Triangles — Practice Quiz

A Geometry cheat sheet for Triangles — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Pythagorean Theorem (Right Triangle)

$$a^2 + b^2 = c^2$$

Where: \(a, b\) = legs, \(c\) = hypotenuse

Area of Triangle

$$A = \frac{1}{2}bh$$

Where: \(b\) = base, \(h\) = height

Law of Sines

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$

Where: \(R\) = circumradius

Law of Cosines

$$c^2 = a^2 + b^2 - 2ab\cos C$$

Heron's Formula

$$A = \sqrt{s(s-a)(s-b)(s-c)}, \quad s = \frac{a+b+c}{2}$$

Where: \(s\) = semi-perimeter

Equilateral Triangle

$$h = \frac{\sqrt{3}}{2}a, \quad A = \frac{\sqrt{3}}{4}a^2$$

Where: \(a\) = side length

Inradius and Circumradius

$$r = \frac{A}{s}, \quad R = \frac{abc}{4A}$$

Where: \(s\) = semi-perimeter, \(A\) = area

Median to Side a

$$m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}$$

Altitude to Side a

$$h_a = \frac{2A}{a}$$

Practice quiz

  1. If a right triangle has legs of length $3$ cm and $4$ cm, what is the length of its hypotenuse?

    • $5$ cm
    • $6$ cm
    • $7$ cm
    • $8$ cm

    Answer: $5$ cm

  2. A triangle has a base of $10$ cm and a height of $8$ cm. What is its area?

    • $40$ cm$^2$
    • $80$ cm$^2$
    • $20$ cm$^2$
    • $18$ cm$^2$

    Answer: $40$ cm$^2$

  3. In triangle $ABC$, if side $a = 10$, angle $A = 30^\circ$, and angle $B = 45^\circ$, what is the length of side $b$?

    • $10\sqrt{2}$
    • $5\sqrt{2}$
    • $10$
    • $20$

    Answer: $10\sqrt{2}$

  4. In triangle $ABC$, if $a = 5$, $b = 8$, and angle $C = 60^\circ$, what is the length of side $c$?

    • $\sqrt{39}$
    • $7$
    • $\sqrt{89}$
    • $13$

    Answer: $7$

  5. A triangle has side lengths $3$, $4$, and $5$. What is its area using Heron's formula?

    • $6$
    • $12$
    • $7.5$
    • $10$

    Answer: $6$

  6. An equilateral triangle has a side length of $6$ cm. What is its area?

    • $9\sqrt{3}$ cm$^2$
    • $18\sqrt{3}$ cm$^2$
    • $36$ cm$^2$
    • $9$ cm$^2$

    Answer: $9\sqrt{3}$ cm$^2$

  7. A triangle has an area of $24$ cm$^2$ and a semi-perimeter of $12$ cm. What is its inradius?

    • $2$ cm
    • $4$ cm
    • $1$ cm
    • $3$ cm

    Answer: $2$ cm

  8. For a triangle with sides $a=6$, $b=8$, $c=10$ and area $A=24$, what is its circumradius $R$?

    • $5$
    • $10$
    • $2.5$
    • $12.5$

    Answer: $5$

  9. In a triangle with sides $a=10$, $b=13$, and $c=13$, what is the length of the median to side $a$ ($m_a$)?

    • $10$
    • $12$
    • $13$
    • $15$

    Answer: $12$

  10. A triangle has an area of $30$ cm$^2$ and side $a$ of length $10$ cm. What is the length of the altitude to side $a$ ($h_a$)?

    • $3$ cm
    • $6$ cm
    • $5$ cm
    • $10$ cm

    Answer: $6$ cm

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