How do I choose the correct area formula?

Identify the actual shape and the measurements supplied before substituting numbers. A triangle needs a base with its matching perpendicular height, or three valid side lengths for Heron's formula. A circle needs a radius or diameter. For a composite region, divide it into known shapes without overlap, or subtract a fully contained cutout from a larger shape. An appearance alone does not establish parallel sides or right angles.

Q: Is a triangle's height the same as a side length?

Only when that side is perpendicular to the chosen base, as happens for the legs of a right triangle. In general, height is the shortest distance from the opposite vertex to the infinite line containing the base. It may lie inside or outside the triangle. A sloping side usually measures a different distance, so using it as height changes the calculated area.

Q: Can I use a circle's diameter directly?

Yes, use A=π d^2/4, or first find r=d/2 and then use A=π r^2. Both methods agree. Substituting a diameter into the radius formula without halving it makes the result four times too large. Keep the exact multiple of π during the work and round only the final decimal if your problem asks for one.

Q: Why must I square the unit conversion factor?

Area includes two length dimensions. Because 1 m=100 cm, a square meter covers one hundred rows of one hundred square centimeters, giving 1 m^2=10000 cm^2. Dividing a square-centimeter value by only one hundred converts just one dimension and leaves the answer wrong. Either convert all lengths first or convert the completed area with the squared factor, but do not mix those methods halfway.

Q: Is the rhombus diagonal formula valid for any quadrilateral?

No. The simple product divided by two works for a rhombus because its diagonals are perpendicular; it also applies to other suitable perpendicular-diagonal quadrilaterals. A general quadrilateral needs additional information about the angle between its diagonals. Knowing only two diagonal lengths does not usually determine its area. Use the shape's established properties rather than assuming a sketch has a right angle.

Q: What if Heron's formula gives zero or a negative radicand?

For an ordinary triangle, every side must be positive and the sum of any two sides must exceed the third. Equality describes a flattened, zero-area boundary case, not a nondegenerate triangle. A violation cannot form a triangle. Recheck the measurements and units instead of taking an absolute value inside the square root; that would conceal an invalid shape and create a misleading result.

Q: Does doubling the perimeter double the area?

There is no universal area rule from perimeter alone: different shapes can enclose different areas with the same boundary length. If every length of one shape is multiplied by a common positive scale factor k, the new shape is similar, its perimeter scales by k, and its area scales by k^2. Thus doubling all lengths quadruples the area, but changing just one measurement requires a fresh calculation.

Q: How can I print and use this reference?

Open your browser's Print command and select paper or Save as PDF. The reference table has a print layout, and the diagrams remain vector drawings. Expand any practice answers or FAQs you want included before printing. These are teaching figures, not calibrated construction templates: the printed centimeter lengths are labels for the examples, not a guarantee that a ruler will measure those lengths on the page.

Find the space inside a flat shape. Use this printable formula reference, labeled diagrams and worked examples to choose the right measurements, handle square units and calculate composite areas.

Ten formula methods · Original worked examples · Exact and rounded answers

Open area calculator

Area formula reference

In every row, $A$ means area. Use one common length unit for all dimensions, then give the result in its squared unit. For example, centimeter inputs give $\mathrm{cm}^2$ . All shapes below are flat; surface area and volume of solids require different formulas.

Swipe across the table to read the dimension notes. With a keyboard, focus the table and use the arrow keys.

Choose the shape, then check what each dimension measures
Shape or method	Area formula	Dimensions and conditions
Rectangle	$A=\ell w$	$\ell$ is length and $w$ is perpendicular width.
Square	$A=s^2$	$s$ is one side; all sides are equal and adjacent sides are perpendicular.
Triangle	$A=\frac{bh}{2}$	$b$ is the chosen base; $h$ is the perpendicular distance to its line from the opposite vertex.
Parallelogram	$A=bh$	$b$ is a base and $h$ is the perpendicular separation of the parallel bases.
Trapezoid	$A=\frac{(b_1+b_2)h}{2}$	$b_1$ and $b_2$ are the parallel sides; $h$ is their perpendicular separation.
Circle	$A=\pi r^2=\frac{\pi d^2}{4}$	$r$ is radius and $d=2r$ is diameter.
Ellipse	$A=\pi ab$	$a$ and $b$ are the perpendicular semiaxes, each half a full axis length.
Rhombus	$A=\frac{d_1d_2}{2}$	$d_1$ and $d_2$ are the complete diagonals of the rhombus, which meet at right angles.
Circular ring	$A=\pi(R^2-r^2)$	$R$ and $r$ are the outer and inner radii of concentric circles, with $R>r\ge0$ .
Triangle: three sides	$A=\sqrt{p(p-a)(p-b)(p-c)}$	$p=(a+b+c)/2$ is the semiperimeter. Positive sides $a,b,c$ must satisfy the strict triangle inequalities.

The shape links open a filled example in the Area Calculator. Formula symbols are defined separately in each row. The basic relationships are cross-checked against OpenStax's geometry reference; the additional methods are explained below.

Rectangle, triangle, circle and trapezoid area formulas, with perpendicular height guides and a circle radius. — Four common area formulas. Dashed guides mark perpendicular heights and a circle radius; dimensions are defined in the reference table.

A reliable method for any area problem

1. Identify the region
Mark the boundary you want to measure. Separate a hole from the surrounding material, and check which sides are parallel or perpendicular. Do not infer a right angle from an approximate sketch. If the region has several parts, outline each part before choosing formulas.
2. Match the dimensions
Convert every length to one unit. Pair a base with its own perpendicular height, distinguish a radius from a diameter, and identify full diagonals or semiaxes where needed. Write the formula before inserting values so that a missing measurement remains obvious.
3. Calculate and check
Substitute, simplify and attach square units. Keep exact fractions or multiples of pi until the final decimal. Compare with a containing rectangle or another decomposition when possible. A result for a contained region cannot exceed the area of its container.

For a rectangle with $\ell=12\,\mathrm{cm}$ and $w=8\,\mathrm{cm}$ , $A=12\times8=96\,\mathrm{cm}^2$ . A square with side $s=6\,\mathrm{cm}$ has $A=6^2=36\,\mathrm{cm}^2$ . These are the simplest applications: multiply two perpendicular lengths. The perimeter answers would use linear units and a different calculation. Compare the Perimeter Calculator.

Example one: triangle and parallelogram

Both figures have base $10\,\mathrm{cm}$ and perpendicular height $6\,\mathrm{cm}$ . The dashed segment measures height. The sloping edge does not, because it is not perpendicular to the base. Equal horizontal and vertical drawing scales preserve the right-angle relationship.

Triangle

b=10\,\mathrm{cm}

h=6\,\mathrm{cm}

A=30\,\mathrm{cm}^2

The dashed height meets the base at a right angle. A tilted side is a different measurement.

Parallelogram

b=10\,\mathrm{cm}

h=6\,\mathrm{cm}

A=60\,\mathrm{cm}^2

The dashed height meets the base at a right angle. A tilted side is a different measurement.

1. Pair the base and its height
Choose the lower horizontal side. Measure from the opposite vertex of the triangle, or the parallel upper side of the parallelogram, straight down to the base line. The right-angle marker identifies that direction.
2. Multiply, then halve only the triangle
$A_{\text{triangle}}=\frac{10\times6}{2}=30\,\mathrm{cm}^2$ $A_{\text{parallelogram}}=10\times6=60\,\mathrm{cm}^2$
3. Check the relationship
A diagonal divides this parallelogram into two congruent triangles. Each triangle takes half its area. Sliding the upper edge sideways while preserving the base and perpendicular height changes the outline but preserves the area.

In an obtuse triangle, the altitude for a chosen base can land on an extension of that base outside the triangle. The perpendicular distance still works; an interior sloping side is not a substitute.

Check the triangle Check the parallelogram Solve other triangle dimensions

Example two: circle area from a diameter

Circle

d=10\,\mathrm{cm}

r=5\,\mathrm{cm}

A=25\pi\,\mathrm{cm}^2

The radius is half the diameter. The circle remains circular at every screen size.

A circular disk has diameter $10\,\mathrm{cm}$ . Find the area of one flat face, not its circumference or the surface area of a sphere.

1. Halve the diameter
$r=\frac{d}{2}=\frac{10}{2}=5\,\mathrm{cm}$
2. Square the radius
$A=\pi r^2=\pi\times5^2=25\pi\,\mathrm{cm}^2$
3. Round the final value
$A\approx78.54\,\mathrm{cm}^2$
The decimal is rounded to two places. The multiple of pi is exact for the stated ideal dimensions.

Check the diameter example

Using $10$ as the radius would give $100\pi\,\mathrm{cm}^2$ , four times the correct area. The diameter formula $A=\pi d^2/4$ provides a direct cross-check. For circumference and radius conversions, use the Circle Calculator.

Example three: trapezoid area

Trapezoid

b_1=8\,\mathrm{cm}

b_2=14\,\mathrm{cm}

h=5\,\mathrm{cm}

The dashed height meets the base at a right angle. A tilted side is a different measurement.

The parallel bases measure $8\,\mathrm{cm}$ and $14\,\mathrm{cm}$ . Their perpendicular separation is $5\,\mathrm{cm}$ . Average the parallel lengths, then multiply by the height.

1. Average the bases
$\frac{b_1+b_2}{2}=\frac{8+14}{2}=11\,\mathrm{cm}$
2. Multiply by height
$A=11\times5=55\,\mathrm{cm}^2$
3. Check the size
$40<55<70$
The answer lies between the areas $8\times5$ and $14\times5$ , measured in square centimeters. Adding the two sloping sides instead of the parallel bases would not use the stated formula.

Check the trapezoid

Example four: area of an L-shaped region

Start with an outer rectangle $12\,\mathrm{cm}$ wide and $8\,\mathrm{cm}$ high. Remove a top-right rectangle $5\,\mathrm{cm}$ wide and $3\,\mathrm{cm}$ high. All corners are right angles. The remaining gold shape can be measured in two independent ways.

L-shaped region

\text{Outer: }12\times8\,\mathrm{cm}^2

\text{Cutout: }5\times3\,\mathrm{cm}^2

\text{Retained: }81\,\mathrm{cm}^2

Subtract the dashed cutout, or add the two non-overlapping gold rectangles. All lengths use the same scale.

Method A: subtract the cutout

Calculate the complete outer rectangle, then remove only the missing region.

A_{\text{outer}}=12\times8=96\,\mathrm{cm}^2

A_{\text{cutout}}=5\times3=15\,\mathrm{cm}^2

A=96-15=81\,\mathrm{cm}^2

Method B: add non-overlapping parts

The lower rectangle has height $8-3=5\,\mathrm{cm}$ . The upper-left rectangle has width $12-5=7\,\mathrm{cm}$ . They share an edge but have no overlapping interior.

A=(12\times5)+(7\times3)

A=60+21=81\,\mathrm{cm}^2

Both methods must agree. Do not add a full-height left strip to a full-width bottom strip without subtracting their overlap: that would count some material twice. For curved cutouts, apply the same subtraction idea using the appropriate circle or semicircle area.

Check the outer rectangle Check the removed rectangle Check the lower rectangle Check the upper-left rectangle

Sketch your partition on square graph paper. The Letter graph paper guide explains paper dimensions and printing. Grid counting can check a simple drawing, but partial boundary cells need attention and do not replace exact measurements.

Example five: mixed units and area scaling

A rectangular panel measures $1.2\,\mathrm{m}$ by $80\,\mathrm{cm}$ . Multiplying the displayed numbers without matching units would mix meters and centimeters. Choose meters first, then calculate.

1. Convert the width
$80\,\mathrm{cm}=0.8\,\mathrm{m}$
Now both dimensions use meters.
2. Find the area
$A=1.2\times0.8=0.96\,\mathrm{m}^2$
The result belongs in square meters.
3. Cross-check in centimeters
$A=120\times80=9600\,\mathrm{cm}^2$ $\frac{9600}{100^2}=0.96$
Both routes describe the same panel.

Because $1\,\mathrm{m}=100\,\mathrm{cm}$ , the area relation is $1\,\mathrm{m}^2=10000\,\mathrm{cm}^2$ . Squaring applies to the conversion factor, not to an area you have already calculated. NIST explains square SI units. Check the converted rectangle, or use the Area Converter for an existing area value.

Similar shapes: scale lengths before predicting area

A_{\text{new}}=k^2A_{\text{original}},\qquad k>0

If every length doubles, the area becomes four times as large. For the original $12\,\mathrm{cm}$ by $8\,\mathrm{cm}$ rectangle, the enlarged dimensions are $24\,\mathrm{cm}$ by $16\,\mathrm{cm}$ , giving $384\,\mathrm{cm}^2=4\times96\,\mathrm{cm}^2$ . Doubling only the width would instead double the area. State what changes before applying a scale factor.

More area methods and their assumptions

Example six: Heron's formula from three sides

A triangle has side lengths $13\,\mathrm{cm}$ , $14\,\mathrm{cm}$ and $15\,\mathrm{cm}$ . The longest side is shorter than the sum of the other two, so these positive lengths form a triangle. When height is not supplied, first calculate the semiperimeter $p$ .

p=\frac{13+14+15}{2}=21\,\mathrm{cm}

A=\sqrt{21(21-13)(21-14)(21-15)}

A=\sqrt{21\times8\times7\times6}=\sqrt{7056}=84\,\mathrm{cm}^2

The radicand has fourth-power length units, so taking its square root gives square units. As a check, the corresponding altitude to the $14\,\mathrm{cm}$ side is $2A/b=12\,\mathrm{cm}$ . The letter $p$ here means semiperimeter, not area or a side length.

Check the three-side triangle Read the Heron formula reference

Ellipse: use semiaxes

An ellipse with full axis lengths $10\,\mathrm{cm}$ and $6\,\mathrm{cm}$ has semiaxes $a=5\,\mathrm{cm}$ and $b=3\,\mathrm{cm}$ . Halve both full lengths before substitution.

A=\pi\times5\times3=15\pi\,\mathrm{cm}^2

Thus $A\approx47.12\,\mathrm{cm}^2$ . Multiplying the two full axes instead would make the area four times too large.

Check the ellipse

Rhombus: full diagonals

For complete diagonal lengths $10\,\mathrm{cm}$ and $6\,\mathrm{cm}$ , multiply and divide by two. The perpendicular diagonals split the rhombus into four right triangles.

A=\frac{10\times6}{2}=30\,\mathrm{cm}^2

Do not halve the diagonals first and then substitute those half-lengths into the same formula.

Check the rhombus

Ring: subtract circle areas

A concentric ring has outer radius $R=5\,\mathrm{cm}$ and inner radius $r=3\,\mathrm{cm}$ . Square each radius before subtracting.

A=\pi(5^2-3^2)=16\pi\,\mathrm{cm}^2

Thus $A\approx50.27\,\mathrm{cm}^2$ . Squaring the difference of the radii would give a different, incorrect expression.

Check the circular ring

The ellipse relationship is explained by Harvey Mudd College's stretched-circle construction. The ring uses the same subtraction principle as the L-shaped example. For a wider reference across geometry and algebra, visit Math Formulas.

Practice before revealing the answers

Write the formula and identify each dimension before doing the arithmetic. Each answer includes the key step so you can check the method as well as the number.

Triangle: base

9\,\mathrm{cm}

, perpendicular height

4\,\mathrm{cm}

. Find its area.

A=\frac{9\times4}{2}=18\,\mathrm{cm}^2

Halving the base-height product is essential for a triangle.

Circle: diameter

6\,\mathrm{m}

. Give its exact area.

r=3\,\mathrm{m},\qquad A=9\pi\,\mathrm{m}^2

The given measurement crosses the whole circle; halve it to get the radius.

Convert

2500\,\mathrm{cm}^2

to square meters.

2500\div10000=0.25\,\mathrm{m}^2

Use the square conversion factor, not the length factor.

FAQ

Frequently Asked Questions

How do I choose the correct area formula?

Identify the actual shape and the measurements supplied before substituting numbers. A triangle needs a base with its matching perpendicular height, or three valid side lengths for Heron's formula. A circle needs a radius or diameter. For a composite region, divide it into known shapes without overlap, or subtract a fully contained cutout from a larger shape. An appearance alone does not establish parallel sides or right angles.

Is a triangle's height the same as a side length?

Only when that side is perpendicular to the chosen base, as happens for the legs of a right triangle. In general, height is the shortest distance from the opposite vertex to the infinite line containing the base. It may lie inside or outside the triangle. A sloping side usually measures a different distance, so using it as height changes the calculated area.

Can I use a circle's diameter directly?

Yes, use

A=\pi d^2/4

, or first find

r=d/2

and then use

A=\pi r^2

. Both methods agree. Substituting a diameter into the radius formula without halving it makes the result four times too large. Keep the exact multiple of

\pi

during the work and round only the final decimal if your problem asks for one.

Why must I square the unit conversion factor?

Area includes two length dimensions. Because

1\,\mathrm{m}=100\,\mathrm{cm}

, a square meter covers one hundred rows of one hundred square centimeters, giving

1\,\mathrm{m}^2=10000\,\mathrm{cm}^2

. Dividing a square-centimeter value by only one hundred converts just one dimension and leaves the answer wrong. Either convert all lengths first or convert the completed area with the squared factor, but do not mix those methods halfway.

Is the rhombus diagonal formula valid for any quadrilateral?

No. The simple product divided by two works for a rhombus because its diagonals are perpendicular; it also applies to other suitable perpendicular-diagonal quadrilaterals. A general quadrilateral needs additional information about the angle between its diagonals. Knowing only two diagonal lengths does not usually determine its area. Use the shape's established properties rather than assuming a sketch has a right angle.

What if Heron's formula gives zero or a negative radicand?

For an ordinary triangle, every side must be positive and the sum of any two sides must exceed the third. Equality describes a flattened, zero-area boundary case, not a nondegenerate triangle. A violation cannot form a triangle. Recheck the measurements and units instead of taking an absolute value inside the square root; that would conceal an invalid shape and create a misleading result.

Does doubling the perimeter double the area?

There is no universal area rule from perimeter alone: different shapes can enclose different areas with the same boundary length. If every length of one shape is multiplied by a common positive scale factor

k

, the new shape is similar, its perimeter scales by

k

, and its area scales by

k^2

. Thus doubling all lengths quadruples the area, but changing just one measurement requires a fresh calculation.

How can I print and use this reference?

Open your browser's Print command and select paper or Save as PDF. The reference table has a print layout, and the diagrams remain vector drawings. Expand any practice answers or FAQs you want included before printing. These are teaching figures, not calibrated construction templates: the printed centimeter lengths are labels for the examples, not a guarantee that a ruler will measure those lengths on the page.

Sources and calculation notes

These references support the definitions and formula conditions. The numerical examples, partitions and diagrams in this guide were created for Toolarithm and independently checked. Calculator links restore the dimensions shown; the calculator may choose a different compatible drawing when the dimensions do not uniquely determine the outline.

OpenStax: Geometric Formulas
Rectangle, square, triangle, parallelogram, trapezoid and circle formulas with dimension definitions.
OpenStax: Area
Area in square units, quadrilaterals and decomposing geometric regions.
OpenStax: Heron's formula
Finding triangle area from three side lengths using the semiperimeter.
Harvey Mudd College: Area of an Ellipse
The semiaxis formula and its interpretation as a stretched circle.
NIST: SI Units - Area
Square-meter units and the square relationship between metric length and area conversions.

Exact relationships assume the stated ideal shapes. For measured objects, the accuracy of the result depends on the measurements and how well the model fits the boundary. A calculator's extra decimal places do not make an approximate measurement more precise.

Continue with related geometry tools

Area Calculator
Enter dimensions and compare exact areas, conversions and worked steps.
Perimeter Calculator
Measure the boundary instead of the enclosed region.
Triangle Calculator
Find missing sides, angles and triangle measurements.
Slope and Intercept Explained
Review coordinate diagrams and the difference between vertical rise and sloping distance.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. The method was last reviewed on October 29, 2026. No independent professional review is claimed.

Methodology

The editorial team checks definitions, examples, linked tools, and cited references. Dates shown here come from the page's maintained publication record.

Read the editorial methodology

Dates and sources

Published June 6, 2026. Updated October 29, 2026.

Area formula reference

A reliable method for any area problem

1. Identify the region

2. Match the dimensions

3. Calculate and check

Example one: triangle and parallelogram

1. Pair the base and its height

2. Multiply, then halve only the triangle

3. Check the relationship

Example two: circle area from a diameter

1. Halve the diameter

2. Square the radius

3. Round the final value

Example three: trapezoid area

1. Average the bases

2. Multiply by height

3. Check the size

Example four: area of an L-shaped region

Method A: subtract the cutout

Method B: add non-overlapping parts

Example five: mixed units and area scaling

1. Convert the width

2. Find the area

3. Cross-check in centimeters

Similar shapes: scale lengths before predicting area

More area methods and their assumptions

Example six: Heron's formula from three sides

Ellipse: use semiaxes

Rhombus: full diagonals

Ring: subtract circle areas

Practice before revealing the answers

Frequently Asked Questions

Sources and calculation notes

Continue with related geometry tools

Who maintains this page

Ownership and review

Methodology

Dates and sources