List of top Mathematics Questions on Area of a Triangle - by Heron’s Formula

The perimeter of an equilateral triangle whose area is \(4\sqrt{3}\,\text{cm}^2\) is equal to
Let $f: [0, 3] \to A$ be defined by $f(x) = 2x^3 - 15x^2 + 36x + 7$ and $g: [0, \infty) \to B$ be defined by $g(x) = \frac{x}{x^{2025} + 1}$. If both functions are onto and $S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \}$, then $n(S)$ is equal to:

Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta > 0. \) If \( B = P A P^T \), \( C = P^T B P \), and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where gcd(m, n) = 1, then \( m + n \) is:

Consider the region \[ R = \left\{ (x, y): x \leq y \leq 9 - \frac{11}{3} x^2, x \geq 0 \right\}. \] The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \) is:
Let A, B, C be three points in the xy-plane, whose position vectors are given by \[ \sqrt{3} \hat{i} + \hat{j}, \quad \hat{i} + \sqrt{3} \hat{j}, \quad \text{and} \quad a\hat{i} + (1-a) \hat{j} \] respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{9}{\sqrt{2}} \), then the sum of all possible values of \( a \) is:
Let \( f \) be a real-valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^3) = x^6 + x^7 \), then the value of \( \sum_{r=1}^{15} f(r^3) \) is:
The area of the region bounded by the curves \( x(1 + y^2) = 1 \) and \( y^2 = 2x \) is:
The maximum area of a triangle whose one vertex is at \( (0, 0) \) and the other two vertices lie on the curve \( y = -2x^2 + 54 \) at points \( (x, y) \) and \( (-x, y) \) where \( y > 0 \) is:
The length of sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm. The area of the triangle is:
A horse, a cow and a goat are tied, each by ropes of length 14 m, at the corners A, B and C respectively, of a grassy triangular field ABC with sides of lengths 35 m, 40 m and 50 m. Find the total area of grass field that can be grazed by them
A horse, a cow and a goat are tied, each by ropes of length 14 m, at the corners A, B and C respectively, of a grassy triangular field ABC with sides of lengths 35 m, 40 m and 50 m. Find the total area of grass field that can be grazed by them
Let \(\Delta\) be the area of the region \(\{(x,y)∈R^2:x^2+y^2≤21,y^2≤4x,x≥1\}\). Then \(\frac{1}{2}(\Delta-21\text{ sin}^{-1} (\frac{2}{\sqrt7}))\) is equal to
Let the co-ordinates of one vertex of \(\Delta ABC\) be \(A(0, 2, \alpha)\) and the other two vertices lie on the line \(x+\frac{\alpha}{5}=y-\frac{1}{2}=z+\frac{4}{3}\) For \(\alpha∈Z\), if the area of \(\Delta ABC\) is 21 sq. units and the line segment BC has length \(2\sqrt{21}\) units, then α2 is equal to ___________.
A rectangle ABCD with ABCD with AB = 2 and BC = 4 is inscribed in rectangle PQRS such that vertices of ABCD lie on sides of PQRS then maximum possible area(in sq. unit) of rectangle PQRS is :
A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘a’. Find the area of the signal board, using Heron’s formula. If its perimeter is 180 cm, what will be the area of the signal board?
The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m, and 120 m (see figure below). The advertisements yield an earning of ₹ 5000 per m2 per year. A company hired one of its walls for 3 months. How much rent did it pay?
triangular side walls of a flyover
There is a slide in a park. One of its side walls has been painted in some colour with a message “KEEP THE PARK GREEN AND CLEAN” (see Fig below). If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.
slide in a park
Find the area of a triangle two sides of which are 18cm and 10cm and the perimeter is 42cm.
Sides of a triangle are in the ratio of 12: 17: 25 and its perimeter is 540cm. Find its area.
An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.