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Triangles: Area & Pythagorean

Triangle area, Pythagorean theorem, special right triangles, triangle inequality

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Triangles: Area & PythagoreanAMC 10 Step by Step

Name: Date: October 28, 2026

20 problems, easiest first

What you need to know

Area. Area $= \frac12 bh$ , with $h$ the height to base $b$ . With two sides $a$ , $b$ and the angle $C$ between them, area $= \frac12 ab\sin C$ . An equilateral triangle with side $s$ has area $\frac{\sqrt3}{4}s^2$ . Example: sides $6$ and $8$ with a $30^\circ$ angle between them give area $\frac12\cdot6\cdot8\cdot\frac12 = 12$ .
Pythagorean Theorem. In a right triangle with legs $a$ , $b$ and hypotenuse $c$ , $a^2+b^2=c^2$ . The converse is true too: if $a^2+b^2=c^2$ , the angle across from $c$ is a right angle. Know the triples $3$ - $4$ - $5$ , $5$ - $12$ - $13$ , $8$ - $15$ - $17$ , $7$ - $24$ - $25$ , and their multiples, such as $9$ - $12$ - $15$ .
Special right triangles. A $45$ - $45$ - $90$ triangle has sides in the ratio $1:1:\sqrt2$ . A $30$ - $60$ - $90$ triangle has sides $1:\sqrt3:2$ , with the short leg across from $30^\circ$ . Example: a $30$ - $60$ - $90$ triangle with hypotenuse $10$ has legs $5$ and $5\sqrt3$ .
Triangle Inequality. Any two sides add up to more than the third: $a+b>c$ , strictly. Example: $4$ , $5$ , $9$ is not a triangle, since $4+5 = 9$ .
Acute, right, or obtuse? Let $c$ be the longest side. If $a^2+b^2 > c^2$ , the triangle is acute; if equal, right; if less, obtuse. Example: $5$ , $6$ , $8$ is obtuse, since $25+36 = 61 < 64$ .
Heron's Formula. Area $=\sqrt{s(s-a)(s-b)(s-c)}$ , where $s$ is the semiperimeter (half the perimeter). Example: sides $5$ , $5$ , $6$ give $s = 8$ and area $\sqrt{8\cdot3\cdot3\cdot2} = 12$ .

How AMC 10 tests it

Problems 3–10: a figure of squares and right triangles; find a length or area by dropping one perpendicular.
"How many integers $x$ make a triangle with sides $a$ , $b$ , $x$ ?" or "...an obtuse triangle?"
Mid-paper: two sides and an altitude, or a $60^\circ$ or $120^\circ$ angle. The altitude splits the triangle into two right triangles that share a leg.
Shaded regions from overlapping triangles, or a triangle cut from a rectangle, found by subtraction.

Standard approaches

Drop an altitude and write a Pythagorean equation on each side of it; subtracting them removes the altitude.
Look for special right triangles or Pythagorean triples before working with square roots.
Find the area in two ways (two base–height pairs) to get an unknown height. Example: the $3$ - $4$ - $5$ triangle has area $6$ , so the height $h$ to the hypotenuse has $\frac12\cdot5\cdot h = 6$ , and $h = \frac{12}{5}$ .
For shaded regions, subtract simple pieces from a rectangle or triangle around the region.

Worked example

In triangle $ABC$ , $AB = 13$ , $AC = 15$ , and the altitude from $A$ to line $BC$ has length $12$ . What is the sum of all possible values of $BC$ ?

(A) $14$ (B) $16$ (C) $18$ (D) $20$ (E) $22$

Solution. Let $D$ be the foot of the altitude, so $AD = 12$ .

Step 1. Right triangle $ABD$ has hypotenuse $13$ and leg $12$ , so it is a $5$ - $12$ - $13$ triangle and $BD = 5$ .

Step 2. Right triangle $ACD$ has hypotenuse $15$ and leg $12$ , so it is a $9$ - $12$ - $15$ triangle and $CD = 9$ .

Step 3. If $D$ lies between $B$ and $C$ , then $BC = 5 + 9 = 14$ . If $D$ lies outside segment $BC$ , beyond $B$ (so angle $B$ is obtuse), then $BC = 9 - 5 = 4$ . Both triangles exist, so both values count.

Step 4. The sum is $14 + 4 = 18$ .

The answer is $\boxed{\textbf{(C)}\ 18}$ .

Pitfalls

Forgetting the obtuse picture, where the foot of an altitude falls outside the triangle. In the example, that picture gives $BC = 4$ .
Mixing up the legs of a $30$ - $60$ - $90$ triangle. The longer leg is across from $60^\circ$ and is $\sqrt3$ times the shorter leg.
Allowing $a+b=c$ (a flat triangle) when counting integer side lengths. Example: with sides $3$ and $7$ , the third side can be $5$ , $6$ , $7$ , $8$ , or $9$ , but not $4$ or $10$ .

Traps that recur

Counting 50 or 51 values per coordinate (including boundary points, which are not interior), or forgetting to subtract the points that are also 60-ray partitional. (2011 AMC 10A #25)
Reading 'the altitude to side AB' as a segment drawn from B, which would put P at B; the altitude meant is the one from the opposite vertex C, and its foot H is what fixes BP. (2025 AMC 10A #23)
Misreading the kite: its angles are 60°, 90°, 120°, 90° (the two √3 sides meet at 60°, the two unit sides at 120°), and its long diagonal is 2. (2024 AMC 10A #22)
Guessing 1:1 because AC = CD and DE = EB look symmetric, or setting AD:DB = AC:DE = 4:3 as if it were an angle bisector. (2019 AMC 10B #16)

1.2017 AMC 10A #7
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
1. A) $30\%$
2. B) $40\%$
3. C) $50\%$
4. D) $60\%$
5. E) $70\%$
2.2013 AMC 10A #3
Square $ABCD$ has side length $10$ . Point $E$ is on $\overline{BC}$ , and the area of $\triangle ABE$ is $40$ . What is $BE$ ?
1. A) $4$
2. B) $5$
3. C) $6$
4. D) $7$
5. E) $8$
3.2005 AMC 10A #4
A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle?
1. A) $\frac{1}{4}x^2$
2. B) $\frac{2}{5}x^2$
3. C) $\frac{1}{2}x^2$
4. D) $x^2$
5. E) $\frac{3}{2}x^2$
4.2004 AMC 10B #8
Minneapolis-St. Paul International Airport is $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
1. A) $13$
2. B) $14$
3. C) $15$
4. D) $16$
5. E) $17$
5.2003 AMC 10B #6
Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $4:3$ . The horizontal length of a " $27$ -inch" television screen is closest, in inches, to which of the following?
1. A) $20$
2. B) $20.5$
3. C) $21$
4. D) $21.5$
5. E) $22$
6.2025 AMC 10A #3
How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length $2025$ ?
1. A) $2025$
2. B) $2026$
3. C) $3012$
4. D) $3037$
5. E) $4050$
7.2022 AMC 10A #5
Square $ABCD$ has side length $1$ . Points $P$ , $Q$ , $R$ , and $S$ each lie on a side of $ABCD$ such that $APQCRS$ is an equilateral convex hexagon with side length $s$ . What is $s$ ?
1. A) $\frac{\sqrt{2}}{3}$
2. B) $\frac{1}{2}$
3. C) $2 - \sqrt{2}$
4. D) $1 - \frac{\sqrt{2}}{4}$
5. E) $\frac{2}{3}$
8.2017 AMC 10A #10
Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
1. A) $16$
2. B) $17$
3. C) $18$
4. D) $19$
5. E) $20$
9.2015 AMC 10A #11
The ratio of the length to the width of a rectangle is $4$ : $3$ . If the rectangle has diagonal of length $d$ , then the area may be expressed as $kd^2$ for some constant $k$ . What is $k$ ?
1. A) $\frac{2}{7}$
2. B) $\frac{3}{7}$
3. C) $\frac{12}{25}$
4. D) $\frac{16}{25}$
5. E) $\frac{3}{4}$
10.2015 AMC 10B #13
The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
1. A) $20$
2. B) $\dfrac{360}{17}$
3. C) $\dfrac{107}{5}$
4. D) $\dfrac{43}{2}$
5. E) $\dfrac{281}{13}$
11.2014 AMC 10A #9
The two legs of a right triangle, which are altitudes, have lengths $2\sqrt3$ and $6$ . How long is the third altitude of the triangle?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
12.2013 AMC 10A #15
Two sides of a triangle have lengths $10$ and $15$ . The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?
1. A) $6$
2. B) $8$
3. C) $9$
4. D) $12$
5. E) $18$
13.2013 AMC 10B #7
Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle?
1. A) $\frac{\sqrt{3}}{3}$
2. B) $\frac{\sqrt{3}}{2}$
3. C) $1$
4. D) $\sqrt{2}$
5. E) $\text{2}$
14.2012 AMC 10B #12
Point $B$ is due east of point $A$ . Point $C$ is due north of point $B$ . The distance between points $A$ and $C$ is $10\sqrt 2$ , and $\angle BAC = 45^\circ$ . Point $D$ is $20$ meters due north of point $C$ . The distance $AD$ is between which two integers?
1. A) $30 \text{and} 31$
2. B) $31 \text{and} 32$
3. C) $32 \text{and} 33$
4. D) $33 \text{and} 34$
5. E) $34 \text{and} 35$
15.2011 AMC 10A #11
Square $EFGH$ has one vertex on each side of square $ABCD$ . Point $E$ is on $\overline{AB}$ with $AE=7\cdot EB$ . What is the ratio of the area of $EFGH$ to the area of $ABCD$ ?
{{}}
1. A) $\frac{49}{64}$
2. B) $\frac{25}{32}$
3. C) $\frac78$
4. D) $\frac{5\sqrt{2}}{8}$
5. E) $\frac{\sqrt{14}}{4}$
16.2010 AMC 10B #7
A triangle has side lengths $10$ , $10$ , and $12$ . A rectangle has width $4$ and area equal to the area of the triangle. What is the perimeter of this rectangle?
1. A) $16$
2. B) $24$
3. C) $28$
4. D) $32$
5. E) $36$
17.2010 AMC 10A #7
Crystal has a running course marked out for her daily run. She begins this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles, is this last portion of her run?
1. A) $1$
2. B) $\sqrt{2}$
3. C) $\sqrt{3}$
4. D) $2$
5. E) $2\sqrt{2}$
18.2009 AMC 10B #12
Distinct points $A$ , $B$ , $C$ , and $D$ lie on a line, with $AB=BC=CD=1$ . Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$ . A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?
1. A) $3$
2. B) $4$
3. C) $5$
4. D) $6$
5. E) $7$
19.2009 AMC 10B #10
A flagpole is originally $5$ meters tall. A hurricane snaps the flagpole at a point $x$ meters above the ground so that the upper part, still attached to the stump, touches the ground $1$ meter away from the base. What is $x$ ?
1. A) $2.0$
2. B) $2.1$
3. C) $2.2$
4. D) $2.3$
5. E) $2.4$
20.2009 AMC 10A #12
In quadrilateral $ABCD$ , $AB = 5$ , $BC = 17$ , $CD = 5$ , $DA = 9$ , and $BD$ is an integer. What is $BD$ ?
1. A) $11$
2. B) $12$
3. C) $13$
4. D) $14$
5. E) $15$

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.