What do the Grade 10 Mathematics cards cover?

29 cards across 6 chapters of the national textbook: Polynomial Functions, Exponential and Logarithmic Functions, Trigonometric functions, Circle, Solid Figures and Coordinate Geometry. You can take any chapter one card at a time on the page itself.

Is there a national exam in Grade 10?

No. Ethiopia sets national exams in Grade 6, Grade 8 and Grade 12 only. These cards are for your school's own exams, and for the national exam that comes a few years later.

Where do these cards come from?

They are drafted from Grade 10 Mathematics, the Ministry of Education textbook for this grade. A second pass that cannot see the chapter then re-derives every formula, constant and table row, and anything it cannot confirm is held back instead of published.

Yes. Every card here is free to read and the printable sheet is free to download. Neither needs an account.

When was this last checked?

6 September 2026. Cards arrive chapter by chapter, and the line under each one says when that card was last read through.

Free short notes and revision cards

Grade 10 Mathematics

Grade 10 Mathematics on Temari has 29 revision cards, arranged by the chapters of the Ethiopian national curriculum. Every card says when the rule applies, what each symbol in it stands for, and the mistake students most often make with it. They are free to read and need no account.

29
Cards: 6
Chapters: 17
Formulas: 6
Reference tables: Grade 10 Mathematics
Textbook: Free
Price: 6 September 2026
Last checked

Everything29Formula17Reference table6Common mistake4Method1Rule1

Polynomial Functions

End Behavior of Polynomial GraphsReference tableOdd · Even

Degree	Leading Coefficient	Left End Behavior	Right End Behavior
Odd	$a_n > 0$	Downwards	Upwards
Odd	$a_n < 0$	Upwards	Downwards
Even	$a_n > 0$	Upwards	Upwards
Even	$a_n < 0$	Downwards	Downwards

When you use it

Use to determine how the ends of a polynomial graph behave far to the left and far to the right based on degree and leading coefficient.

Watch out

Remember that polynomial graphs are always smooth and continuous, having no sharp corners, jumps, or holes.

Drafted from Grade 10 Mathematics, pages 66-116, then checked twice before it went up

Polynomial Division TheoremFormula

f(x) = d(x)q(x) + r(x)

f(x) = d(x)q(x) + r(x)

$f(x)$: dividend polynomial $dimensionless$
$d(x)$: non zero divisor polynomial $dimensionless$
$q(x)$: quotient polynomial $dimensionless$
$r(x)$: remainder polynomial whose degree is less than d(x) or is zero $dimensionless$

When you use it

Use to express a polynomial in terms of a divisor, quotient, and remainder during polynomial long division.

Watch out

Stop the division process only when the remainder is zero or its degree is strictly less than the divisor degree.

Drafted from Grade 10 Mathematics, pages 66-116, then checked twice before it went up

Rational Root TestMethod

When you use it

Use to find all potential rational zeros p/q for a polynomial function with integer coefficients.

Watch out

The numerator p must be a factor of the constant term a_0, and the denominator q must be a factor of the leading coefficient a_n, not the reverse.

Drafted from Grade 10 Mathematics, pages 66-116, then checked twice before it went up

Remainder and Factor TheoremsRule

When you use it

Use to find the remainder when f(x) is divided by x - c, or to confirm that x - c is a factor by checking if f(c) = 0.

Watch out

When dividing by x + c, evaluate f(-c) because the standard divisor form is x - (-c).

Drafted from Grade 10 Mathematics, pages 66-116, then checked twice before it went up

Special products and factoring identitiesFormula

x^{3} + a^{3} = (x+a)(x^{2} - ax + a^{2})

x^{3} + a^{3} = (x+a)(x^{2} - ax + a^{2})

$x$: the variable, or any expression standing in its place
$a$: a constant, or an expression, the same one everywhere in the identity
$b$: the second constant in the monic quadratic rule
$n$: a positive whole number, in the even powers rule

Expression	Its factorised form
$x^{2} - a^{2}$	$(x+a)(x-a)$
$x^{2} + 2ax + a^{2}$	$(x+a)^{2}$
$x^{2} - 2ax + a^{2}$	$(x-a)^{2}$
$x^{2} + (a+b)x + ab$	$(x+a)(x+b)$
$x^{3} + 3ax^{2} + 3a^{2}x + a^{3}$	$(x+a)^{3}$
$x^{3} + a^{3}$	$(x+a)(x^{2} - ax + a^{2})$
$x^{3} - a^{3}$	$(x-a)(x^{2} + ax + a^{2})$
$x^{2n} - a^{2n}$	$(x^{n} - a^{n})(x^{n} + a^{n})$

When you use it

Reach for these when an expression matches one of the patterns and you need it written as a product: solving a polynomial equation, cancelling a fraction, or simplifying before you differentiate. Read the expression against the list first, because a pattern that matches saves the whole trial and error search. The even powers line is the difference of squares with $x^{n}$ in place of $x$ , and each of its two factors will often break down further.

Watch out

The sign inside the trinomial is the opposite of the sign in the bracket. Students copy the binomial's sign across and write $x^{3} + a^{3} = (x+a)(x^{2} + ax + a^{2})$ , which is wrong: put $x = 2$ and $a = 3$ into it and you get 95 instead of 35. The last term is $+a^{2}$ in both cubes. Two more that cost marks on the same page. A sum of squares $x^{2} + a^{2}$ has no factorisation over the real numbers, so it is never $(x+a)^{2}$ . And the $x^{2} + (a+b)x + ab$ rule needs the coefficient of $x^{2}$ to be 1, so it does nothing for $2x^{2} + 5x + 3$ until you take the 2 out or split the middle term.

Exponential and Logarithmic Functions

False Logarithmic IdentitiesCommon mistake

When you use it

Reach for this to verify log operations and prevent combining rules incorrectly.

Watch out

Do not distribute logarithms across addition: $\log_a(M+N)$ is not $\log_a M + \log_a N$ . Similarly, $\log_a(MN)$ is not $(\log_a M)(\log_a N)$ , and $\frac{\log_a M}{\log_a N}$ is not $\log_a(M-N)$ .

Drafted from Grade 10 Mathematics, pages 117-186, then checked twice before it went up

Graphs of Exponential and Logarithmic FunctionsReference tableDomain · Range · Intercept · +3

Feature	Exponential $f(x) = a^x$ ($a > 1$)	Logarithmic $f(x) = \log_a x$ ($a > 1$)
Domain	$(-\infty, \infty)$	$(0, \infty)$
Range	$(0, \infty)$	$(-\infty, \infty)$
Intercept	$y$ -intercept at $(0, 1)$	$x$ -intercept at $(1, 0)$
Asymptote	Horizontal asymptote $y = 0$	Vertical asymptote $x = 0$
Monotonicity	Increasing on $(-\infty, \infty)$	Increasing on $(0, \infty)$
Relationship	Inverse of $\log_a x$	Inverse of $a^x$ (reflection across $y = x$ )

When you use it

Use when finding the domain, range, intercepts, and asymptotes of exponential and logarithmic functions.

Watch out

The exponential function never touches $y = 0$ , and the logarithmic function is undefined for $x \le 0$ .

Drafted from Grade 10 Mathematics, pages 117-186, then checked twice before it went up

Laws of ExponentsReference tableProduct of powers · Quotient of powers · Power of a power · +5

Law Name	Formula	Condition
Product of powers	$a^m \cdot a^n = a^{m+n}$	$a$ is a real number
Quotient of powers	$\frac{a^m}{a^n} = a^{m-n}$	$a$ is not equal to 0
Power of a power	$(a^m)^n = a^{mn}$	$m, n$ are integers
Power of a product	$(ab)^n = a^n b^n$	$a, b$ are real numbers
Power of a quotient	(\frac{a}{b})^n = \frac{a^n}{b^n}	$b$ is not equal to 0
Zero exponent	$a^0 = 1$	$a$ is not equal to 0
Negative exponent	$a^{-n} = \frac{1}{a^n}$	$a$ is not equal to 0
Rational exponent	$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$	$a > 0$ or $n$ is odd

When you use it

Use when simplifying algebraic expressions with integer or rational powers, or solving exponential expressions.

Watch out

The base in a negative expression matters: $(-3)^2 = 9$ but $-3^2 = -9$ . Also, $0^0$ is undefined.

Drafted from Grade 10 Mathematics, pages 117-186, then checked twice before it went up

Laws of LogarithmsReference tableProduct law · Quotient law · Power law · +4

Law Name	Formula	Condition
Product law	$\log_a(MN) = \log_a M + \log_a N$	$M, N > 0, a > 0, a$ not equal to 1
Quotient law	$\log_a(\frac{M}{N}) = \log_a M - \log_a N$	$M, N > 0, a > 0, a$ not equal to 1
Power law	$\log_a(M^r) = r \log_a M$	$M > 0, r$ is real
Change of base	$\log_a M = \frac{\log_b M}{\log_b a}$	$a, b > 0$ , both not equal to 1
Log of base	$\log_a a = 1$	$a > 0, a$ not equal to 1
Log of one	$\log_a 1 = 0$	$a > 0, a$ not equal to 1
Inverse identity	$a^{\log_a M} = M$	$M > 0, a > 0, a$ not equal to 1

When you use it

Use to expand, condense, or evaluate logarithmic expressions and to change the base of a logarithm.

Watch out

Logarithms are only defined for positive arguments and positive bases not equal to 1.

Drafted from Grade 10 Mathematics, pages 117-186, then checked twice before it went up

Trigonometric functions

Arc Length FormulaFormula

l = r \theta

l = r \theta

$l$: arc length $\text{m}$
$r$: radius of the circle $\text{m}$
$\theta$: central angle in radians $\text{rad}$

When you use it

Use this formula to find the length of an arc subtended by a central angle in a circle of a given radius.

Watch out

The central angle must be converted to radians before calculating. Using degrees directly will give an incorrect answer.

Drafted from Grade 10 Mathematics, pages 187-237, then checked twice before it went up

Degree and Radian Confusion in Arc CalculationsCommon mistake

When you use it

Check this when solving problems involving arc length, circular motion, or angle conversion.

Watch out

Never insert angle values in degrees into the formula l = r theta without multiplying by pi / 180 first.

Drafted from Grade 10 Mathematics, pages 187-237, then checked twice before it went up

Double Angle Sine FormulaFormula

\sin 2\theta = 2\sin\theta\cos\theta

\sin 2\theta = 2\sin\theta\cos\theta

$\theta$: angle $\text{rad}$

When you use it

Use this formula to calculate the sine of twice an angle using the sine and cosine of the single angle.

Watch out

Sine of two theta is not equal to two times sine of theta. You must multiply two by both sine theta and cosine theta.

Drafted from Grade 10 Mathematics, pages 187-237, then checked twice before it went up

Pythagorean Trigonometric IdentityFormula

\sin^2\theta + \cos^2\theta = 1

\sin^2\theta + \cos^2\theta = 1

$\theta$: angle $\text{rad}$

When you use it

Use this identity to find the value of sine given cosine or cosine given sine for any angle.

Watch out

Remember to determine the correct positive or negative sign of the resulting trigonometric value based on the quadrant where the angle terminates.

Drafted from Grade 10 Mathematics, pages 187-237, then checked twice before it went up

Reference Angles by QuadrantReference tableQuadrant I · Quadrant II · Quadrant III · +1

Quadrant	Angle Range	Reference Angle Formula
Quadrant I	$0^\circ < \theta < 90^\circ$	$\theta_R = \theta$
Quadrant II	$90^\circ < \theta < 180^\circ$	$\theta_R = 180^\circ - \theta$
Quadrant III	$180^\circ < \theta < 270^\circ$	$\theta_R = \theta - 180^\circ$
Quadrant IV	$270^\circ < \theta < 360^\circ$	$\theta_R = 360^\circ - \theta$

When you use it

Use this table to find the acute reference angle for any non-quadrantal angle in standard position.

Watch out

Always subtract to or from the horizontal x-axis (180 degrees or 360 degrees), never the vertical y-axis (90 degrees or 270 degrees).

Drafted from Grade 10 Mathematics, pages 187-237, then checked twice before it went up

Sine rule, cosine rule and the area formulaFormula

a^{2} = b^{2} + c^{2} - 2bc\,\cos A

a^{2} = b^{2} + c^{2} - 2bc\,\cos A

$a,\,b,\,c$: the lengths of the three sides
$A,\,B,\,C$: the angles. Each one faces the side written with the same letter, so A faces a and sits between b and c

Formula	Reach for it when	Where it holds
$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$	You know two angles and any one side, or two sides and an angle opposite one of them.	Every triangle.
$a^{2} = b^{2} + c^{2} - 2bc\,\cos A$	You know two sides and the angle between them, and you want the third side.	Every triangle.
$\cos A = \dfrac{b^{2} + c^{2} - a^{2}}{2bc}$	You know all three sides and you want an angle.	Every triangle.
$\text{Area} = \tfrac{1}{2}\,ab\,\sin C$	You know two sides and the angle between them. The same triangle also gives $\tfrac{1}{2}\,bc\,\sin A$ and $\tfrac{1}{2}\,ac\,\sin B$ .	Every triangle.
$\sin A = \dfrac{\text{opposite}}{\text{hypotenuse}}$	You know the side facing the angle and the hypotenuse.	Right-angled triangles only.
$\cos A = \dfrac{\text{adjacent}}{\text{hypotenuse}}$	You know the side beside the angle and the hypotenuse.	Right-angled triangles only.
$\tan A = \dfrac{\text{opposite}}{\text{adjacent}}$	You know the side facing the angle and the side beside it.	Right-angled triangles only.
$a^{2} + b^{2} = c^{2}$	You know two sides and you want the third. Here $c$ is the hypotenuse, the side facing the right angle.	Right-angled triangles only. It is the cosine rule with the angle set to 90 degrees, where the cosine is zero.

When you use it

Use these when a triangle question mixes sides and angles and the triangle has no right angle. Match what you are given against the table: two angles and any one side calls for the sine rule, while two sides with the angle between them calls for the cosine rule, and that same pair hands you the area straight away.

Watch out

Two traps, and both are about which angle belongs with which sides. The cosine rule subtracts a single product: work out $2bc\,\cos A$ as one quantity and take it away from $b^{2} + c^{2}$ . Working out $b^{2} + c^{2} - 2bc$ first and then multiplying by $\cos A$ gives a different number every time. Second, the angle has to be the one enclosed by the two sides you multiply. $\tfrac{1}{2}\,ab\,\sin C$ is the area only because $C$ sits between $a$ and $b$ , and $\cos A$ in the cosine rule is the angle between $b$ and $c$ . One more: when the sine rule gives you an angle, the calculator returns only the acute answer, so check whether 180 degrees minus that angle also fits the triangle before you accept it.

Circle

Angle of Interior Intersecting ChordsFormula

\theta = \frac{1}{2}(\alpha + \beta)

\theta = \frac{1}{2}(\alpha + \beta)

$\theta$: interior angle between intersecting chords $deg$
$\alpha$: measure of first intercepted arc $deg$
$\beta$: measure of second intercepted arc $deg$

When you use it

Use this formula when two chords intersect inside a circle to find the angle formed from their intercepted arcs.

Watch out

Add the two intercepted arcs together, do not subtract them.

Drafted from Grade 10 Mathematics, pages 238-271, then checked twice before it went up

Arc LengthFormula

s = \frac{\theta}{360^{\circ}} \cdot 2\pi r

s = \frac{\theta}{360^{\circ}} \cdot 2\pi r

$s$: arc length $m$
$\theta$: central angle $deg$
$r$: radius $m$
$\pi$: pi constant $unitless$

When you use it

Use this formula to calculate the length of a circular arc when the radius and the central angle in degrees are known.

Watch out

Ensure the angle theta is the central angle in degrees before dividing by 360 degrees.

Drafted from Grade 10 Mathematics, pages 238-271, then checked twice before it went up

Area of a SectorFormula

A = \frac{\theta}{360^{\circ}} \pi r^2

A = \frac{\theta}{360^{\circ}} \pi r^2

$A$: area of sector $m^2$
$\theta$: central angle $deg$
$r$: radius $m$
$\pi$: pi constant $unitless$

When you use it

Use this formula to find the area of a circle sector bounded by two radii and an arc.

Watch out

Do not square the product of pi and r, only the radius r is squared.

Drafted from Grade 10 Mathematics, pages 238-271, then checked twice before it went up

Chord LengthFormula

c = 2r \sin\left(\frac{\theta}{2}\right)

c = 2r \sin\left(\frac{\theta}{2}\right)

$c$: chord length $m$
$r$: radius $m$
$\theta$: central angle subtended by chord $deg$

When you use it

Use this formula to calculate the length of a chord subtended by a central angle theta.

Watch out

Divide the angle theta by 2 before evaluating the sine function.

Drafted from Grade 10 Mathematics, pages 238-271, then checked twice before it went up

Perimeter of a Sector MistakeCommon mistake

When you use it

Reach for this reminder whenever asked to calculate the perimeter of a circular sector.

Watch out

Calculating only the arc length gives the wrong answer. The perimeter of a sector must include both straight edges, P = s + 2r.

Drafted from Grade 10 Mathematics, pages 238-271, then checked twice before it went up

Solid Figures

Cross-Section Area RatioFormula

\frac{A'}{A} = \frac{k^2}{h^2}

\frac{A'}{A} = \frac{k^2}{h^2}

$A'$: Area of the horizontal cross-section $m^2$
$A$: Area of the base of the pyramid or cone $m^2$
$k$: Distance from the vertex to the cross-section plane $m$
$h$: Altitude of the pyramid or cone $m$

When you use it

Use when finding the area of a parallel slice or cross-section at a known distance from the vertex of a pyramid or cone.

Watch out

Distance k is measured from the vertex, not from the base. If distance from the base is given, subtract it from h first.

Drafted from Grade 10 Mathematics, pages 272-327, then checked twice before it went up

Lateral Surface Area of a Frustum of a ConeFormula

LSA = \pi l(r + r')

LSA = \pi l(r + r')

$LSA$: Lateral surface area of the cone frustum $m^2$
$l$: Slant height of the frustum $m$
$r$: Radius of the larger base $m$
$r'$: Radius of the smaller base $m$

When you use it

Use this to compute the curved surface area of a frustum of a right circular cone.

Watch out

This formula gives only the lateral area. To find the total surface area, add both circular base areas.

Drafted from Grade 10 Mathematics, pages 272-327, then checked twice before it went up

Surface Area of Composed SolidsCommon mistake

When you use it

Reach for this reminder whenever finding the total surface area of an object made by joining two solids together.

Watch out

Do not simply add the total surface areas of the individual solids. The interior touching faces are covered and must not be included.

Drafted from Grade 10 Mathematics, pages 272-327, then checked twice before it went up

Volume of a Frustum of a PyramidFormula

V = \frac{1}{3} h(A + A' + \sqrt{A A'})

V = \frac{1}{3} h(A + A' + \sqrt{A A'})

$V$: Volume of the frustum of a pyramid $m^3$
$h$: Altitude of the frustum $m$
$A$: Area of the lower base $m^2$
$A'$: Area of the upper base $m^2$

When you use it

Use to calculate the volume of a pyramid frustum when the upper and lower base areas and altitude are known.

Watch out

Remember to include the geometric mean term sqrt(A A') inside the parentheses.

Drafted from Grade 10 Mathematics, pages 272-327, then checked twice before it went up

Coordinate Geometry

Area of a Triangle in Coordinate PlaneFormula

A = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|

A = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|

$A$: Area of the triangle $\text{square units}$
$x_1, y_1$: Coordinates of the first vertex $\text{units}$
$x_2, y_2$: Coordinates of the second vertex $\text{units}$
$x_3, y_3$: Coordinates of the third vertex $\text{units}$

When you use it

Use this formula to calculate the area of a triangle when the coordinates of its three vertices are given.

Watch out

Always apply the absolute value at the end. Area can never be negative, and if it evaluates to zero the three points are collinear.

Drafted from Grade 10 Mathematics, page 328 onwards, then checked twice before it went up

Distance FormulaFormula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

$d$: Distance between two points $\text{units}$
$x_1, y_1$: Coordinates of the first point $\text{units}$
$x_2, y_2$: Coordinates of the second point $\text{units}$

When you use it

Use this formula to compute the straight line distance between any two points in the Cartesian plane.

Watch out

Do not mix coordinates from different points. Subtract x from x, and y from y, before squaring.

Drafted from Grade 10 Mathematics, page 328 onwards, then checked twice before it went up

Parallel and Perpendicular LinesReference tableParallel lines · Perpendicular lines

Line Relationship	Slope Condition	Geometric Meaning
Parallel lines	$m_1 = m_2$	Equal inclination angles
Perpendicular lines	$m_1 \cdot m_2 = -1$	Intersect at right angle

When you use it

Use these conditions to check whether two non-vertical lines are parallel or perpendicular, or to construct perpendicular and parallel line equations.

Watch out

The perpendicular condition only applies to non-vertical and non-horizontal lines. A vertical line is perpendicular to a horizontal line.

Drafted from Grade 10 Mathematics, page 328 onwards, then checked twice before it went up

Section FormulaFormula

(x_0, y_0) = \left(\frac{p x_2 + q x_1}{p + q}, \frac{p y_2 + q y_1}{p + q}\right)

(x_0, y_0) = \left(\frac{p x_2 + q x_1}{p + q}, \frac{p y_2 + q y_1}{p + q}\right)

$x_0, y_0$: Coordinates of the dividing point $\text{units}$
$p : q$: Ratio of internal division $\text{dimensionless}$
$x_1, y_1$: Coordinates of the first endpoint $\text{units}$
$x_2, y_2$: Coordinates of the second endpoint $\text{units}$

When you use it

Use this formula to find the coordinates of a point dividing a line segment internally in a given ratio p:q.

Watch out

Multiply the first ratio part p with the second point coordinates, and the second ratio part q with the first point coordinates.

Drafted from Grade 10 Mathematics, page 328 onwards, then checked twice before it went up

Slope and InclinationFormula

m = \frac{y_2 - y_1}{x_2 - x_1} = \tan\theta

m = \frac{y_2 - y_1}{x_2 - x_1} = \tan\theta

$m$: Slope of the line $\text{dimensionless}$
$\theta$: Angle of inclination $^{\circ}$
$x_1, y_1$: Coordinates of the first point $\text{units}$
$x_2, y_2$: Coordinates of the second point $\text{units}$

When you use it

Use this formula to compute the gradient of a straight line from two known points or from its angle of inclination with the positive x axis.

Watch out

A vertical line where x1 equals x2 has an undefined slope, not zero slope. A horizontal line has slope zero.

Drafted from Grade 10 Mathematics, page 328 onwards, then checked twice before it went up

The same subject in other years

An exam paper keeps asking for what the year below taught. Those cards are here too.

Your other Grade 10 subjects

Everything else in this year that has cards so far.

Questions students ask

What do the Grade 10 Mathematics cards cover?: 29 cards across 6 chapters of the national textbook: Polynomial Functions, Exponential and Logarithmic Functions, Trigonometric functions, Circle, Solid Figures and Coordinate Geometry. You can take any chapter one card at a time on the page itself.
Is there a national exam in Grade 10?: No. Ethiopia sets national exams in Grade 6, Grade 8 and Grade 12 only. These cards are for your school's own exams, and for the national exam that comes a few years later.
Where do these cards come from?: They are drafted from Grade 10 Mathematics, the Ministry of Education textbook for this grade. A second pass that cannot see the chapter then re-derives every formula, constant and table row, and anything it cannot confirm is held back instead of published.
Is this free?: Yes. Every card here is free to read and the printable sheet is free to download. Neither needs an account.
When was this last checked?: 6 September 2026. Cards arrive chapter by chapter, and the line under each one says when that card was last read through.

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